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math-functions 0.1.7.0 → 0.3.4.4

raw patch · 48 files changed

Files

Numeric/MathFunctions/Comparison.hs view
@@ -19,6 +19,7 @@       -- * Ulps-based comparison     , addUlps     , ulpDistance+    , ulpDelta     , within     ) where @@ -36,7 +37,7 @@ -- | -- Calculate relative error of two numbers: ----- > |a - b| / max |a| |b|+-- \[ \frac{|a - b|}{\max(|a|,|b|)} \] -- -- It lies in [0,1) interval for numbers with same sign and (1,2] for -- numbers with different sign. If both arguments are zero or negative@@ -49,9 +50,9 @@  -- | Check that relative error between two numbers @a@ and @b@. If -- 'relativeError' returns NaN it returns @False@.-eqRelErr :: Double -- ^ @eps@ relative error should be in [0,1) range-         -> Double -- ^ @a@-         -> Double -- ^ @b@+eqRelErr :: Double -- ^ /eps/ relative error should be in [0,1) range+         -> Double -- ^ /a/+         -> Double -- ^ /b/          -> Bool eqRelErr eps a b = relativeError a b < eps @@ -80,7 +81,8 @@  -- | -- Measure distance between two @Double@s in ULPs (units of least--- precision).+-- precision). Note that it's different from @abs (ulpDelta a b)@+-- since it returns correct result even when 'ulpDelta' overflows. ulpDistance :: Double             -> Double             -> Word64@@ -100,6 +102,31 @@       d  | ai > bi   = ai - bi          | otherwise = bi - ai   return $! d++-- |+-- Measure signed distance between two @Double@s in ULPs (units of least+-- precision). Note that unlike 'ulpDistance' it can overflow.+--+-- > >>> ulpDelta 1 (1 + m_epsilon)+-- > 1+ulpDelta :: Double+         -> Double+         -> Int64+ulpDelta a b = runST $ do+  buf <- newByteArray 8+  ai0 <- writeByteArray buf 0 a >> readByteArray buf 0+  bi0 <- writeByteArray buf 0 b >> readByteArray buf 0+  -- IEEE754 floats use most significant bit as sign bit (not+  -- 2-complement) and we need to rearrange representations of float+  -- number so that they could be compared lexicographically as+  -- Word64.+  let big     = 0x8000000000000000 :: Word64+      order i | i < big   = i + big+              | otherwise = maxBound - i+      ai = order ai0+      bi = order bi0+  return $! fromIntegral $ bi - ai+  -- | Compare two 'Double' values for approximate equality, using -- Dawson's method.
Numeric/MathFunctions/Constants.hs view
@@ -13,12 +13,15 @@     (       -- * IEE754 constants       m_epsilon+    , m_sqrt_eps     , m_huge     , m_tiny     , m_max_exp     , m_pos_inf     , m_neg_inf     , m_NaN+    , m_max_log+    , m_min_log       -- * Mathematical constants     , m_1_sqrt_2     , m_2_sqrt_pi@@ -32,11 +35,12 @@ -- IEE754 constants ---------------------------------------------------------------- --- | A very large number.+-- | Largest representable finite value. m_huge :: Double m_huge = 1.7976931348623157e308 {-# INLINE m_huge #-} +-- | The smallest representable positive normalized value. m_tiny :: Double m_tiny = 2.2250738585072014e-308 {-# INLINE m_tiny #-}@@ -61,8 +65,17 @@ m_NaN = 0/0 {-# INLINE m_NaN #-} +-- | Maximum possible finite value of @log x@+m_max_log :: Double+m_max_log = 709.782712893384+{-# INLINE m_max_log #-} +-- | Logarithm of smallest normalized double ('m_tiny')+m_min_log :: Double+m_min_log = -708.3964185322641+{-# INLINE m_min_log #-} + ---------------------------------------------------------------- -- Mathematical constants ----------------------------------------------------------------@@ -91,6 +104,12 @@ m_epsilon :: Double m_epsilon = encodeFloat (signif+1) expo - 1.0     where (signif,expo) = decodeFloat (1.0::Double)++-- | @sqrt m_epsilon@+--+-- @since 0.3.3.0+m_sqrt_eps :: Double+m_sqrt_eps = 1.4901161193847656e-8  -- | @log(sqrt((2*pi))@ m_ln_sqrt_2_pi :: Double
Numeric/Polynomial.hs view
@@ -69,7 +69,7 @@ -- $lists -- -- When all coefficients are known statically it's more convenient to--- pass coefficient in a list instad of vector. Functions below+-- pass coefficient in a list instead of vector. Functions below -- provide just that functionality. If list is known statically it -- will be inlined anyway.
Numeric/Polynomial/Chebyshev.hs view
@@ -27,9 +27,12 @@ -- A Chebyshev polynomial of the first kind is defined by the -- following recurrence: ----- > t 0 _ = 1--- > t 1 x = x--- > t n x = 2 * x * t (n-1) x - t (n-2) x+-- \[\begin{aligned}+-- T_0(x)     &= 1 \\+-- T_1(x)     &= x \\+-- T_{n+1}(x) &= 2xT_n(x) - T_{n-1}(x) \\+-- \end{aligned}+-- \]  data C = C {-# UNPACK #-} !Double {-# UNPACK #-} !Double @@ -68,7 +71,7 @@ -- -- * Broucke, R. (1973) Algorithm 446: Ten subroutines for the --   manipulation of Chebyshev series. /Communications of the ACM/---   16(4):254&#8211;256.  <http://doi.acm.org/10.1145/362003.362037>+--   16(4):254–256.  <http://doi.acm.org/10.1145/362003.362037> -- -- * Clenshaw, C.W. (1962) Chebyshev series for mathematical --   functions. /National Physical Laboratory Mathematical Tables 5/,
+ Numeric/RootFinding.hs view
@@ -0,0 +1,387 @@+{-# LANGUAGE BangPatterns       #-}+{-# LANGUAGE DeriveDataTypeable #-}+{-# LANGUAGE DeriveFoldable     #-}+{-# LANGUAGE DeriveGeneric      #-}+{-# LANGUAGE DeriveTraversable  #-}+{-# LANGUAGE TypeFamilies       #-}+-- |+-- Module    : Numeric.RootFinding+-- Copyright : (c) 2011 Bryan O'Sullivan, 2018 Alexey Khudyakov+-- License   : BSD3+--+-- Maintainer  : bos@serpentine.com+-- Stability   : experimental+-- Portability : portable+--+-- Haskell functions for finding the roots of real functions of real+-- arguments. These algorithms are iterative so we provide both+-- function returning root (or failure to find root) and list of+-- iterations.+module Numeric.RootFinding+    ( -- * Data types+      Root(..)+    , fromRoot+    , Tolerance(..)+    , withinTolerance+    , IterationStep(..)+    , findRoot+    -- * Ridders algorithm+    , RiddersParam(..)+    , ridders+    , riddersIterations+    , RiddersStep(..)+    -- * Newton-Raphson algorithm+    , NewtonParam(..)+    , newtonRaphson+    , newtonRaphsonIterations+    , NewtonStep(..)+    -- * References+    -- $references+    ) where++import Control.Applicative              (Alternative(..))+import Control.Monad                    (MonadPlus(..), ap)+import Control.DeepSeq                  (NFData(..))+import Data.Data                        (Data, Typeable)+import Data.Default.Class+import GHC.Generics                     (Generic)+import Numeric.MathFunctions.Comparison (within,eqRelErr)+import Numeric.MathFunctions.Constants  (m_epsilon)++++----------------------------------------------------------------+-- Data types+----------------------------------------------------------------++-- | The result of searching for a root of a mathematical function.+data Root a+  = NotBracketed+    -- ^ The function does not have opposite signs when+    -- evaluated at the lower and upper bounds of the search.+  | SearchFailed+    -- ^ The search failed to converge to within the given+    -- error tolerance after the given number of iterations.+  | Root !a+    -- ^ A root was successfully found.+  deriving (Eq, Read, Show, Typeable, Data, Foldable, Traversable, Functor, Generic)++instance (NFData a) => NFData (Root a) where+    rnf NotBracketed = ()+    rnf SearchFailed = ()+    rnf (Root a)     = rnf a++instance Applicative Root where+    pure  = Root+    (<*>) = ap++instance Monad Root where+    NotBracketed >>= _ = NotBracketed+    SearchFailed >>= _ = SearchFailed+    Root a       >>= f = f a+    return = pure++instance MonadPlus Root where+    mzero = empty+    mplus = (<|>)++instance Alternative Root where+    empty = NotBracketed+    r@Root{}     <|> _            = r+    _            <|> r@Root{}     = r+    NotBracketed <|> r            = r+    r            <|> NotBracketed = r+    _            <|> r            = r++-- | Returns either the result of a search for a root, or the default+-- value if the search failed.+fromRoot :: a                 -- ^ Default value.+         -> Root a            -- ^ Result of search for a root.+         -> a+fromRoot _ (Root a) = a+fromRoot a _        = a+++-- | Error tolerance for finding root. It describes when root finding+--   algorithm should stop trying to improve approximation.+data Tolerance+  = RelTol !Double+    -- ^ Relative error tolerance. Given @RelTol ε@ two values are+    --   considered approximately equal if+    --   \[ \frac{|a - b|}{|\operatorname{max}(a,b)} < \varepsilon \]+  | AbsTol !Double+    -- ^ Absolute error tolerance. Given @AbsTol δ@ two values are+    --   considered approximately equal if \[ |a - b| < \delta \].+    --   Note that @AbsTol 0@ could be used to require to find+    --   approximation within machine precision.+  deriving (Eq, Read, Show, Typeable, Data, Generic)++-- | Check that two values are approximately equal. In addition to+--   specification values are considered equal if they're within 1ulp+--   of precision. No further improvement could be done anyway.+withinTolerance :: Tolerance -> Double -> Double -> Bool+withinTolerance _ a b+  | within 1 a b = True+withinTolerance (RelTol eps) a b = eqRelErr eps a b+withinTolerance (AbsTol tol) a b = abs (a - b) < tol++-- | Type class for checking whether iteration converged already.+class IterationStep a where+  -- | Return @Just root@ is current iteration converged within+  --   required error tolerance. Returns @Nothing@ otherwise.+  matchRoot :: Tolerance -> a -> Maybe (Root Double)++-- | Find root in lazy list of iterations.+findRoot :: IterationStep a+  => Int                        -- ^ Maximum+  -> Tolerance                  -- ^ Error tolerance+  -> [a]+  -> Root Double+findRoot maxN tol = go 0+  where+    go !i _  | i >= maxN = SearchFailed+    go !_ []             = SearchFailed+    go  i (x:xs)  = case matchRoot tol x of+      Just r  -> r+      Nothing -> go (i+1) xs+{-# INLINABLE  findRoot #-}+{-# SPECIALIZE findRoot :: Int -> Tolerance -> [RiddersStep] -> Root Double #-}+{-# SPECIALIZE findRoot :: Int -> Tolerance -> [NewtonStep]  -> Root Double #-}+++----------------------------------------------------------------+-- Attaching information to roots+----------------------------------------------------------------++-- | Parameters for 'ridders' root finding+data RiddersParam = RiddersParam+  { riddersMaxIter :: !Int+    -- ^ Maximum number of iterations. Default = 100+  , riddersTol     :: !Tolerance+    -- ^ Error tolerance for root approximation. Default is relative+    --   error 4·ε, where ε is machine precision.+  }+  deriving (Eq, Read, Show, Typeable, Data, Generic)++instance Default RiddersParam where+  def = RiddersParam+        { riddersMaxIter = 100+        , riddersTol     = RelTol (4 * m_epsilon)+        }++-- | Single Ridders step. It's a bracket of root+data RiddersStep+  = RiddersStep   !Double !Double+  -- ^ Ridders step. Parameters are bracket for the root+  | RiddersBisect !Double !Double+  -- ^ Bisection step. It's fallback which is taken when Ridders+  --   update takes us out of bracket+  | RiddersRoot   !Double+  -- ^ Root found+  | RiddersNoBracket+  -- ^ Root is not bracketed+  deriving (Eq, Read, Show, Typeable, Data, Generic)++instance NFData RiddersStep where+  rnf x = x `seq` ()++instance IterationStep RiddersStep where+  matchRoot tol r = case r of+    RiddersRoot x               -> Just $ Root x+    RiddersNoBracket            -> Just NotBracketed+    RiddersStep a b+      | withinTolerance tol a b -> Just $ Root ((a + b) / 2)+      | otherwise               -> Nothing+    RiddersBisect a b+      | withinTolerance tol a b -> Just $ Root ((a + b) / 2)+      | otherwise               -> Nothing+++-- | Use the method of Ridders[Ridders1979] to compute a root of a+--   function. It doesn't require derivative and provide quadratic+--   convergence (number of significant digits grows quadratically+--   with number of iterations).+--+--   The function must have opposite signs when evaluated at the lower+--   and upper bounds of the search (i.e. the root must be+--   bracketed). If there's more that one root in the bracket+--   iteration will converge to some root in the bracket.+ridders+  :: RiddersParam               -- ^ Parameters for algorithms. @def@+                                --   provides reasonable defaults+  -> (Double,Double)            -- ^ Bracket for root+  -> (Double -> Double)         -- ^ Function to find roots+  -> Root Double+ridders p bracket fun+  = findRoot (riddersMaxIter p) (riddersTol p)+  $ riddersIterations bracket fun++-- | List of iterations for Ridders methods. See 'ridders' for+--   documentation of parameters+riddersIterations :: (Double,Double) -> (Double -> Double) -> [RiddersStep]+riddersIterations (lo,hi) f+  | flo == 0    = [RiddersRoot lo]+  | fhi == 0    = [RiddersRoot hi]+    -- root is not bracketed+  | flo*fhi > 0 = [RiddersNoBracket]+    -- Ensure that a<b in iterations+  | lo < hi     = RiddersStep lo hi : go lo flo hi fhi+  | otherwise   = RiddersStep lo hi : go hi fhi lo flo+  where+    flo = f lo+    fhi = f hi+    --+    go !a !fa !b !fb+      | fm == 0       = [RiddersRoot m]+      | fn == 0       = [RiddersRoot n]+      -- Ridder's approximation coincide with one of old bounds or+      -- went out of (a,b) range due to numerical problems. Revert+      -- to bisection+      | n <= a || n >= b   = case () of+          _| fm*fa < 0 -> recBisect a fa m fm+           | otherwise -> recBisect m fm b fb+      | fn*fm < 0          = recRidders n fn m fm+      | fn*fa < 0          = recRidders a fa n fn+      | otherwise          = recRidders n fn b fb+      where+        recBisect  x fx y fy = RiddersBisect x y : go x fx y fy+        recRidders x fx y fy = RiddersStep   x y : go x fx y fy+        --+        dm  = (b - a) * 0.5+        -- Mean point+        m   = (a + b) / 2+        fm  = f m+        -- Ridders update+        n   = m - signum (fb - fa) * dm * fm / sqrt(fm*fm - fa*fb)+        fn  = f n++++----------------------------------------------------------------+-- Newton-Raphson algorithm+----------------------------------------------------------------++-- | Parameters for 'ridders' root finding+data NewtonParam = NewtonParam+  { newtonMaxIter :: !Int+    -- ^ Maximum number of iterations. Default = 50+  , newtonTol     :: !Tolerance+    -- ^ Error tolerance for root approximation. Default is relative+    --   error 4·ε, where ε is machine precision+  }+  deriving (Eq, Read, Show, Typeable, Data, Generic)++instance Default NewtonParam where+  def = NewtonParam+        { newtonMaxIter = 50+        , newtonTol     = RelTol (4 * m_epsilon)+        }++-- | Steps for Newton iterations+data NewtonStep+  = NewtonStep         !Double !Double+  -- ^ Normal Newton-Raphson update. Parameters are: old guess, new guess+  | NewtonBisection    !Double !Double+  -- ^ Bisection fallback when Newton-Raphson iteration doesn't+  --   work. Parameters are bracket on root+  | NewtonRoot         !Double+  -- ^ Root is found+  | NewtonNoBracket+  -- ^ Root is not bracketed+  deriving (Eq, Read, Show, Typeable, Data, Generic)++instance NFData NewtonStep where+  rnf x = x `seq` ()++instance IterationStep NewtonStep where+  matchRoot tol r = case r of+    NewtonRoot x                 -> Just (Root x)+    NewtonNoBracket              -> Just NotBracketed+    NewtonStep x x'+      | withinTolerance tol x x' -> Just (Root x')+      | otherwise                -> Nothing+    NewtonBisection a b+      | withinTolerance tol a b  -> Just (Root ((a + b) / 2))+      | otherwise                -> Nothing+  {-# INLINE matchRoot #-}+++-- | Solve equation using Newton-Raphson iterations.+--+--   This method require both initial guess and bounds for root. If+--   Newton step takes us out of bounds on root function reverts to+--   bisection.+newtonRaphson+  :: NewtonParam                 -- ^ Parameters for algorithm. @def@+                                 --   provide reasonable defaults.+  -> (Double,Double,Double)      -- ^ Triple of @(low bound, initial+                                 --   guess, upper bound)@. If initial+                                 --   guess if out of bracket middle+                                 --   of bracket is taken as+                                 --   approximation+  -> (Double -> (Double,Double)) -- ^ Function to find root of. It+                                 --   returns pair of function value and+                                 --   its first derivative+  -> Root Double+newtonRaphson p guess fun+  = findRoot (newtonMaxIter p) (newtonTol p)+  $ newtonRaphsonIterations guess fun++-- | List of iteration for Newton-Raphson algorithm. See documentation+--   for 'newtonRaphson' for meaning of parameters.+newtonRaphsonIterations :: (Double,Double,Double) -> (Double -> (Double,Double)) -> [NewtonStep]+newtonRaphsonIterations (lo,guess,hi) function+  | flo == 0    = [NewtonRoot lo]+  | fhi == 0    = [NewtonRoot hi]+  | flo*fhi > 0 = [NewtonNoBracket]+    -- Ensure that function value on low bound is negative+  | flo > 0     = go hi guess' lo+  | otherwise   = go lo guess hi+  where+    (flo,_) = function lo+    (fhi,_) = function hi+    -- Ensure that initial guess is within bracket+    guess'+      | guess >= lo && guess <= hi = guess+      | guess >= hi && guess <= lo = guess+      | otherwise                  = (lo + hi) / 2+    -- Newton iterations. Invariant:+    --   > f xA < 0+    --   > f xB > 0+    go xA x xB+      | f  == 0   = [NewtonRoot x]+      | f' == 0   = bisectionStep+      -- Accept Newton step since it stays within bracket.+      | (x' - xA) * (x' - xB) < 0 = newtonStep+      -- Otherwise bracket root and pick new approximation as+      -- midpoint.+      | otherwise                 = bisectionStep+      where+        -- Calculate Newton step+        (f,f') = function x+        x'   = x - f / f'+        -- Newton step+        newtonStep+          | f > 0     = NewtonStep x x' : go xA x' x+          | otherwise = NewtonStep x x' : go x  x' xB+        -- Fallback bisection step+        bisectionStep+          | f > 0     = NewtonBisection xA x : go xA ((xA + x) / 2) x+          | otherwise = NewtonBisection x xB : go x  ((x + xB) / 2) xB++++----------------------------------------------------------------+-- Internal functions+----------------------------------------------------------------++-- $references+--+-- * Ridders, C.F.J. (1979) A new algorithm for computing a single+--   root of a real continuous function.+--   /IEEE Transactions on Circuits and Systems/ 26:979&#8211;980.+--+-- * Press W.H.; Teukolsky S.A.; Vetterling W.T.; Flannery B.P.+--   (2007). \"Section 9.2.1. Ridders' Method\". /Numerical Recipes: The+--   Art of Scientific Computing (3rd ed.)./ New York: Cambridge+--   University Press. ISBN 978-0-521-88068-8.
+ Numeric/Series.hs view
@@ -0,0 +1,180 @@+{-# LANGUAGE BangPatterns              #-}+{-# LANGUAGE ExistentialQuantification #-}+-- |+-- Module    : Numeric.Series+-- Copyright : (c) 2016 Alexey Khudyakov+-- License   : BSD3+--+-- Maintainer  : alexey.skladnoy@gmail.com, bos@serpentine.com+-- Stability   : experimental+-- Portability : portable+--+-- Functions for working with infinite sequences. In particular+-- summation of series and evaluation of continued fractions.+module Numeric.Series (+    -- * Data type for infinite sequences.+    Sequence(..)+    -- * Constructors+  , enumSequenceFrom+  , enumSequenceFromStep+  , scanSequence+    -- * Summation of series+  , sumSeries+  , sumPowerSeries+  , sequenceToList+    -- * Evaluation of continued fractions+  , evalContFractionB+  ) where++import Control.Applicative+import Data.List (unfoldr)++import Numeric.MathFunctions.Constants (m_epsilon)+++----------------------------------------------------------------++-- | Infinite series. It's represented as opaque state and step+--   function.+data Sequence a = forall s. Sequence s (s -> (a,s))++instance Functor Sequence where+  fmap f (Sequence s0 step) = Sequence s0 (\s -> let (a,s') = step s in (f a, s'))+  {-# INLINE fmap #-}++instance Applicative Sequence where+  pure a = Sequence () (\() -> (a,()))+  Sequence sA fA <*> Sequence sB fB = Sequence (sA,sB) $ \(!sa,!sb) ->+    let (a,sa') = fA sa+        (b,sb') = fB sb+    in (a b, (sa',sb'))+  {-# INLINE pure  #-}+  {-# INLINE (<*>) #-}++-- | Elementwise operations with sequences+instance Num a => Num (Sequence a) where+  (+) = liftA2 (+)+  (*) = liftA2 (*)+  (-) = liftA2 (-)+  {-# INLINE (+) #-}+  {-# INLINE (*) #-}+  {-# INLINE (-) #-}+  abs         = fmap abs+  signum      = fmap signum+  fromInteger = pure . fromInteger+  {-# INLINE abs         #-}+  {-# INLINE signum      #-}+  {-# INLINE fromInteger #-}++-- | Elementwise operations with sequences+instance Fractional a => Fractional (Sequence a) where+  (/)          = liftA2 (/)+  recip        = fmap recip+  fromRational = pure . fromRational+  {-# INLINE (/)          #-}+  {-# INLINE recip        #-}+  {-# INLINE fromRational #-}++++----------------------------------------------------------------+-- Constructors+----------------------------------------------------------------++-- | @enumSequenceFrom x@ generate sequence:+--+-- \[ a_n = x + n \]+enumSequenceFrom :: Num a => a -> Sequence a+enumSequenceFrom i = Sequence i (\n -> (n,n+1))+{-# INLINE enumSequenceFrom #-}++-- | @enumSequenceFromStep x d@ generate sequence:+--+-- \[ a_n = x + nd \]+enumSequenceFromStep :: Num a => a -> a -> Sequence a+enumSequenceFromStep n d = Sequence n (\i -> (i,i+d))+{-# INLINE enumSequenceFromStep #-}++-- | Analog of 'scanl' for sequence.+scanSequence :: (b -> a -> b) -> b -> Sequence a -> Sequence b+{-# INLINE scanSequence #-}+scanSequence f b0 (Sequence s0 step) = Sequence (b0,s0) $ \(b,s) ->+  let (a,s') = step s+      b'     = f b a+  in (b,(b',s'))+++----------------------------------------------------------------+-- Evaluation of series+----------------------------------------------------------------++-- | Calculate sum of series+--+-- \[ \sum_{i=0}^\infty a_i \]+--+-- Summation is stopped when+--+-- \[ a_{n+1} < \varepsilon\sum_{i=0}^n a_i \]+--+-- where ε is machine precision ('m_epsilon')+sumSeries :: Sequence Double -> Double+{-# INLINE sumSeries #-}+sumSeries (Sequence sInit step)+  = go x0 s0+  where +    (x0,s0) = step sInit+    go x s | abs (d/x) >= m_epsilon = go x' s'+           | otherwise              = x'+      where (d,s') = step s+            x'     = x + d++-- | Calculate sum of series+--+-- \[ \sum_{i=0}^\infty x^ia_i \]+--+-- Calculation is stopped when next value in series is less than+-- ε·sum.+sumPowerSeries :: Double -> Sequence Double -> Double+sumPowerSeries x ser = sumSeries $ liftA2 (*) (scanSequence (*) 1 (pure x)) ser+{-# INLINE sumPowerSeries #-}++-- | Convert series to infinite list+sequenceToList :: Sequence a -> [a]+sequenceToList (Sequence s f) = unfoldr (Just . f) s++++----------------------------------------------------------------+-- Evaluation of continued fractions+----------------------------------------------------------------++-- |+-- Evaluate continued fraction using modified Lentz algorithm.+-- Sequence contain pairs (a[i],b[i]) which form following expression:+--+-- \[+-- b_0 + \frac{a_1}{b_1+\frac{a_2}{b_2+\frac{a_3}{b_3 + \cdots}}}+-- \]+--+-- Modified Lentz algorithm is described in Numerical recipes 5.2+-- "Evaluation of Continued Fractions"+evalContFractionB :: Sequence (Double,Double) -> Double+{-# INLINE evalContFractionB #-}+evalContFractionB (Sequence sInit step)+  = let ((_,b0),s0) = step sInit+        f0          = maskZero b0+    in  go f0 f0 0 s0+  where+    tiny = 1e-60+    maskZero 0 = tiny+    maskZero x = x+    +    go f c d s+      | abs (delta - 1) >= m_epsilon = go f' c' d' s'+      | otherwise                    = f'+      where+          ((a,b),s') = step s+          d'    = recip $ maskZero $ b + a*d+          c'    = maskZero $ b + a/c +          delta = c'*d'+          f'    = f*delta
Numeric/SpecFunctions.hs view
@@ -1,4 +1,3 @@-{-# LANGUAGE BangPatterns, ScopedTypeVariables #-} -- | -- Module    : Numeric.SpecFunctions -- Copyright : (c) 2009, 2011, 2012 Bryan O'Sullivan@@ -29,8 +28,12 @@     -- * Sinc   , sinc     -- * Logarithm+    -- $log1p   , log1p+  , log1pmx   , log2+    -- * Exponent+  , expm1     -- * Factorial   , factorial   , logFactorial@@ -44,6 +47,13 @@  import Numeric.SpecFunctions.Internal +-- $log1p+--+-- Base starting from @4.9.0@ (GHC 8.0) provides 'log1p' and 'expm1'+-- as method of class 'Floating'. In this case we simply reexport+-- these function. Otherwise we provide our own with more restrictive+-- signature @Double → Double@.+ -- $references -- -- * Bernardo, J. (1976) Algorithm AS 103: Psi (digamma)@@ -81,6 +91,11 @@ --   Vol. 22, No. 3 (1973), pp. 411-414 --   <http://www.jstor.org/pss/2346798> ----- * Shea, B. (1988) Algorithm AS 239: Chi-squared and incomplete---   gamma integral. /Applied Statistics/---   37(3):466&#8211;473. <http://www.jstor.org/stable/2347328>+-- * Temme, N.M. (1992) Asymptotic inversion of the incomplete beta+--   function. /Journal of Computational and Applied Mathematics+--   41(1992) 145-157.+--+-- * Temme, N.M. (1994) A set of algorithms for the incomplete gamma+--   functions. /Probability in the Engineering and Informational+--   Sciences/, 8, 1994, 291-307. Printed in the U.S.A.+
+ Numeric/SpecFunctions/Compat.hs view
@@ -0,0 +1,167 @@+{-# LANGUAGE CPP                      #-}+{-# LANGUAGE ForeignFunctionInterface #-}+-- |+-- Functions which have different implementations on different platforms+module Numeric.SpecFunctions.Compat (+    erf+  , erfc+  , log1p+  , expm1+  ) where++#if !defined(USE_SYSTEM_ERF) || !defined(USE_SYSTEM_EXPM1)+import qualified Data.Vector.Unboxed as U+#endif++#if !defined(USE_SYSTEM_ERF)+import Numeric.Polynomial.Chebyshev    (chebyshev)+import Numeric.Polynomial              (evaluateOddPolynomial)+#endif++#if !defined(USE_SYSTEM_EXPM1)+import Control.Applicative             (liftA2)+import Numeric.Polynomial.Chebyshev    (chebyshevBroucke)+import Numeric.Series                  (scanSequence,sumSeries,enumSequenceFrom)+import Numeric.MathFunctions.Constants+#endif++#if defined(USE_SYSTEM_EXPM1)+import GHC.Float (log1p,expm1)+#endif+++----------------------------------------------------------------+-- erf & erfc+--+-- We provide pure haskell implementation for GHCJS and accessible on+-- GHC via flag+----------------------------------------------------------------++#if defined(USE_SYSTEM_ERF)++erf :: Double -> Double+erf = c_erf+{-# INLINE erf #-}++erfc :: Double -> Double+erfc = c_erfc+{-# INLINE erfc #-}++foreign import ccall unsafe "erf"  c_erf  :: Double -> Double+foreign import ccall unsafe "erfc" c_erfc :: Double -> Double++#else++erf :: Double -> Double+erf x+  -- Computing erf as 1-erfc loses precision near 0 so we switch to+  -- Taylor expansion here+  | abs x < 0.1 = 0.56418958354775629+                * evaluateOddPolynomial x erfTaylorSeries+  | x < 0       = (-1) + erfcCheb (-x)+  | otherwise   =   1  - erfcCheb x++erfTaylorSeries :: U.Vector Double+{-# NOINLINE erfTaylorSeries #-}+erfTaylorSeries = U.fromList+  [  2+  , -2/3+  ,  1/5+  , -1/21+  ,  1/108+  , -1/660+  ,  1/4680+  ]++erfc :: Double -> Double+erfc x | x < 0     = 2 - erfcCheb (-x)+       | otherwise = erfcCheb x++-- Adapted from Numerical Recipes §6.2.2+erfcCheb :: Double -> Double+erfcCheb z+  = t * exp( -z * z + chebyshev ty erfcCoef )+  where+    -- We're using approximation:+    --+    --   erfc(z) ≈ t·exp(-z² + P(t))+    --   t       = 2 / (2 + z)+    t  = 2 / (2 + z)+    ty = 2 * t - 1+++erfcCoef :: U.Vector Double+{-# NOINLINE erfcCoef #-}+erfcCoef = U.fromList+  [ -0.6513268598908546   ,  6.4196979235649026e-1 ,  1.9476473204185836e-2+  , -9.561514786808631e-3 , -9.46595344482036e-4   ,  3.66839497852761e-4+  ,  4.2523324806907e-5   , -2.0278578112534e-5    , -1.624290004647e-6+  ,  1.303655835580e-6    ,  1.5626441722e-8       , -8.5238095915e-8+  ,  6.529054439e-9       ,  5.059343495e-9        , -9.91364156e-10+  , -2.27365122e-10       ,  9.6467911e-11         ,  2.394038e-12+  , -6.886027e-12         ,  8.94487e-13           ,  3.13092e-13+  , -1.12708e-13          ,  3.81e-16              ,  7.106e-15+  , -1.523e-15            , -9.4e-17               ,  1.21e-16+  , -2.8e-17+  ]++#endif+++----------------------------------------------------------------+-- expm1 & log1p+--+-- We use one provided by base of for GHCJS use hand-coded one+----------------------------------------------------------------++#if !defined(USE_SYSTEM_EXPM1)++-- | Compute @exp x - 1@ without loss of accuracy for x near zero.+expm1 :: Double -> Double+-- NOTE: this is simplest implementation and not terribly efficient.+expm1 x+  | x < (-37.42994775023705) = -1+  | x > m_max_log            = m_pos_inf+  | abs x > 0.5              = exp x - 1+  | otherwise                = sumSeries $ liftA2 (*) (scanSequence (*) x (pure x))+                                                      (1 / scanSequence (*) 1 (enumSequenceFrom 2))+-- | Compute the natural logarithm of 1 + @x@.  This is accurate even+--   for values of @x@ near zero, where use of @log(1+x)@ would lose+--   precision.+log1p :: Double -> Double+log1p x+    | x == 0               = 0+    | x == -1              = m_neg_inf+    | x < -1               = m_NaN+    | x' < m_epsilon * 0.5 = x+    | (x >= 0 && x < 1e-8) || (x >= -1e-9 && x < 0)+                           = x * (1 - x * 0.5)+    | x' < 0.375           = x * (1 - x * chebyshevBroucke (x / 0.375) coeffs)+    | otherwise            = log (1 + x)+  where+    x' = abs x+    coeffs = U.fromList [+               0.10378693562743769800686267719098e+1,+              -0.13364301504908918098766041553133e+0,+               0.19408249135520563357926199374750e-1,+              -0.30107551127535777690376537776592e-2,+               0.48694614797154850090456366509137e-3,+              -0.81054881893175356066809943008622e-4,+               0.13778847799559524782938251496059e-4,+              -0.23802210894358970251369992914935e-5,+               0.41640416213865183476391859901989e-6,+              -0.73595828378075994984266837031998e-7,+               0.13117611876241674949152294345011e-7,+              -0.23546709317742425136696092330175e-8,+               0.42522773276034997775638052962567e-9,+              -0.77190894134840796826108107493300e-10,+               0.14075746481359069909215356472191e-10,+              -0.25769072058024680627537078627584e-11,+               0.47342406666294421849154395005938e-12,+              -0.87249012674742641745301263292675e-13,+               0.16124614902740551465739833119115e-13,+              -0.29875652015665773006710792416815e-14,+               0.55480701209082887983041321697279e-15,+              -0.10324619158271569595141333961932e-15+             ]+#endif
Numeric/SpecFunctions/Extra.hs view
@@ -12,10 +12,12 @@     bd0   , chooseExact   , logChooseFast+  , logGammaAS245+  , logGammaCorrection   ) where -import Numeric.MathFunctions.Constants (m_NaN)-import Numeric.SpecFunctions.Internal  (chooseExact,logChooseFast)+import Numeric.MathFunctions.Constants (m_NaN,m_pos_inf)+import Numeric.SpecFunctions.Internal  (chooseExact,logChooseFast,logGammaCorrection)  -- | Evaluate the deviance term @x log(x/np) + np - x@. bd0 :: Double                   -- ^ @x@@@ -34,3 +36,61 @@     loop j ej s = case s + ej/(2*j+1) of                     s' | s' == s   -> s'  -- FIXME: Comparing Doubles for equality!                        | otherwise -> loop (j+1) (ej*vv) s'++++-- | Compute the logarithm of the gamma function Γ(/x/).  Uses+-- Algorithm AS 245 by Macleod.+--+-- Gives an accuracy of 10-12 significant decimal digits, except+-- for small regions around /x/ = 1 and /x/ = 2, where the function+-- goes to zero.  For greater accuracy, use 'logGammaL'.+--+-- Returns ∞ if the input is outside of the range (0 < /x/ ≤ 1e305).+logGammaAS245 :: Double -> Double+-- Adapted from http://people.sc.fsu.edu/~burkardt/f_src/asa245/asa245.html+logGammaAS245 x+    | x <= 0    = m_pos_inf+    -- Handle positive infinity. logGamma overflows before 1e308 so+    -- it's safe+    | x > 1e308 = m_pos_inf+    -- Normal cases+    | x < 1.5   = a + c *+                  ((((r1_4 * b + r1_3) * b + r1_2) * b + r1_1) * b + r1_0) /+                  ((((b + r1_8) * b + r1_7) * b + r1_6) * b + r1_5)+    | x < 4     = (x - 2) *+                  ((((r2_4 * x + r2_3) * x + r2_2) * x + r2_1) * x + r2_0) /+                  ((((x + r2_8) * x + r2_7) * x + r2_6) * x + r2_5)+    | x < 12    = ((((r3_4 * x + r3_3) * x + r3_2) * x + r3_1) * x + r3_0) /+                  ((((x + r3_8) * x + r3_7) * x + r3_6) * x + r3_5)+    | x > 3e6   = k+    | otherwise = k + x1 *+                  ((r4_2 * x2 + r4_1) * x2 + r4_0) /+                  ((x2 + r4_4) * x2 + r4_3)+  where+    (a , b , c)+        | x < 0.5   = (-y , x + 1 , x)+        | otherwise = (0  , x     , x - 1)++    y      = log x+    k      = x * (y-1) - 0.5 * y + alr2pi+    alr2pi = 0.918938533204673++    x1 = 1 / x+    x2 = x1 * x1++    r1_0 =  -2.66685511495;   r1_1 =  -24.4387534237;    r1_2 = -21.9698958928+    r1_3 =  11.1667541262;    r1_4 =    3.13060547623;   r1_5 =   0.607771387771+    r1_6 =  11.9400905721;    r1_7 =   31.4690115749;    r1_8 =  15.2346874070++    r2_0 = -78.3359299449;    r2_1 = -142.046296688;     r2_2 = 137.519416416+    r2_3 =  78.6994924154;    r2_4 =    4.16438922228;   r2_5 =  47.0668766060+    r2_6 = 313.399215894;     r2_7 =  263.505074721;     r2_8 =  43.3400022514++    r3_0 =  -2.12159572323e5; r3_1 =    2.30661510616e5; r3_2 =   2.74647644705e4+    r3_3 =  -4.02621119975e4; r3_4 =   -2.29660729780e3; r3_5 =  -1.16328495004e5+    r3_6 =  -1.46025937511e5; r3_7 =   -2.42357409629e4; r3_8 =  -5.70691009324e2++    r4_0 = 0.279195317918525;  r4_1 = 0.4917317610505968;+    r4_2 = 0.0692910599291889; r4_3 = 3.350343815022304+    r4_4 = 6.012459259764103
Numeric/SpecFunctions/Internal.hs view
@@ -1,813 +1,1381 @@ {-# LANGUAGE BangPatterns, ScopedTypeVariables #-}--- |--- Module    : Numeric.SpecFunctions.Internal--- Copyright : (c) 2009, 2011, 2012 Bryan O'Sullivan--- License   : BSD3------ Maintainer  : bos@serpentine.com--- Stability   : experimental--- Portability : portable------ Internal module with implementation of special functions.-module Numeric.SpecFunctions.Internal where--import Data.Bits       ((.&.), (.|.), shiftR)-import Data.Int        (Int64)-import qualified Data.Number.Erf     as Erf (erfc,erf)-import qualified Data.Vector.Unboxed as U--import Numeric.Polynomial.Chebyshev    (chebyshevBroucke)-import Numeric.Polynomial              (evaluateEvenPolynomialL,evaluateOddPolynomialL)-import Numeric.MathFunctions.Constants ( m_epsilon, m_NaN, m_neg_inf, m_pos_inf-                                       , m_sqrt_2_pi, m_ln_sqrt_2_pi, m_sqrt_2-                                       , m_eulerMascheroni-                                       )-import Text.Printf---------------------------------------------------------------------- Error function--------------------------------------------------------------------- | Error function.------ > erf -∞ = -1--- > erf  0 =  0--- > erf +∞ =  1-erf :: Double -> Double-{-# INLINE erf #-}-erf = Erf.erf---- | Complementary error function.------ > erfc -∞ = 2--- > erfc  0 = 1--- > errc +∞ = 0-erfc :: Double -> Double-{-# INLINE erfc #-}-erfc = Erf.erfc----- | Inverse of 'erf'.-invErf :: Double -- ^ /p/ ∈ [-1,1]-       -> Double-invErf p = invErfc (1 - p)---- | Inverse of 'erfc'.-invErfc :: Double -- ^ /p/ ∈ [0,2]-        -> Double-invErfc p-  | p == 2        = m_neg_inf-  | p == 0        = m_pos_inf-  | p >0 && p < 2 = if p <= 1 then r else -r-  | otherwise     = modErr $ "invErfc: p must be in [0,2] got " ++ show p-  where-    pp = if p <= 1 then p else 2 - p-    t  = sqrt $ -2 * log( 0.5 * pp)-    -- Initial guess-    x0 = -0.70711 * ((2.30753 + t * 0.27061) / (1 + t * (0.99229 + t * 0.04481)) - t)-    r  = loop 0 x0-    ---    loop :: Int -> Double -> Double-    loop !j !x-      | j >= 2    = x-      | otherwise = let err = erfc x - pp-                        x'  = x + err / (1.12837916709551257 * exp(-x * x) - x * err) -- // Halley-                    in loop (j+1) x'----------------------------------------------------------------------- Gamma function--------------------------------------------------------------------- Adapted from http://people.sc.fsu.edu/~burkardt/f_src/asa245/asa245.html---- | Compute the logarithm of the gamma function Γ(/x/).  Uses--- Algorithm AS 245 by Macleod.------ Gives an accuracy of 10-12 significant decimal digits, except--- for small regions around /x/ = 1 and /x/ = 2, where the function--- goes to zero.  For greater accuracy, use 'logGammaL'.------ Returns ∞ if the input is outside of the range (0 < /x/ ≤ 1e305).-logGamma :: Double -> Double-logGamma x-    | x <= 0    = m_pos_inf-    -- Handle positive infinity. logGamma overflows before 1e308 so-    -- it's safe-    | x > 1e308 = m_pos_inf-    -- Normal cases-    | x < 1.5   = a + c *-                  ((((r1_4 * b + r1_3) * b + r1_2) * b + r1_1) * b + r1_0) /-                  ((((b + r1_8) * b + r1_7) * b + r1_6) * b + r1_5)-    | x < 4     = (x - 2) *-                  ((((r2_4 * x + r2_3) * x + r2_2) * x + r2_1) * x + r2_0) /-                  ((((x + r2_8) * x + r2_7) * x + r2_6) * x + r2_5)-    | x < 12    = ((((r3_4 * x + r3_3) * x + r3_2) * x + r3_1) * x + r3_0) /-                  ((((x + r3_8) * x + r3_7) * x + r3_6) * x + r3_5)-    | x > 3e6   = k-    | otherwise = k + x1 *-                  ((r4_2 * x2 + r4_1) * x2 + r4_0) /-                  ((x2 + r4_4) * x2 + r4_3)-  where-    (a , b , c)-        | x < 0.5   = (-y , x + 1 , x)-        | otherwise = (0  , x     , x - 1)--    y      = log x-    k      = x * (y-1) - 0.5 * y + alr2pi-    alr2pi = 0.918938533204673--    x1 = 1 / x-    x2 = x1 * x1--    r1_0 =  -2.66685511495;   r1_1 =  -24.4387534237;    r1_2 = -21.9698958928-    r1_3 =  11.1667541262;    r1_4 =    3.13060547623;   r1_5 =   0.607771387771-    r1_6 =  11.9400905721;    r1_7 =   31.4690115749;    r1_8 =  15.2346874070--    r2_0 = -78.3359299449;    r2_1 = -142.046296688;     r2_2 = 137.519416416-    r2_3 =  78.6994924154;    r2_4 =    4.16438922228;   r2_5 =  47.0668766060-    r2_6 = 313.399215894;     r2_7 =  263.505074721;     r2_8 =  43.3400022514--    r3_0 =  -2.12159572323e5; r3_1 =    2.30661510616e5; r3_2 =   2.74647644705e4-    r3_3 =  -4.02621119975e4; r3_4 =   -2.29660729780e3; r3_5 =  -1.16328495004e5-    r3_6 =  -1.46025937511e5; r3_7 =   -2.42357409629e4; r3_8 =  -5.70691009324e2--    r4_0 = 0.279195317918525;  r4_1 = 0.4917317610505968;-    r4_2 = 0.0692910599291889; r4_3 = 3.350343815022304-    r4_4 = 6.012459259764103----data L = L {-# UNPACK #-} !Double {-# UNPACK #-} !Double---- | Compute the logarithm of the gamma function, &#915;(/x/).  Uses a--- Lanczos approximation.------ This function is slower than 'logGamma', but gives 14 or more--- significant decimal digits of accuracy, except around /x/ = 1 and--- /x/ = 2, where the function goes to zero.------ Returns &#8734; if the input is outside of the range (0 < /x/--- &#8804; 1e305).-logGammaL :: Double -> Double-logGammaL x-    | x <= 0    = m_pos_inf-    -- Lanroz approximation loses precision for small arguments-    | x <= 1e-3 = logGamma x-    | otherwise = fini . U.foldl' go (L 0 (x+7)) $ a-    where fini (L l _) = log (l+a0) + log m_sqrt_2_pi - x65 + (x-0.5) * log x65-          go (L l t) k = L (l + k / t) (t-1)-          x65 = x + 6.5-          a0  = 0.9999999999995183-          a   = U.fromList [ 0.1659470187408462e-06-                           , 0.9934937113930748e-05-                           , -0.1385710331296526-                           , 12.50734324009056-                           , -176.6150291498386-                           , 771.3234287757674-                           , -1259.139216722289-                           , 676.5203681218835-                           ]------ | Compute the log gamma correction factor for @x@ &#8805; 10.  This--- correction factor is suitable for an alternate (but less--- numerically accurate) definition of 'logGamma':------ >lgg x = 0.5 * log(2*pi) + (x-0.5) * log x - x + logGammaCorrection x-logGammaCorrection :: Double -> Double-logGammaCorrection x-    | x < 10    = m_NaN-    | x < big   = chebyshevBroucke (t * t * 2 - 1) coeffs / x-    | otherwise = 1 / (x * 12)-  where-    big    = 94906265.62425156-    t      = 10 / x-    coeffs = U.fromList [-               0.1666389480451863247205729650822e+0,-              -0.1384948176067563840732986059135e-4,-               0.9810825646924729426157171547487e-8,-              -0.1809129475572494194263306266719e-10,-               0.6221098041892605227126015543416e-13,-              -0.3399615005417721944303330599666e-15,-               0.2683181998482698748957538846666e-17-             ]------ | Compute the normalized lower incomplete gamma function--- γ(/s/,/x/). Normalization means that--- γ(/s/,∞)=1. Uses Algorithm AS 239 by Shea.-incompleteGamma :: Double       -- ^ /s/ ∈ (0,∞)-                -> Double       -- ^ /x/ ∈ (0,∞)-                -> Double-incompleteGamma p x-    | isNaN p || isNaN x = m_NaN-    | x < 0 || p <= 0    = m_pos_inf-    | x == 0             = 0-    -- For very large `p' normal approximation gives <1e-10 error-    | p >= 2e5           = norm (3 * sqrt p * ((x/p) ** (1/3) + 1/(9*p) - 1))-    | p >= 500           = approx-    -- Dubious approximation-    | x >= 1e8           = 1-    | x <= 1 || x < p    = let a = p * log x - x - logGamma (p + 1)-                               g = a + log (pearson p 1 1)-                           in if g > limit then exp g else 0-    | otherwise          = let g = p * log x - x - logGamma p + log cf-                           in if g > limit then 1 - exp g else 1-  where-    -- CDF for standard normal distributions-    norm a = 0.5 * erfc (- a / m_sqrt_2)-    -- For large values of `p' we use 18-point Gauss-Legendre-    -- integration.-    approx-      | ans > 0   = 1 - ans-      | otherwise = -ans-      where-        -- Set upper limit for integration-        xu | x > p1    =         (p1 + 11.5*sqrtP1) `max` (x + 6*sqrtP1)-           | otherwise = max 0 $ (p1 -  7.5*sqrtP1) `min` (x - 5*sqrtP1)-        s = U.sum $ U.zipWith go coefY coefW-        go y w = let t = x + (xu - x)*y-                 in w * exp( -(t-p1) + p1*(log t - lnP1) )-        ans = s * (xu - x) * exp( p1 * (lnP1 - 1) - logGamma p)-        ---        p1     = p - 1-        lnP1   = log  p1-        sqrtP1 = sqrt p1-    ---    pearson !a !c !g-        | c' <= tolerance = g'-        | otherwise       = pearson a' c' g'-        where a' = a + 1-              c' = c * x / a'-              g' = g + c'-    cf = let a = 1 - p-             b = a + x + 1-             p3 = x + 1-             p4 = x * b-         in contFrac a b 0 1 x p3 p4 (p3/p4)-    contFrac !a !b !c !p1 !p2 !p3 !p4 !g-        | abs (g - rn) <= min tolerance (tolerance * rn) = g-        | otherwise = contFrac a' b' c' (f p3) (f p4) (f p5) (f p6) rn-        where a' = a + 1-              b' = b + 2-              c' = c + 1-              an = a' * c'-              p5 = b' * p3 - an * p1-              p6 = b' * p4 - an * p2-              rn = p5 / p6-              f n | abs p5 > overflow = n / overflow-                  | otherwise         = n-    limit     = -88-    tolerance = 1e-14-    overflow  = 1e37------ Adapted from Numerical Recipes §6.2.1---- | Inverse incomplete gamma function. It's approximately inverse of---   'incompleteGamma' for the same /s/. So following equality---   approximately holds:------ > invIncompleteGamma s . incompleteGamma s = id-invIncompleteGamma :: Double    -- ^ /s/ ∈ (0,∞)-                   -> Double    -- ^ /p/ ∈ [0,1]-                   -> Double-invIncompleteGamma a p-  | a <= 0         =-      modErr $ printf "invIncompleteGamma: a must be positive. a=%g p=%g" a p-  | p < 0 || p > 1 =-      modErr $ printf "invIncompleteGamma: p must be in [0,1] range. a=%g p=%g" a p-  | p == 0         = 0-  | p == 1         = 1 / 0-  | otherwise      = loop 0 guess-  where-    -- Solve equation γ(a,x) = p using Halley method-    loop :: Int -> Double -> Double-    loop i x-      | i >= 12           = x'-      -- For small s derivative becomes approximately 1/x*exp(-x) and-      -- skyrockets for small x. If it happens correct answer is 0.-      | isInfinite f'     = 0-      | abs dx < eps * x' = x'-      | otherwise         = loop (i + 1) x'-      where-        -- Value of γ(a,x) - p-        f    = incompleteGamma a x - p-        -- dγ(a,x)/dx-        f'   | a > 1     = afac * exp( -(x - a1) + a1 * (log x - lna1))-             | otherwise = exp( -x + a1 * log x - gln)-        u    = f / f'-        -- Halley correction to Newton-Rapson step-        corr = u * (a1 / x - 1)-        dx   = u / (1 - 0.5 * min 1.0 corr)-        -- New approximation to x-        x'   | x < dx    = 0.5 * x -- Do not go below 0-             | otherwise = x - dx-    -- Calculate inital guess for root-    guess-      ---      | a > 1   =-         let t  = sqrt $ -2 * log(if p < 0.5 then p else 1 - p)-             x1 = (2.30753 + t * 0.27061) / (1 + t * (0.99229 + t * 0.04481)) - t-             x2 = if p < 0.5 then -x1 else x1-         in max 1e-3 (a * (1 - 1/(9*a) - x2 / (3 * sqrt a)) ** 3)-      -- For a <= 1 use following approximations:-      --   γ(a,1) ≈ 0.253a + 0.12a²-      ---      --   γ(a,x) ≈ γ(a,1)·x^a                               x <  1-      --   γ(a,x) ≈ γ(a,1) + (1 - γ(a,1))(1 - exp(1 - x))    x >= 1-      | otherwise =-         let t = 1 - a * (0.253 + a*0.12)-         in if p < t-            then (p / t) ** (1 / a)-            else 1 - log( 1 - (p-t) / (1-t))-    -- Constants-    a1   = a - 1-    lna1 = log a1-    afac = exp( a1 * (lna1 - 1) - gln )-    gln  = logGamma a-    eps  = 1e-8----------------------------------------------------------------------- Beta function--------------------------------------------------------------------- | Compute the natural logarithm of the beta function.-logBeta :: Double -> Double -> Double-logBeta a b-    | p < 0     = m_NaN-    | p == 0    = m_pos_inf-    | p >= 10   = log q * (-0.5) + m_ln_sqrt_2_pi + logGammaCorrection p + c +-                  (p - 0.5) * log ppq + q * log1p(-ppq)-    | q >= 10   = logGamma p + c + p - p * log pq + (q - 0.5) * log1p(-ppq)-    | otherwise = logGamma p + logGamma q - logGamma pq-    where-      p   = min a b-      q   = max a b-      ppq = p / pq-      pq  = p + q-      c   = logGammaCorrection q - logGammaCorrection pq---- | Regularized incomplete beta function. Uses algorithm AS63 by--- Majumder and Bhattachrjee and quadrature approximation for large--- /p/ and /q/.-incompleteBeta :: Double -- ^ /p/ > 0-               -> Double -- ^ /q/ > 0-               -> Double -- ^ /x/, must lie in [0,1] range-               -> Double-incompleteBeta p q = incompleteBeta_ (logBeta p q) p q---- | Regularized incomplete beta function. Same as 'incompleteBeta'--- but also takes logarithm of beta function as parameter.-incompleteBeta_ :: Double -- ^ logarithm of beta function for given /p/ and /q/-                -> Double -- ^ /p/ > 0-                -> Double -- ^ /q/ > 0-                -> Double -- ^ /x/, must lie in [0,1] range-                -> Double-incompleteBeta_ beta p q x-  | p <= 0 || q <= 0            =-      modErr $ printf "incompleteBeta_: p <= 0 || q <= 0. p=%g q=%g x=%g" p q x-  | x <  0 || x >  1 || isNaN x =-      modErr $ printf "incompletBeta_: x out of [0,1] range. p=%g q=%g x=%g" p q x-  | x == 0 || x == 1            = x-  | p >= (p+q) * x   = incompleteBetaWorker beta p q x-  | otherwise        = 1 - incompleteBetaWorker beta q p (1 - x)----- Approximation of incomplete beta by quandrature.------ Note that x =< p/(p+q)-incompleteBetaApprox :: Double -> Double -> Double -> Double -> Double-incompleteBetaApprox beta p q x-  | ans > 0   = 1 - ans-  | otherwise = -ans-  where-    -- Constants-    p1    = p - 1-    q1    = q - 1-    mu    = p / (p + q)-    lnmu  = log mu-    lnmuc = log (1 - mu)-    -- Upper limit for integration-    xu = max 0 $ min (mu - 10*t) (x - 5*t)-       where-         t = sqrt $ p*q / ( (p+q) * (p+q) * (p + q + 1) )-    -- Calculate incomplete beta by quadrature-    go y w = let t = x + (xu - x) * y-             in  w * exp( p1 * (log t - lnmu) + q1 * (log(1-t) - lnmuc) )-    s   = U.sum $ U.zipWith go coefY coefW-    ans = s * (xu - x) * exp( p1 * lnmu + q1 * lnmuc - beta )----- Worker for incomplete beta function. It is separate function to--- avoid confusion with parameter during parameter swapping-incompleteBetaWorker :: Double -> Double -> Double -> Double -> Double-incompleteBetaWorker beta p q x-  -- For very large p and q this method becomes very slow so another-  -- method is used.-  | p > 3000 && q > 3000 = incompleteBetaApprox beta p q x-  | otherwise            = loop (p+q) (truncate $ q + cx * (p+q)) 1 1 1-  where-    -- Constants-    eps = 1e-15-    cx  = 1 - x-    -- Loop-    loop !psq (ns :: Int) ai term betain-      | done      = betain' * exp( p * log x + (q - 1) * log cx - beta) / p-      | otherwise = loop psq' (ns - 1) (ai + 1) term' betain'-      where-        -- New values-        term'   = term * fact / (p + ai)-        betain' = betain + term'-        fact | ns >  0   = (q - ai) * x/cx-             | ns == 0   = (q - ai) * x-             | otherwise = psq * x-        -- Iterations are complete-        done = db <= eps && db <= eps*betain' where db = abs term'-        psq' = if ns < 0 then psq + 1 else psq------ | Compute inverse of regularized incomplete beta function. Uses--- initial approximation from AS109, AS64 and Halley method to solve--- equation.-invIncompleteBeta :: Double     -- ^ /p/ > 0-                  -> Double     -- ^ /q/ > 0-                  -> Double     -- ^ /a/ ∈ [0,1]-                  -> Double-invIncompleteBeta p q a-  | p <= 0 || q <= 0 =-      modErr $ printf "invIncompleteBeta p <= 0 || q <= 0.  p=%g q=%g a=%g" p q a-  | a <  0 || a >  1 =-      modErr $ printf "invIncompleteBeta x must be in [0,1].  p=%g q=%g a=%g" p q a-  | a == 0 || a == 1 = a-  | a > 0.5          = 1 - invIncompleteBetaWorker (logBeta p q) q p (1 - a)-  | otherwise        =     invIncompleteBetaWorker (logBeta p q) p q  a---invIncompleteBetaWorker :: Double -> Double -> Double -> Double -> Double--- NOTE: p <= 0.5.-invIncompleteBetaWorker beta a b p = loop (0::Int) guess-  where-    a1 = a - 1-    b1 = b - 1-    -- Solve equation using Halley method-    loop !i !x-      -- We cannot continue at this point so we simply return `x'-      | x == 0 || x == 1             = x-      -- When derivative becomes infinite we cannot continue-      -- iterations. It can only happen in vicinity of 0 or 1. It's-      -- hardly possible to get good answer in such circumstances but-      -- `x' is already reasonable.-      | isInfinite f'                = x-      -- Iterations limit reached. Most of the time solution will-      -- converge to answer because of discreteness of Double. But-      -- solution have good precision already.-      | i >= 10                      = x-      -- Solution converges-      | abs dx <= 16 * m_epsilon * x = x'-      | otherwise                    = loop (i+1) x'-      where-        -- Calculate Halley step.-        f   = incompleteBeta_ beta a b x - p-        f'  = exp $ a1 * log x + b1 * log (1 - x) - beta-        u   = f / f'-        dx  = u / (1 - 0.5 * min 1 (u * (a1 / x - b1 / (1 - x))))-        -- Next approximation. If Halley step leads us out of [0,1]-        -- range we revert to bisection.-        x'  | z < 0     = x / 2-            | z > 1     = (x + 1) / 2-            | otherwise = z-            where z = x - dx-    -- Calculate initial guess. Approximations from AS64, AS109 and-    -- Numerical recipes are used.-    ---    -- Equations are referred to by name of paper and number e.g. [AS64 2]-    -- In AS64 papers equations are not numbered so they are refered-    -- to by number of appearance starting from definition of-    -- incomplete beta.-    guess-      -- In this region we use approximation from AS109 (Carter-      -- approximation). It's reasonably good (2 iterations on-      -- average)-      | a > 1 && b > 1 =-          let r = (y*y - 3) / 6-              s = 1 / (2*a - 1)-              t = 1 / (2*b - 1)-              h = 2 / (s + t)-              w = y * sqrt(h + r) / h - (t - s) * (r + 5/6 - 2 / (3 * h))-          in a / (a + b * exp(2 * w))-      -- Otherwise we revert to approximation from AS64 derived from-      -- [AS64 2] when it's applicable.-      ---      -- It slightly reduces average number of iterations when `a' and-      -- `b' have different magnitudes.-      | chi2 > 0 && ratio > 1 = 1 - 2 / (ratio + 1)-      -- If all else fails we use approximation from "Numerical-      -- Recipes". It's very similar to approximations [AS64 4,5] but-      -- it never goes out of [0,1] interval.-      | otherwise = case () of-          _| p < t / w  -> (a * p * w) ** (1/a)-           | otherwise  -> 1 - (b * (1 - p) * w) ** (1/b)-           where-             lna = log $ a / (a+b)-             lnb = log $ b / (a+b)-             t   = exp( a * lna ) / a-             u   = exp( b * lnb ) / b-             w   = t + u-      where-        -- Formula [2]-        ratio = (4*a + 2*b - 2) / chi2-        -- Quantile of chi-squared distribution. Formula [3].-        chi2 = 2 * b * (1 - t + y * sqrt t) ** 3-          where-            t   = 1 / (9 * b)-        -- `y' is Hasting's approximation of p'th quantile of standard-        -- normal distribution.-        y   = r - ( 2.30753 + 0.27061 * r )-                  / ( 1.0 + ( 0.99229 + 0.04481 * r ) * r )-          where-            r = sqrt $ - 2 * log p----------------------------------------------------------------------- Sinc function--------------------------------------------------------------------- | Compute sinc function @sin(x)\/x@-sinc :: Double -> Double-sinc x-  | ax < eps_0 = 1-  | ax < eps_2 = 1 - x2/6-  | ax < eps_4 = 1 - x2/6 + x2*x2/120-  | otherwise  = sin x / x-  where-    ax    = abs x-    x2    = x*x-    -- For explanation of choice see `doc/sinc.hs'-    eps_0 = 1.8250120749944284e-8 -- sqrt (6ε/4)-    eps_2 = 1.4284346431400855e-4 --   (30ε)**(1/4) / 2-    eps_4 = 4.043633626430947e-3  -- (1206ε)**(1/6) / 2---------------------------------------------------------------------- Logarithm--------------------------------------------------------------------- | Compute the natural logarithm of 1 + @x@.  This is accurate even--- for values of @x@ near zero, where use of @log(1+x)@ would lose--- precision.-log1p :: Double -> Double-log1p x-    | x == 0               = 0-    | x == -1              = m_neg_inf-    | x < -1               = m_NaN-    | x' < m_epsilon * 0.5 = x-    | (x >= 0 && x < 1e-8) || (x >= -1e-9 && x < 0)-                           = x * (1 - x * 0.5)-    | x' < 0.375           = x * (1 - x * chebyshevBroucke (x / 0.375) coeffs)-    | otherwise            = log (1 + x)-  where-    x' = abs x-    coeffs = U.fromList [-               0.10378693562743769800686267719098e+1,-              -0.13364301504908918098766041553133e+0,-               0.19408249135520563357926199374750e-1,-              -0.30107551127535777690376537776592e-2,-               0.48694614797154850090456366509137e-3,-              -0.81054881893175356066809943008622e-4,-               0.13778847799559524782938251496059e-4,-              -0.23802210894358970251369992914935e-5,-               0.41640416213865183476391859901989e-6,-              -0.73595828378075994984266837031998e-7,-               0.13117611876241674949152294345011e-7,-              -0.23546709317742425136696092330175e-8,-               0.42522773276034997775638052962567e-9,-              -0.77190894134840796826108107493300e-10,-               0.14075746481359069909215356472191e-10,-              -0.25769072058024680627537078627584e-11,-               0.47342406666294421849154395005938e-12,-              -0.87249012674742641745301263292675e-13,-               0.16124614902740551465739833119115e-13,-              -0.29875652015665773006710792416815e-14,-               0.55480701209082887983041321697279e-15,-              -0.10324619158271569595141333961932e-15-             ]----- | /O(log n)/ Compute the logarithm in base 2 of the given value.-log2 :: Int -> Int-log2 v0-    | v0 <= 0   = modErr $ "log2: nonpositive input, got " ++ show v0-    | otherwise = go 5 0 v0-  where-    go !i !r !v | i == -1        = r-                | v .&. b i /= 0 = let si = U.unsafeIndex sv i-                                   in go (i-1) (r .|. si) (v `shiftR` si)-                | otherwise      = go (i-1) r v-    b = U.unsafeIndex bv-    !bv = U.fromList [0x2, 0xc, 0xf0, 0xff00, 0xffff0000, 0xffffffff00000000]-    !sv = U.fromList [1,2,4,8,16,32]---------------------------------------------------------------------- Factorial--------------------------------------------------------------------- | Compute the factorial function /n/!.  Returns +∞ if the--- input is above 170 (above which the result cannot be represented by--- a 64-bit 'Double').-factorial :: Int -> Double-factorial n-    | n < 0     = error "Numeric.SpecFunctions.factorial: negative input"-    | n <= 1    = 1-    | n <= 170  = U.product $ U.map fromIntegral $ U.enumFromTo 2 n-    | otherwise = m_pos_inf---- | Compute the natural logarithm of the factorial function.  Gives--- 16 decimal digits of precision.-logFactorial :: Integral a => a -> Double-logFactorial n-    | n <  0    = error "Numeric.SpecFunctions.logFactorial: negative input"-    | n <= 14   = log $ factorial $ fromIntegral n-    -- N.B. Γ(n+1) = n!-    ---    -- We use here asymptotic series for gamma function. See-    -- http://mathworld.wolfram.com/StirlingsSeries.html-    | otherwise = (x - 0.5) * log x - x-                + m_ln_sqrt_2_pi-                + evaluateOddPolynomialL (1/x) [1/12, -1/360, 1/1260, -1/1680]-    where x = fromIntegral n + 1-{-# SPECIALIZE logFactorial :: Int -> Double #-}---- | Calculate the error term of the Stirling approximation.  This is--- only defined for non-negative values.------ > stirlingError @n@ = @log(n!) - log(sqrt(2*pi*n)*(n/e)^n)-stirlingError :: Double -> Double-stirlingError n-  | n <= 15.0   = case properFraction (n+n) of-                    (i,0) -> sfe `U.unsafeIndex` i-                    _     -> logGamma (n+1.0) - (n+0.5) * log n + n --                             m_ln_sqrt_2_pi-  | n > 500     = evaluateOddPolynomialL (1/n) [s0,-s1]-  | n > 80      = evaluateOddPolynomialL (1/n) [s0,-s1,s2]-  | n > 35      = evaluateOddPolynomialL (1/n) [s0,-s1,s2,-s3]-  | otherwise   = evaluateOddPolynomialL (1/n) [s0,-s1,s2,-s3,s4]-  where-    s0 = 0.083333333333333333333        -- 1/12-    s1 = 0.00277777777777777777778      -- 1/360-    s2 = 0.00079365079365079365079365   -- 1/1260-    s3 = 0.000595238095238095238095238  -- 1/1680-    s4 = 0.0008417508417508417508417508 -- 1/1188-    sfe = U.fromList [ 0.0,-                0.1534264097200273452913848,   0.0810614667953272582196702,-                0.0548141210519176538961390,   0.0413406959554092940938221,-                0.03316287351993628748511048,  0.02767792568499833914878929,-                0.02374616365629749597132920,  0.02079067210376509311152277,-                0.01848845053267318523077934,  0.01664469118982119216319487,-                0.01513497322191737887351255,  0.01387612882307074799874573,-                0.01281046524292022692424986,  0.01189670994589177009505572,-                0.01110455975820691732662991,  0.010411265261972096497478567,-                0.009799416126158803298389475, 0.009255462182712732917728637,-                0.008768700134139385462952823, 0.008330563433362871256469318,-                0.007934114564314020547248100, 0.007573675487951840794972024,-                0.007244554301320383179543912, 0.006942840107209529865664152,-                0.006665247032707682442354394, 0.006408994188004207068439631,-                0.006171712263039457647532867, 0.005951370112758847735624416,-                0.005746216513010115682023589, 0.005554733551962801371038690 ]---------------------------------------------------------------------- Combinatorics--------------------------------------------------------------------- |--- Quickly compute the natural logarithm of /n/ @`choose`@ /k/, with--- no checking.------ Less numerically stable:------ > exp $ lg (n+1) - lg (k+1) - lg (n-k+1)--- >   where lg = logGamma . fromIntegral-logChooseFast :: Double -> Double -> Double-logChooseFast n k = -log (n + 1) - logBeta (n - k + 1) (k + 1)---- | Calculate binomial coefficient using exact formula-chooseExact :: Int -> Int -> Double-n `chooseExact` k-  = U.foldl' go 1 $ U.enumFromTo 1 k-  where-    go a i      = a * (nk + j) / j-        where j = fromIntegral i :: Double-    nk = fromIntegral (n - k)---- | Compute logarithm of the binomial coefficient.-logChoose :: Int -> Int -> Double-n `logChoose` k-    | k  > n    = (-1) / 0-      -- For very large N exact algorithm overflows double so we-      -- switch to beta-function based one-    | k' < 50 && (n < 20000000) = log $ chooseExact n k'-    | otherwise                 = logChooseFast (fromIntegral n) (fromIntegral k)-  where-    k' = min k (n-k)---- | Compute the binomial coefficient /n/ @\``choose`\`@ /k/. For--- values of /k/ > 50, this uses an approximation for performance--- reasons.  The approximation is accurate to 12 decimal places in the--- worst case------ Example:------ > 7 `choose` 3 == 35-choose :: Int -> Int -> Double-n `choose` k-    | k  > n         = 0-    | k' < 50        = chooseExact n k'-    | approx < max64 = fromIntegral . round64 $ approx-    | otherwise      = approx-  where-    k'             = min k (n-k)-    approx         = exp $ logChooseFast (fromIntegral n) (fromIntegral k')-    max64          = fromIntegral (maxBound :: Int64)-    round64 x      = round x :: Int64---- | Compute ψ0(/x/), the first logarithmic derivative of the gamma--- function. Uses Algorithm AS 103 by Bernardo, based on Minka's C--- implementation.-digamma :: Double -> Double-digamma x-    | isNaN x || isInfinite x                  = m_NaN-    -- FIXME:-    --   This is ugly. We are testing here that number is in fact-    --   integer. It's somewhat tricky question to answer. When ε for-    --   given number becomes 1 or greater every number is represents-    --   an integer. We also must make sure that excess precision-    --   won't bite us.-    | x <= 0 && fromIntegral (truncate x :: Int64) == x = m_neg_inf-    -- Jeffery's reflection formula-    | x < 0     = digamma (1 - x) + pi / tan (negate pi * x)-    | x <= 1e-6 = - γ - 1/x + trigamma1 * x-    | x' < c    = r-    -- De Moivre's expansion-    | otherwise = let s = 1/x'-                  in  evaluateEvenPolynomialL s-                        [   r + log x' - 0.5 * s-                        , - 1/12-                        ,   1/120-                        , - 1/252-                        ,   1/240-                        , - 1/132-                        ,  391/32760-                        ]-  where-    γ  = m_eulerMascheroni-    c  = 12-    -- Reduce to digamma (x + n) where (x + n) >= c-    (r, x') = reduce 0 x-      where-        reduce !s y-          | y < c     = reduce (s - 1 / y) (y + 1)-          | otherwise = (s, y)----------------------------------------------------------------------- Constants--------------------------------------------------------------------- Coefficients for 18-point Gauss-Legendre integration. They are--- used in implementation of incomplete gamma and beta functions.-coefW,coefY :: U.Vector Double-coefW = U.fromList [ 0.0055657196642445571, 0.012915947284065419, 0.020181515297735382-                   , 0.027298621498568734,  0.034213810770299537, 0.040875750923643261-                   , 0.047235083490265582,  0.053244713977759692, 0.058860144245324798-                   , 0.064039797355015485,  0.068745323835736408, 0.072941885005653087-                   , 0.076598410645870640,  0.079687828912071670, 0.082187266704339706-                   , 0.084078218979661945,  0.085346685739338721, 0.085983275670394821-                   ]-coefY = U.fromList [ 0.0021695375159141994, 0.011413521097787704, 0.027972308950302116-                   , 0.051727015600492421,  0.082502225484340941, 0.12007019910960293-                   , 0.16415283300752470,   0.21442376986779355,  0.27051082840644336-                   , 0.33199876341447887,   0.39843234186401943,  0.46931971407375483-                   , 0.54413605556657973,   0.62232745288031077,  0.70331500465597174-                   , 0.78649910768313447,   0.87126389619061517,  0.95698180152629142-                   ]-{-# NOINLINE coefW #-}-{-# NOINLINE coefY #-}--trigamma1 :: Double-trigamma1 = 1.6449340668482264365 -- pi**2 / 6--modErr :: String -> a-modErr msg = error $ "Numeric.SpecFunctions." ++ msg+{-# OPTIONS_HADDOCK hide #-}+-- |+-- Module    : Numeric.SpecFunctions.Internal+-- Copyright : (c) 2009, 2011, 2012 Bryan O'Sullivan+-- License   : BSD3+--+-- Maintainer  : bos@serpentine.com+-- Stability   : experimental+-- Portability : portable+--+-- Internal module with implementation of special functions.+module Numeric.SpecFunctions.Internal+    ( module Numeric.SpecFunctions.Internal+    , Compat.log1p+    , Compat.expm1+    ) where++import Data.Bits          ((.&.), (.|.), shiftR)+import Data.Int           (Int64)+import Data.Default.Class+import qualified Data.Vector.Unboxed as U+import           Data.Vector.Unboxed   ((!))+import Text.Printf++import Numeric.Polynomial.Chebyshev    (chebyshevBroucke)+import Numeric.Polynomial              (evaluatePolynomial, evaluatePolynomialL, evaluateEvenPolynomialL+                                       ,evaluateOddPolynomialL)+import Numeric.RootFinding             (Root(..), newtonRaphson, NewtonParam(..), Tolerance(..))+import Numeric.Series+import Numeric.MathFunctions.Constants+import Numeric.SpecFunctions.Compat (log1p)+import qualified Numeric.SpecFunctions.Compat as Compat++----------------------------------------------------------------+-- Error function+----------------------------------------------------------------++-- | Error function.+--+-- \[+-- \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} \exp(-t^2) dt+-- \]+--+-- Function limits are:+--+-- \[+-- \begin{aligned}+--  &\operatorname{erf}(-\infty) &=& -1 \\+--  &\operatorname{erf}(0)       &=& \phantom{-}\,0 \\+--  &\operatorname{erf}(+\infty) &=& \phantom{-}\,1 \\+-- \end{aligned}+-- \]+erf :: Double -> Double+erf = Compat.erf+{-# INLINE erf #-}++-- | Complementary error function.+--+-- \[+-- \operatorname{erfc}(x) = 1 - \operatorname{erf}(x)+-- \]+--+-- Function limits are:+--+-- \[+-- \begin{aligned}+--  &\operatorname{erf}(-\infty) &=&\, 2 \\+--  &\operatorname{erf}(0)       &=&\, 1 \\+--  &\operatorname{erf}(+\infty) &=&\, 0 \\+-- \end{aligned}+-- \]+erfc :: Double -> Double+erfc = Compat.erfc+{-# INLINE erfc #-}++-- | Inverse of 'erf'.+invErf :: Double -- ^ /p/ ∈ [-1,1]+       -> Double+invErf p+  | p ==  1         = m_pos_inf+  | p == -1         = m_neg_inf+  | p < 1 && p > -1 = if p > 0 then r else -r+  | otherwise       = error "invErf: p must in [-1,1] range"+  where+    -- We solve equation same as in invErfc. We're able to ruse same+    -- Halley step by solving equation:+    --   > pp - erf x = 0+    -- instead of+    --   > erf x - pp = 0+    pp     = abs p+    r      = step $ step $ guessInvErfc $ 1 - pp+    step x = invErfcHalleyStep (pp - erf x) x++-- | Inverse of 'erfc'.+invErfc :: Double -- ^ /p/ ∈ [0,2]+        -> Double+invErfc p+  | p == 2        = m_neg_inf+  | p == 0        = m_pos_inf+  | p >0 && p < 2 = if p <= 1 then r else -r+  | otherwise     = modErr $ "invErfc: p must be in [0,2] got " ++ show p+  where+    pp | p <= 1    = p+       | otherwise = 2 - p+    -- We perform 2 Halley steps in order to get to solution+    r      = step $ step $ guessInvErfc pp+    step x = invErfcHalleyStep (erfc x - pp) x++-- Initial guess for invErfc & invErf+guessInvErfc :: Double -> Double+guessInvErfc p+  = -0.70711 * ((2.30753 + t * 0.27061) / (1 + t * (0.99229 + t * 0.04481)) - t)+  where+    t = sqrt $ -2 * log( 0.5 * p)++-- Halley step for solving invErfc+invErfcHalleyStep :: Double -> Double -> Double+invErfcHalleyStep err x+  = x + err / (1.12837916709551257 * exp(-x * x) - x * err)++----------------------------------------------------------------+-- Gamma function+----------------------------------------------------------------++data L = L {-# UNPACK #-} !Double {-# UNPACK #-} !Double++-- | Compute the logarithm of the gamma function, Γ(/x/).+--+-- \[+-- \Gamma(x) = \int_0^{\infty}t^{x-1}e^{-t}\,dt = (x - 1)!+-- \]+--+-- This implementation uses Lanczos approximation. It gives 14 or more+-- significant decimal digits, except around /x/ = 1 and /x/ = 2,+-- where the function goes to zero.+--+-- Returns &#8734; if the input is outside of the range (0 < /x/+-- &#8804; 1e305).+logGamma :: Double -> Double+logGamma z+  | z <= 0    = m_pos_inf+  -- For very small values z we can just use Laurent expansion+  | z < m_sqrt_eps = log (1/z - m_eulerMascheroni)+  -- For z<1 we use recurrence. Γ(z+1) = z·Γ(z) Note that in order to+  -- avoid precision loss we have to compute parameter to+  -- approximations here:+  --+  -- > (z + 1) - 1 = z+  -- > (z + 1) - 2 = z - 1+  --+  -- Simple passing (z + 1) to piecewise approximations and computing+  -- difference leads to bad loss of precision near 1.+  -- This is reason lgamma1_15 & lgamma15_2 have three parameters+  | z < 0.5   = lgamma1_15 z (z - 1) - log z+  | z < 1     = lgamma15_2 z (z - 1) - log z+  -- Piecewise polynomial approximations+  | z <= 1.5  = lgamma1_15 (z - 1) (z - 2)+  | z < 2     = lgamma15_2 (z - 1) (z - 2)+  | z < 15    = lgammaSmall z+  -- Otherwise we switch to Lanczos approximation+  | otherwise = lanczosApprox z+++-- | Synonym for 'logGamma'. Retained for compatibility+logGammaL :: Double -> Double+logGammaL = logGamma+{-# DEPRECATED logGammaL "Use logGamma instead" #-}++++-- Polynomial expansion used in interval (1,1.5]+--+-- > logΓ(z) = (z-1)(z-2)(Y + R(z-1))+lgamma1_15 :: Double -> Double -> Double+lgamma1_15 zm1 zm2+   = r * y + r * ( evaluatePolynomial zm1 tableLogGamma_1_15P+                 / evaluatePolynomial zm1 tableLogGamma_1_15Q+                 )+   where+     r = zm1 * zm2+     y = 0.52815341949462890625++tableLogGamma_1_15P,tableLogGamma_1_15Q :: U.Vector Double+tableLogGamma_1_15P = U.fromList+  [  0.490622454069039543534e-1+  , -0.969117530159521214579e-1+  , -0.414983358359495381969e0+  , -0.406567124211938417342e0+  , -0.158413586390692192217e0+  , -0.240149820648571559892e-1+  , -0.100346687696279557415e-2+  ]+{-# NOINLINE tableLogGamma_1_15P #-}+tableLogGamma_1_15Q = U.fromList+  [ 1+  , 0.302349829846463038743e1+  , 0.348739585360723852576e1+  , 0.191415588274426679201e1+  , 0.507137738614363510846e0+  , 0.577039722690451849648e-1+  , 0.195768102601107189171e-2+  ]+{-# NOINLINE tableLogGamma_1_15Q #-}++++-- Polynomial expansion used in interval (1.5,2)+--+-- > logΓ(z) = (2-z)(1-z)(Y + R(2-z))+lgamma15_2 :: Double -> Double -> Double+lgamma15_2 zm1 zm2+   = r * y + r * ( evaluatePolynomial (-zm2) tableLogGamma_15_2P+                 / evaluatePolynomial (-zm2) tableLogGamma_15_2Q+                 )+   where+     r = zm1 * zm2+     y = 0.452017307281494140625++tableLogGamma_15_2P,tableLogGamma_15_2Q :: U.Vector Double+tableLogGamma_15_2P = U.fromList+  [ -0.292329721830270012337e-1+  ,  0.144216267757192309184e0+  , -0.142440390738631274135e0+  ,  0.542809694055053558157e-1+  , -0.850535976868336437746e-2+  ,  0.431171342679297331241e-3+  ]+{-# NOINLINE tableLogGamma_15_2P #-}+tableLogGamma_15_2Q = U.fromList+  [  1+  , -0.150169356054485044494e1+  ,  0.846973248876495016101e0+  , -0.220095151814995745555e0+  ,  0.25582797155975869989e-1+  , -0.100666795539143372762e-2+  , -0.827193521891290553639e-6+  ]+{-# NOINLINE tableLogGamma_15_2Q #-}++++-- Polynomial expansion used in interval (2,3)+--+-- > logΓ(z) = (z - 2)(z + 1)(Y + R(z-2))+lgamma2_3 :: Double -> Double+lgamma2_3 z+  = r * y + r * ( evaluatePolynomial zm2 tableLogGamma_2_3P+                / evaluatePolynomial zm2 tableLogGamma_2_3Q+                )+  where+    r   = zm2 * (z + 1)+    zm2 = z - 2+    y   = 0.158963680267333984375e0+++tableLogGamma_2_3P,tableLogGamma_2_3Q :: U.Vector Double+tableLogGamma_2_3P = U.fromList+  [ -0.180355685678449379109e-1+  ,  0.25126649619989678683e-1+  ,  0.494103151567532234274e-1+  ,  0.172491608709613993966e-1+  , -0.259453563205438108893e-3+  , -0.541009869215204396339e-3+  , -0.324588649825948492091e-4+  ]+{-# NOINLINE tableLogGamma_2_3P #-}+tableLogGamma_2_3Q = U.fromList+  [  1+  ,  0.196202987197795200688e1+  ,  0.148019669424231326694e1+  ,  0.541391432071720958364e0+  ,  0.988504251128010129477e-1+  ,  0.82130967464889339326e-2+  ,  0.224936291922115757597e-3+  , -0.223352763208617092964e-6+  ]+{-# NOINLINE tableLogGamma_2_3Q #-}++++-- For small z we can just use Gamma function recurrence and reduce+-- problem to interval [2,3] and use polynomial approximation+-- there. Surprisingly it gives very good precision+lgammaSmall :: Double -> Double+lgammaSmall = go 0+  where+    go acc z | z < 3     = acc + lgamma2_3 z+             | otherwise = go (acc + log zm1) zm1+             where+               zm1 = z - 1+++-- Lanczos approximation for gamma function.+--+-- > Γ(z) = sqrt(2π)(z + g - 0.5)^(z - 0.5)·exp{-(z + g - 0.5)}·A_g(z)+--+-- Coefficients are taken from boost. Constants are absorbed into+-- polynomial's coefficients.+lanczosApprox :: Double -> Double+lanczosApprox z+  = (log (z + g - 0.5) - 1) * (z - 0.5)+  + log (evalRatio tableLanczos z)+  where+    g = 6.024680040776729583740234375++tableLanczos :: U.Vector (Double,Double)+{-# NOINLINE tableLanczos #-}+tableLanczos = U.fromList+  [ (56906521.91347156388090791033559122686859    , 0)+  , (103794043.1163445451906271053616070238554    , 39916800)+  , (86363131.28813859145546927288977868422342    , 120543840)+  , (43338889.32467613834773723740590533316085    , 150917976)+  , (14605578.08768506808414169982791359218571    , 105258076)+  , (3481712.15498064590882071018964774556468     , 45995730)+  , (601859.6171681098786670226533699352302507    , 13339535)+  , (75999.29304014542649875303443598909137092    , 2637558)+  , (6955.999602515376140356310115515198987526    , 357423)+  , (449.9445569063168119446858607650988409623    , 32670)+  , (19.51992788247617482847860966235652136208    , 1925)+  , (0.5098416655656676188125178644804694509993   , 66)+  , (0.006061842346248906525783753964555936883222 , 1)+  ]++-- Evaluate rational function. Polynomials in both numerator and+-- denominator must have same order. Function seems to be too specific+-- so it's not exposed+--+-- Special care taken in order to avoid overflow for large values of x+evalRatio :: U.Vector (Double,Double) -> Double -> Double+evalRatio coef x+  | x > 1     = fini $ U.foldl' stepL (L 0 0) coef+  | otherwise = fini $ U.foldr' stepR (L 0 0) coef+  where+    fini (L num den) = num / den+    stepR (a,b) (L num den) = L (num * x  + a) (den * x  + b)+    stepL (L num den) (a,b) = L (num * rx + a) (den * rx + b)+    rx = recip x++++-- |+-- Compute the log gamma correction factor for Stirling+-- approximation for @x@ &#8805; 10.  This correction factor is+-- suitable for an alternate (but less numerically accurate)+-- definition of 'logGamma':+--+-- \[+-- \log\Gamma(x) = \frac{1}{2}\log(2\pi) + (x-\frac{1}{2})\log x - x + \operatorname{logGammaCorrection}(x)+-- \]+logGammaCorrection :: Double -> Double+logGammaCorrection x+    | x < 10    = m_NaN+    | x < big   = chebyshevBroucke (t * t * 2 - 1) coeffs / x+    | otherwise = 1 / (x * 12)+  where+    big    = 94906265.62425156+    t      = 10 / x+    coeffs = U.fromList [+               0.1666389480451863247205729650822e+0,+              -0.1384948176067563840732986059135e-4,+               0.9810825646924729426157171547487e-8,+              -0.1809129475572494194263306266719e-10,+               0.6221098041892605227126015543416e-13,+              -0.3399615005417721944303330599666e-15,+               0.2683181998482698748957538846666e-17+             ]++++-- | Compute the normalized lower incomplete gamma function+-- γ(/z/,/x/). Normalization means that γ(/z/,∞)=1+--+-- \[+-- \gamma(z,x) = \frac{1}{\Gamma(z)}\int_0^{x}t^{z-1}e^{-t}\,dt+-- \]+--+-- Uses Algorithm AS 239 by Shea.+incompleteGamma :: Double       -- ^ /z/ ∈ (0,∞)+                -> Double       -- ^ /x/ ∈ (0,∞)+                -> Double+-- Notation used:+--  + P(a,x) - regularized lower incomplete gamma+--  + Q(a,x) - regularized upper incomplete gamma+incompleteGamma a x+  | a <= 0 || x < 0 = error+     $ "incompleteGamma: Domain error z=" ++ show a ++ " x=" ++ show x+  | x == 0          = 0+  | x == m_pos_inf  = 1+  -- For very small x we use following expansion for P:+  --+  -- See http://functions.wolfram.com/GammaBetaErf/GammaRegularized/06/01/05/01/01/+  | x < sqrt m_epsilon && a > 1+    = x**a / a / exp (logGamma a) * (1 - a*x / (a + 1))+  | x < 0.5 = case () of+    _| (-0.4)/log x < a  -> taylorSeriesP+     | otherwise         -> taylorSeriesComplQ+  | x < 1.1 = case () of+    _| 0.75*x < a        -> taylorSeriesP+     | otherwise         -> taylorSeriesComplQ+  | a > 20 && useTemme    = uniformExpansion+  | x - (1 / (3 * x)) < a = taylorSeriesP+  | otherwise             = contFraction+  where+    mu = (x - a) / a+    useTemme = (a > 200 && 20/a > mu*mu)+            || (abs mu < 0.4)+    -- Gautschi's algorithm.+    --+    -- Evaluate series for P(a,x). See [Temme1994] Eq. 5.5 and [NOTE:+    -- incompleteGamma.taylorP]+    factorP+      | a < 10     = x ** a+                   / (exp x * exp (logGamma (a + 1)))+      | a < 1182.5 = (x * exp 1 / a) ** a+                   / exp x+                   / sqrt (2*pi*a)+                   / exp (logGammaCorrection a)+      | otherwise  = (x * exp 1 / a * exp (-x/a)) ** a+                   / sqrt (2*pi*a)+                   / exp (logGammaCorrection a)+    taylorSeriesP+      = sumPowerSeries x (scanSequence (/) 1 $ enumSequenceFrom (a+1))+      * factorP+    -- Series for 1-Q(a,x). See [Temme1994] Eq. 5.5+    taylorSeriesComplQ+      = sumPowerSeries (-x) (scanSequence (/) 1 (enumSequenceFrom 1) / enumSequenceFrom a)+      * x**a / exp(logGamma a)+    -- Legendre continued fractions+    contFraction = 1 - ( exp ( log x * a - x - logGamma a )+                       / evalContFractionB frac+                       )+      where+        frac = (\k -> (k*(a-k), x - a + 2*k + 1)) <$> enumSequenceFrom 0+    -- Evaluation based on uniform expansions. See [Temme1994] 5.2+    uniformExpansion =+      let -- Coefficients f_m in paper+          fm :: U.Vector Double+          fm = U.fromList [ 1.00000000000000000000e+00+                          ,-3.33333333333333370341e-01+                          , 8.33333333333333287074e-02+                          ,-1.48148148148148153802e-02+                          , 1.15740740740740734316e-03+                          , 3.52733686067019369930e-04+                          ,-1.78755144032921825352e-04+                          , 3.91926317852243766954e-05+                          ,-2.18544851067999240532e-06+                          ,-1.85406221071515996597e-06+                          , 8.29671134095308545622e-07+                          ,-1.76659527368260808474e-07+                          , 6.70785354340149841119e-09+                          , 1.02618097842403069078e-08+                          ,-4.38203601845335376897e-09+                          , 9.14769958223679020897e-10+                          ,-2.55141939949462514346e-11+                          ,-5.83077213255042560744e-11+                          , 2.43619480206674150369e-11+                          ,-5.02766928011417632057e-12+                          , 1.10043920319561347525e-13+                          , 3.37176326240098513631e-13+                          ]+          y   = - log1pmx mu+          eta = sqrt (2 * y) * signum mu+          -- Evaluate S_α (Eq. 5.9)+          loop !_  !_  u 0 = u+          loop bm1 bm0 u i = let t  = (fm ! i) + (fromIntegral i + 1)*bm1 / a+                                 u' = eta * u + t+                             in  loop bm0 t u' (i-1)+          s_a = let n = U.length fm+                in  loop (fm ! (n-1)) (fm ! (n-2)) 0 (n-3)+                  / exp (logGammaCorrection a)+      in 1/2 * erfc(-eta*sqrt(a/2)) - exp(-(a*y)) / sqrt (2*pi*a) * s_a++++-- Adapted from Numerical Recipes §6.2.1++-- | Inverse incomplete gamma function. It's approximately inverse of+--   'incompleteGamma' for the same /z/. So following equality+--   approximately holds:+--+-- > invIncompleteGamma z . incompleteGamma z ≈ id+invIncompleteGamma :: Double    -- ^ /z/ ∈ (0,∞)+                   -> Double    -- ^ /p/ ∈ [0,1]+                   -> Double+invIncompleteGamma a p+  | a <= 0         =+      modErr $ printf "invIncompleteGamma: a must be positive. a=%g p=%g" a p+  | p < 0 || p > 1 =+      modErr $ printf "invIncompleteGamma: p must be in [0,1] range. a=%g p=%g" a p+  | p == 0         = 0+  | p == 1         = 1 / 0+  | otherwise      = loop 0 guess+  where+    -- Solve equation γ(a,x) = p using Halley method+    loop :: Int -> Double -> Double+    loop i x+      | i >= 12           = x'+      -- For small s derivative becomes approximately 1/x*exp(-x) and+      -- skyrockets for small x. If it happens correct answer is 0.+      | isInfinite f'     = 0+      | abs dx < eps * x' = x'+      | otherwise         = loop (i + 1) x'+      where+        -- Value of γ(a,x) - p+        f    = incompleteGamma a x - p+        -- dγ(a,x)/dx+        f'   | a > 1     = afac * exp( -(x - a1) + a1 * (log x - lna1))+             | otherwise = exp( -x + a1 * log x - gln)+        u    = f / f'+        -- Halley correction to Newton-Rapson step+        corr = u * (a1 / x - 1)+        dx   = u / (1 - 0.5 * min 1.0 corr)+        -- New approximation to x+        x'   | x < dx    = 0.5 * x -- Do not go below 0+             | otherwise = x - dx+    -- Calculate initial guess for root+    guess+      --+      | a > 1   =+         let t  = sqrt $ -2 * log(if p < 0.5 then p else 1 - p)+             x1 = (2.30753 + t * 0.27061) / (1 + t * (0.99229 + t * 0.04481)) - t+             x2 = if p < 0.5 then -x1 else x1+         in max 1e-3 (a * (1 - 1/(9*a) - x2 / (3 * sqrt a)) ** 3)+      -- For a <= 1 use following approximations:+      --   γ(a,1) ≈ 0.253a + 0.12a²+      --+      --   γ(a,x) ≈ γ(a,1)·x^a                               x <  1+      --   γ(a,x) ≈ γ(a,1) + (1 - γ(a,1))(1 - exp(1 - x))    x >= 1+      | otherwise =+         let t = 1 - a * (0.253 + a*0.12)+         in if p < t+            then (p / t) ** (1 / a)+            else 1 - log( 1 - (p-t) / (1-t))+    -- Constants+    a1   = a - 1+    lna1 = log a1+    afac = exp( a1 * (lna1 - 1) - gln )+    gln  = logGamma a+    eps  = 1e-8++++----------------------------------------------------------------+-- Beta function+----------------------------------------------------------------++-- | Compute the natural logarithm of the beta function.+--+-- \[+-- B(a,b) = \int_0^1 t^{a-1}(1-t)^{b-1}\,dt = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}+-- \]+logBeta+  :: Double                     -- ^ /a/ > 0+  -> Double                     -- ^ /b/ > 0+  -> Double+logBeta a b+  | p < 0     = m_NaN+  | p == 0    = m_pos_inf+  | p >= 10   = allStirling+  | q >= 10   = twoStirling+  -- This order of summands marginally improves precision+  | otherwise = logGamma p + (logGamma q - logGamma pq)+  where+    p   = min a b+    q   = max a b+    ppq = p / pq+    pq  = p + q+    -- When both parameters are large than 10 we can use Stirling+    -- approximation with correction. It's more precise than sum of+    -- logarithms of gamma functions+    allStirling+      = log q * (-0.5)+      + m_ln_sqrt_2_pi+      + logGammaCorrection p+      + (logGammaCorrection q - logGammaCorrection pq)+      + (p - 0.5) * log ppq+      + q * log1p(-ppq)+    -- Otherwise only two of three gamma functions use Stirling+    -- approximation+    twoStirling+      = logGamma p+      + (logGammaCorrection q - logGammaCorrection pq)+      + p+      - p * log pq+      + (q - 0.5) * log1p(-ppq)+++-- | Regularized incomplete beta function.+--+-- \[+-- I(x;a,b) = \frac{1}{B(a,b)} \int_0^x t^{a-1}(1-t)^{b-1}\,dt+-- \]+--+-- Uses algorithm AS63 by Majumder and Bhattachrjee and quadrature+-- approximation for large /p/ and /q/.+incompleteBeta :: Double -- ^ /a/ > 0+               -> Double -- ^ /b/ > 0+               -> Double -- ^ /x/, must lie in [0,1] range+               -> Double+incompleteBeta p q = incompleteBeta_ (logBeta p q) p q++-- | Regularized incomplete beta function. Same as 'incompleteBeta'+-- but also takes logarithm of beta function as parameter.+incompleteBeta_ :: Double -- ^ logarithm of beta function for given /p/ and /q/+                -> Double -- ^ /a/ > 0+                -> Double -- ^ /b/ > 0+                -> Double -- ^ /x/, must lie in [0,1] range+                -> Double+incompleteBeta_ beta p q x+  | p <= 0 || q <= 0            =+      modErr $ printf "incompleteBeta_: p <= 0 || q <= 0. p=%g q=%g x=%g" p q x+  | x <  0 || x >  1 || isNaN x =+      modErr $ printf "incompleteBeta_: x out of [0,1] range. p=%g q=%g x=%g" p q x+  | x == 0 || x == 1            = x+  | p >= (p+q) * x   = incompleteBetaWorker beta p q x+  | otherwise        = 1 - incompleteBetaWorker beta q p (1 - x)+++-- Approximation of incomplete beta by quadrature.+--+-- Note that x =< p/(p+q)+incompleteBetaApprox :: Double -> Double -> Double -> Double -> Double+incompleteBetaApprox beta p q x+  | ans > 0   = 1 - ans+  | otherwise = -ans+  where+    -- Constants+    p1    = p - 1+    q1    = q - 1+    mu    = p / (p + q)+    lnmu  = log     mu+    lnmuc = log1p (-mu)+    -- Upper limit for integration+    xu = max 0 $ min (mu - 10*t) (x - 5*t)+       where+         t = sqrt $ p*q / ( (p+q) * (p+q) * (p + q + 1) )+    -- Calculate incomplete beta by quadrature+    go y w = let t = x + (xu - x) * y+             in  w * exp( p1 * (log t - lnmu) + q1 * (log(1-t) - lnmuc) )+    s   = U.sum $ U.zipWith go coefY coefW+    ans = s * (xu - x) * exp( p1 * lnmu + q1 * lnmuc - beta )+++-- Worker for incomplete beta function. It is separate function to+-- avoid confusion with parameter during parameter swapping+incompleteBetaWorker :: Double -> Double -> Double -> Double -> Double+incompleteBetaWorker beta p q x+  -- For very large p and q this method becomes very slow so another+  -- method is used.+  | p > 3000 && q > 3000 = incompleteBetaApprox beta p q x+  | otherwise            = loop (p+q) (truncate $ q + cx * (p+q)) 1 1 1+  where+    -- Constants+    eps = 1e-15+    cx  = 1 - x+    -- Common multiplies for expansion. Accurate calculation is a bit+    -- tricky. Performing calculation in log-domain leads to slight+    -- loss of precision for small x, while using ** prone to+    -- underflows.+    --+    -- If either beta function of x**p·(1-x)**(q-1) underflows we+    -- switch to log domain. It could waste work but there's no easy+    -- switch criterion.+    factor+      | beta < m_min_log || prod < m_tiny = exp( p * log x + (q - 1) * log cx - beta)+      | otherwise                         = prod / exp beta+      where+        prod =  x**p * cx**(q - 1)+    -- Soper's expansion of incomplete beta function+    loop !psq (ns :: Int) ai term betain+      | done      = betain' * factor / p+      | otherwise = loop psq' (ns - 1) (ai + 1) term' betain'+      where+        -- New values+        term'   = term * fact / (p + ai)+        betain' = betain + term'+        fact | ns >  0   = (q - ai) * x/cx+             | ns == 0   = (q - ai) * x+             | otherwise = psq * x+        -- Iterations are complete+        done = db <= eps && db <= eps*betain' where db = abs term'+        psq' = if ns < 0 then psq + 1 else psq++++-- | Compute inverse of regularized incomplete beta function. Uses+-- initial approximation from AS109, AS64 and Halley method to solve+-- equation.+invIncompleteBeta :: Double     -- ^ /a/ > 0+                  -> Double     -- ^ /b/ > 0+                  -> Double     -- ^ /x/ ∈ [0,1]+                  -> Double+invIncompleteBeta p q a+  | p <= 0 || q <= 0 =+      modErr $ printf "invIncompleteBeta p <= 0 || q <= 0.  p=%g q=%g a=%g" p q a+  | a <  0 || a >  1 =+      modErr $ printf "invIncompleteBeta x must be in [0,1].  p=%g q=%g a=%g" p q a+  | a == 0 || a == 1 = a+  | otherwise        = invIncompleteBetaWorker (logBeta p q) p q  a+++invIncompleteBetaWorker :: Double -> Double -> Double -> Double -> Double+invIncompleteBetaWorker beta a b p = loop (0::Int) (invIncBetaGuess beta a b p)+  where+    a1 = a - 1+    b1 = b - 1+    -- Solve equation using Halley method+    loop !i !x+      -- We cannot continue at this point so we simply return `x'+      | x == 0 || x == 1             = x+      -- When derivative becomes infinite we cannot continue+      -- iterations. It can only happen in vicinity of 0 or 1. It's+      -- hardly possible to get good answer in such circumstances but+      -- `x' is already reasonable.+      | isInfinite f'                = x+      -- Iterations limit reached. Most of the time solution will+      -- converge to answer because of discreteness of Double. But+      -- solution have good precision already.+      | i >= 10                      = x+      -- Solution converges+      | abs dx <= 16 * m_epsilon * x = x'+      | otherwise                    = loop (i+1) x'+      where+        -- Calculate Halley step.+        f   = incompleteBeta_ beta a b x - p+        f'  = exp $ a1 * log x + b1 * log1p (-x) - beta+        u   = f / f'+        -- We bound Halley correction to Newton-Raphson to (-1,1) range+        corr | d > 1     = 1+             | d < -1    = -1+             | isNaN d   = 0+             | otherwise = d+          where+            d = u * (a1 / x - b1 / (1 - x))+        dx  = u / (1 - 0.5 * corr)+        -- Next approximation. If Halley step leads us out of [0,1]+        -- range we revert to bisection.+        x'  | z < 0     = x / 2+            | z > 1     = (x + 1) / 2+            | otherwise = z+            where z = x - dx+++-- Calculate initial guess for inverse incomplete beta function.+invIncBetaGuess :: Double -> Double -> Double -> Double -> Double+-- Calculate initial guess. for solving equation for inverse incomplete beta.+-- It's really hodgepodge of different approximations accumulated over years.+--+-- Equations are referred to by name of paper and number e.g. [AS64 2]+-- In AS64 papers equations are not numbered so they are referred to by+-- number of appearance starting from definition of incomplete beta.+invIncBetaGuess beta a b p+  -- If both a and b are less than 1 incomplete beta have inflection+  -- point.+  --+  -- > x = (1 - a) / (2 - a - b)+  --+  -- We approximate incomplete beta by neglecting one of factors under+  -- integral and then rescaling result of integration into [0,1]+  -- range.+  | a < 1 && b < 1 =+    let x_infl = (1 - a) / (2 - a - b)+        p_infl = incompleteBeta a b x_infl+        x | p < p_infl = let xg = (a * p     * exp beta) ** (1/a) in xg / (1+xg)+          | otherwise  = let xg = (b * (1-p) * exp beta) ** (1/b) in 1 - xg/(1+xg)+    in x+  -- If both a and b larger or equal that 1 but not too big we use+  -- same approximation as above but calculate it a bit differently+  | a+b <= 6 && a>1 && b>1 =+    let x_infl = (a - 1) / (a + b - 2)+        p_infl = incompleteBeta a b x_infl+        x | p < p_infl = exp ((log(p * a) + beta) / a)+          | otherwise  = 1 - exp((log((1-p) * b) + beta) / b)+    in x+  -- For small a and not too big b we use approximation from boost.+  | b < 5 && a <= 1 =+    let x | p**(1/a) < 0.5 = (p ** (a * exp beta)) ** (1/a)+          | otherwise      = 1 - (1 - p ** (b * exp beta))**(1/b)+    in x+  -- When a>>b and both are large approximation from [Temme1992],+  -- section 4 "the incomplete gamma function case" used. In this+  -- region it greatly improves over other approximation (AS109, AS64,+  -- "Numerical Recipes")+  --+  -- FIXME: It could be used when b>>a too but it require inverse of+  --        upper incomplete gamma to be precise enough. In current+  --        implementation it loses precision in horrible way (40+  --        order of magnitude off for sufficiently small p)+  | a+b > 5 &&  a/b > 4 =+    let -- Calculate initial approximation to eta using eq 4.1+        eta0 = invIncompleteGamma b (1-p) / a+        mu   = b / a            -- Eq. 4.3+        -- A lot of helpers for calculation of+        w    = sqrt(1 + mu)     -- Eq. 4.9+        w_2  = w * w+        w_3  = w_2 * w+        w_4  = w_2 * w_2+        w_5  = w_3 * w_2+        w_6  = w_3 * w_3+        w_7  = w_4 * w_3+        w_8  = w_4 * w_4+        w_9  = w_5 * w_4+        w_10 = w_5 * w_5+        d    = eta0 - mu+        d_2  = d * d+        d_3  = d_2 * d+        d_4  = d_2 * d_2+        w1   = w + 1+        w1_2 = w1 * w1+        w1_3 = w1 * w1_2+        w1_4 = w1_2 * w1_2+        -- Evaluation of eq 4.10+        e1 = (w + 2) * (w - 1) / (3 * w)+           + (w_3 + 9 * w_2 + 21 * w + 5) * d+             / (36 * w_2 * w1)+           - (w_4 - 13 * w_3 + 69 * w_2 + 167 * w + 46) * d_2+             / (1620 * w1_2 * w_3)+           - (7 * w_5 + 21 * w_4 + 70 * w_3 + 26 * w_2 - 93 * w - 31) * d_3+             / (6480 * w1_3 * w_4)+           - (75 * w_6 + 202 * w_5 + 188 * w_4 - 888 * w_3 - 1345 * w_2 + 118 * w + 138) * d_4+             / (272160 * w1_4 * w_5)+        e2 = (28 * w_4 + 131 * w_3 + 402 * w_2 + 581 * w + 208) * (w - 1)+             / (1620 * w1 * w_3)+           - (35 * w_6 - 154 * w_5 - 623 * w_4 - 1636 * w_3 - 3983 * w_2 - 3514 * w - 925) * d+             / (12960 * w1_2 * w_4)+           - ( 2132 * w_7 + 7915 * w_6 + 16821 * w_5 + 35066 * w_4 + 87490 * w_3+             + 141183 * w_2 + 95993 * w + 21640+             ) * d_2+             / (816480 * w_5 * w1_3)+           - ( 11053 * w_8 + 53308 * w_7 + 117010 * w_6 + 163924 * w_5 + 116188 * w_4+             - 258428 * w_3 - 677042 * w_2 - 481940 * w - 105497+             ) * d_3+             / (14696640 * w1_4 * w_6)+        e3 = -( (3592 * w_7 + 8375 * w_6 - 1323 * w_5 - 29198 * w_4 - 89578 * w_3+                - 154413 * w_2 - 116063 * w - 29632+                ) * (w - 1)+              )+              / (816480 * w_5 * w1_2)+           - ( 442043 * w_9 + 2054169 * w_8 + 3803094 * w_7 + 3470754 * w_6 + 2141568 * w_5+             - 2393568 * w_4 - 19904934 * w_3 - 34714674 * w_2 - 23128299 * w - 5253353+             ) * d+             / (146966400 * w_6 * w1_3)+           - ( 116932 * w_10 + 819281 * w_9 + 2378172 * w_8 + 4341330 * w_7 + 6806004 * w_6+             + 10622748 * w_5 + 18739500 * w_4 + 30651894 * w_3 + 30869976 * w_2+             + 15431867 * w + 2919016+             ) * d_2+             / (146966400 * w1_4 * w_7)+        eta = evaluatePolynomialL (1/a) [eta0, e1, e2, e3]+        -- Now we solve eq 4.2 to recover x using Newton iterations+        u       = eta - mu * log eta + (1 + mu) * log(1 + mu) - mu+        cross   = 1 / (1 + mu);+        lower   = if eta < mu then cross else 0+        upper   = if eta < mu then 1     else cross+        x_guess = (lower + upper) / 2+        func x  = ( u + log x + mu*log(1 - x)+                  , 1/x - mu/(1-x)+                  )+        Root x0 = newtonRaphson def{newtonTol=RelTol 1e-8} (lower, x_guess, upper) func+    in x0+  -- For large a and b approximation from AS109 (Carter+  -- approximation). It's reasonably good in this region+  | a > 1 && b > 1 =+      let r = (y*y - 3) / 6+          s = 1 / (2*a - 1)+          t = 1 / (2*b - 1)+          h = 2 / (s + t)+          w = y * sqrt(h + r) / h - (t - s) * (r + 5/6 - 2 / (3 * h))+      in a / (a + b * exp(2 * w))+  -- Otherwise we revert to approximation from AS64 derived from+  -- [AS64 2] when it's applicable.+  --+  -- It slightly reduces average number of iterations when `a' and+  -- `b' have different magnitudes.+  | chi2 > 0 && ratio > 1 = 1 - 2 / (ratio + 1)+  -- If all else fails we use approximation from "Numerical+  -- Recipes". It's very similar to approximations [AS64 4,5] but+  -- it never goes out of [0,1] interval.+  | otherwise = case () of+      _| p < t / w  -> (a * p * w) ** (1/a)+       | otherwise  -> 1 - (b * (1 - p) * w) ** (1/b)+       where+         lna = log $ a / (a+b)+         lnb = log $ b / (a+b)+         t   = exp( a * lna ) / a+         u   = exp( b * lnb ) / b+         w   = t + u+  where+    -- Formula [AS64 2]+    ratio = (4*a + 2*b - 2) / chi2+    -- Quantile of chi-squared distribution. Formula [AS64 3].+    chi2 = 2 * b * (1 - t + y * sqrt t) ** 3+      where+        t   = 1 / (9 * b)+    -- `y' is Hasting's approximation of p'th quantile of standard+    -- normal distribution.+    y   = r - ( 2.30753 + 0.27061 * r )+              / ( 1.0 + ( 0.99229 + 0.04481 * r ) * r )+      where+        r = sqrt $ - 2 * log p++++----------------------------------------------------------------+-- Sinc function+----------------------------------------------------------------++-- | Compute sinc function @sin(x)\/x@+sinc :: Double -> Double+sinc x+  | ax < eps_0 = 1+  | ax < eps_2 = 1 - x2/6+  | ax < eps_4 = 1 - x2/6 + x2*x2/120+  | otherwise  = sin x / x+  where+    ax    = abs x+    x2    = x*x+    -- For explanation of choice see `doc/sinc.hs'+    eps_0 = 1.8250120749944284e-8 -- sqrt (6ε/4)+    eps_2 = 1.4284346431400855e-4 --   (30ε)**(1/4) / 2+    eps_4 = 4.043633626430947e-3  -- (1206ε)**(1/6) / 2+++----------------------------------------------------------------+-- Logarithm+----------------------------------------------------------------++-- | Compute log(1+x)-x:+log1pmx :: Double -> Double+log1pmx x+  | x <  -1        = error "Domain error"+  | x == -1        = m_neg_inf+  | ax > 0.95      = log(1 + x) - x+  | ax < m_epsilon = -(x * x) /2+  | otherwise      = - x * x * sumPowerSeries (-x) (recip <$> enumSequenceFrom 2)+  where+   ax = abs x++-- | /O(log n)/ Compute the logarithm in base 2 of the given value.+log2 :: Int -> Int+log2 v0+    | v0 <= 0   = modErr $ "log2: nonpositive input, got " ++ show v0+    | otherwise = go 5 0 v0+  where+    go !i !r !v | i == -1        = r+                | v .&. b i /= 0 = let si = U.unsafeIndex sv i+                                   in go (i-1) (r .|. si) (v `shiftR` si)+                | otherwise      = go (i-1) r v+    b = U.unsafeIndex bv+    !bv = U.fromList [ 0x02, 0x0c, 0xf0, 0xff00+                     , fromIntegral (0xffff0000 :: Word)+                     , fromIntegral (0xffffffff00000000 :: Word)]+    !sv = U.fromList [1,2,4,8,16,32]+++----------------------------------------------------------------+-- Factorial+----------------------------------------------------------------++-- | Compute the factorial function /n/!.  Returns +∞ if the input is+--   above 170 (above which the result cannot be represented by a+--   64-bit 'Double').+factorial :: Int -> Double+factorial n+  | n < 0     = error "Numeric.SpecFunctions.factorial: negative input"+  | n > 170   = m_pos_inf+  | otherwise = U.unsafeIndex factorialTable n++-- | Compute the natural logarithm of the factorial function.  Gives+--   16 decimal digits of precision.+logFactorial :: Integral a => a -> Double+logFactorial n+  | n <  0    = error "Numeric.SpecFunctions.logFactorial: negative input"+  -- For smaller inputs we just look up table+  | n <= 170  = log $ U.unsafeIndex factorialTable (fromIntegral n)+  -- Otherwise we use asymptotic Stirling's series. Number of terms+  -- necessary depends on the argument.+  | n < 1500  = stirling + rx * ((1/12) - (1/360)*rx*rx)+  | otherwise = stirling + (1/12)*rx+  where+    stirling = (x - 0.5) * log x - x + m_ln_sqrt_2_pi+    x        = fromIntegral n + 1+    rx       = 1 / x+{-# SPECIALIZE logFactorial :: Int -> Double #-}+++-- | Calculate the error term of the Stirling approximation.  This is+-- only defined for non-negative values.+--+-- \[+-- \operatorname{stirlingError}(n) = \log(n!) - \log(\sqrt{2\pi n}\frac{n}{e}^n)+-- \]+stirlingError :: Double -> Double+stirlingError n+  | n <= 15.0   = case properFraction (n+n) of+                    (i,0) -> sfe `U.unsafeIndex` i+                    _     -> logGamma (n+1.0) - (n+0.5) * log n + n -+                             m_ln_sqrt_2_pi+  | n > 500     = evaluateOddPolynomialL (1/n) [s0,-s1]+  | n > 80      = evaluateOddPolynomialL (1/n) [s0,-s1,s2]+  | n > 35      = evaluateOddPolynomialL (1/n) [s0,-s1,s2,-s3]+  | otherwise   = evaluateOddPolynomialL (1/n) [s0,-s1,s2,-s3,s4]+  where+    s0 = 0.083333333333333333333        -- 1/12+    s1 = 0.00277777777777777777778      -- 1/360+    s2 = 0.00079365079365079365079365   -- 1/1260+    s3 = 0.000595238095238095238095238  -- 1/1680+    s4 = 0.0008417508417508417508417508 -- 1/1188+    sfe = U.fromList [ 0.0,+                0.1534264097200273452913848,   0.0810614667953272582196702,+                0.0548141210519176538961390,   0.0413406959554092940938221,+                0.03316287351993628748511048,  0.02767792568499833914878929,+                0.02374616365629749597132920,  0.02079067210376509311152277,+                0.01848845053267318523077934,  0.01664469118982119216319487,+                0.01513497322191737887351255,  0.01387612882307074799874573,+                0.01281046524292022692424986,  0.01189670994589177009505572,+                0.01110455975820691732662991,  0.010411265261972096497478567,+                0.009799416126158803298389475, 0.009255462182712732917728637,+                0.008768700134139385462952823, 0.008330563433362871256469318,+                0.007934114564314020547248100, 0.007573675487951840794972024,+                0.007244554301320383179543912, 0.006942840107209529865664152,+                0.006665247032707682442354394, 0.006408994188004207068439631,+                0.006171712263039457647532867, 0.005951370112758847735624416,+                0.005746216513010115682023589, 0.005554733551962801371038690 ]+++----------------------------------------------------------------+-- Combinatorics+----------------------------------------------------------------++-- |+-- Quickly compute the natural logarithm of /n/ @`choose`@ /k/, with+-- no checking.+--+-- Less numerically stable:+--+-- > exp $ lg (n+1) - lg (k+1) - lg (n-k+1)+-- >   where lg = logGamma . fromIntegral+logChooseFast :: Double -> Double -> Double+logChooseFast n k = -log (n + 1) - logBeta (n - k + 1) (k + 1)++-- | Calculate binomial coefficient using exact formula+chooseExact :: Int -> Int -> Double+n `chooseExact` k+  = U.foldl' go 1 $ U.enumFromTo 1 k+  where+    go a i      = a * (nk + j) / j+        where j = fromIntegral i :: Double+    nk = fromIntegral (n - k)++-- | Compute logarithm of the binomial coefficient.+logChoose :: Int -> Int -> Double+n `logChoose` k+    | k  > n    = (-1) / 0+      -- For very large N exact algorithm overflows double so we+      -- switch to beta-function based one+    | k' < 50 && (n < 20000000) = log $ chooseExact n k'+    | otherwise                 = logChooseFast (fromIntegral n) (fromIntegral k)+  where+    k' = min k (n-k)++-- | Compute the binomial coefficient /n/ @\``choose`\`@ /k/. For+-- values of /k/ > 50, this uses an approximation for performance+-- reasons.  The approximation is accurate to 12 decimal places in the+-- worst case+--+-- Example:+--+-- > 7 `choose` 3 == 35+choose :: Int -> Int -> Double+n `choose` k+    | k  > n         = 0+    | k' < 50        = chooseExact n k'+    | approx < max64 = fromIntegral . round64 $ approx+    | otherwise      = approx+  where+    k'             = min k (n-k)+    approx         = exp $ logChooseFast (fromIntegral n) (fromIntegral k')+    max64          = fromIntegral (maxBound :: Int64)+    round64 x      = round x :: Int64++-- | Compute ψ(/x/), the first logarithmic derivative of the gamma+--   function.+--+-- \[+-- \psi(x) = \frac{d}{dx} \ln \left(\Gamma(x)\right) = \frac{\Gamma'(x)}{\Gamma(x)}+-- \]+--+-- Uses Algorithm AS 103 by Bernardo, based on Minka's C implementation.+digamma :: Double -> Double+digamma x+    | isNaN x || isInfinite x                  = m_NaN+    -- FIXME:+    --   This is ugly. We are testing here that number is in fact+    --   integer. It's somewhat tricky question to answer. When ε for+    --   given number becomes 1 or greater every number is represents+    --   an integer. We also must make sure that excess precision+    --   won't bite us.+    | x <= 0 && fromIntegral (truncate x :: Int64) == x = m_neg_inf+    -- Jeffery's reflection formula+    | x < 0     = digamma (1 - x) + pi / tan (negate pi * x)+    | x <= 1e-6 = - γ - 1/x + trigamma1 * x+    | x' < c    = r+    -- De Moivre's expansion+    | otherwise = let s = 1/x'+                  in  evaluateEvenPolynomialL s+                        [   r + log x' - 0.5 * s+                        , - 1/12+                        ,   1/120+                        , - 1/252+                        ,   1/240+                        , - 1/132+                        ,  391/32760+                        ]+  where+    γ  = m_eulerMascheroni+    c  = 12+    -- Reduce to digamma (x + n) where (x + n) >= c+    (r, x') = reduce 0 x+      where+        reduce !s y+          | y < c     = reduce (s - 1 / y) (y + 1)+          | otherwise = (s, y)++++----------------------------------------------------------------+-- Constants+----------------------------------------------------------------++-- Coefficients for 18-point Gauss-Legendre integration. They are+-- used in implementation of incomplete gamma and beta functions.+coefW,coefY :: U.Vector Double+coefW = U.fromList [ 0.0055657196642445571, 0.012915947284065419, 0.020181515297735382+                   , 0.027298621498568734,  0.034213810770299537, 0.040875750923643261+                   , 0.047235083490265582,  0.053244713977759692, 0.058860144245324798+                   , 0.064039797355015485,  0.068745323835736408, 0.072941885005653087+                   , 0.076598410645870640,  0.079687828912071670, 0.082187266704339706+                   , 0.084078218979661945,  0.085346685739338721, 0.085983275670394821+                   ]+coefY = U.fromList [ 0.0021695375159141994, 0.011413521097787704, 0.027972308950302116+                   , 0.051727015600492421,  0.082502225484340941, 0.12007019910960293+                   , 0.16415283300752470,   0.21442376986779355,  0.27051082840644336+                   , 0.33199876341447887,   0.39843234186401943,  0.46931971407375483+                   , 0.54413605556657973,   0.62232745288031077,  0.70331500465597174+                   , 0.78649910768313447,   0.87126389619061517,  0.95698180152629142+                   ]+{-# NOINLINE coefW #-}+{-# NOINLINE coefY #-}++trigamma1 :: Double+trigamma1 = 1.6449340668482264365 -- pi**2 / 6++modErr :: String -> a+modErr msg = error $ "Numeric.SpecFunctions." ++ msg++factorialTable :: U.Vector Double+{-# NOINLINE factorialTable #-}+factorialTable = U.fromListN 171+  [ 1.0+  , 1.0+  , 2.0+  , 6.0+  , 24.0+  , 120.0+  , 720.0+  , 5040.0+  , 40320.0+  , 362880.0+  , 3628800.0+  , 3.99168e7+  , 4.790016e8+  , 6.2270208e9+  , 8.71782912e10+  , 1.307674368e12+  , 2.0922789888e13+  , 3.55687428096e14+  , 6.402373705728e15+  , 1.21645100408832e17+  , 2.43290200817664e18+  , 5.109094217170944e19+  , 1.1240007277776077e21+  , 2.5852016738884974e22+  , 6.204484017332394e23+  , 1.5511210043330984e25+  , 4.032914611266056e26+  , 1.0888869450418352e28+  , 3.0488834461171384e29+  , 8.841761993739702e30+  , 2.6525285981219103e32+  , 8.222838654177922e33+  , 2.631308369336935e35+  , 8.683317618811886e36+  , 2.9523279903960412e38+  , 1.0333147966386144e40+  , 3.719933267899012e41+  , 1.3763753091226343e43+  , 5.23022617466601e44+  , 2.0397882081197442e46+  , 8.159152832478977e47+  , 3.3452526613163803e49+  , 1.4050061177528798e51+  , 6.041526306337383e52+  , 2.6582715747884485e54+  , 1.1962222086548019e56+  , 5.5026221598120885e57+  , 2.5862324151116818e59+  , 1.2413915592536073e61+  , 6.082818640342675e62+  , 3.0414093201713376e64+  , 1.5511187532873822e66+  , 8.065817517094388e67+  , 4.2748832840600255e69+  , 2.308436973392414e71+  , 1.2696403353658275e73+  , 7.109985878048634e74+  , 4.0526919504877214e76+  , 2.3505613312828785e78+  , 1.386831185456898e80+  , 8.32098711274139e81+  , 5.075802138772247e83+  , 3.146997326038793e85+  , 1.9826083154044399e87+  , 1.2688693218588415e89+  , 8.24765059208247e90+  , 5.44344939077443e92+  , 3.647111091818868e94+  , 2.4800355424368305e96+  , 1.711224524281413e98+  , 1.197857166996989e100+  , 8.504785885678623e101+  , 6.1234458376886085e103+  , 4.470115461512684e105+  , 3.307885441519386e107+  , 2.4809140811395396e109+  , 1.88549470166605e111+  , 1.4518309202828586e113+  , 1.1324281178206297e115+  , 8.946182130782974e116+  , 7.15694570462638e118+  , 5.797126020747368e120+  , 4.753643337012841e122+  , 3.9455239697206583e124+  , 3.314240134565353e126+  , 2.81710411438055e128+  , 2.422709538367273e130+  , 2.1077572983795275e132+  , 1.8548264225739844e134+  , 1.650795516090846e136+  , 1.4857159644817613e138+  , 1.352001527678403e140+  , 1.2438414054641305e142+  , 1.1567725070816416e144+  , 1.087366156656743e146+  , 1.0329978488239058e148+  , 9.916779348709496e149+  , 9.619275968248211e151+  , 9.426890448883246e153+  , 9.332621544394413e155+  , 9.332621544394415e157+  , 9.425947759838358e159+  , 9.614466715035125e161+  , 9.902900716486179e163+  , 1.0299016745145626e166+  , 1.0813967582402908e168+  , 1.1462805637347082e170+  , 1.2265202031961378e172+  , 1.3246418194518288e174+  , 1.4438595832024934e176+  , 1.5882455415227428e178+  , 1.7629525510902446e180+  , 1.974506857221074e182+  , 2.2311927486598134e184+  , 2.543559733472187e186+  , 2.9250936934930154e188+  , 3.393108684451898e190+  , 3.9699371608087206e192+  , 4.68452584975429e194+  , 5.574585761207606e196+  , 6.689502913449126e198+  , 8.094298525273443e200+  , 9.875044200833601e202+  , 1.214630436702533e205+  , 1.5061417415111406e207+  , 1.8826771768889257e209+  , 2.372173242880047e211+  , 3.0126600184576594e213+  , 3.856204823625804e215+  , 4.974504222477286e217+  , 6.466855489220473e219+  , 8.471580690878819e221+  , 1.1182486511960041e224+  , 1.4872707060906857e226+  , 1.9929427461615188e228+  , 2.6904727073180504e230+  , 3.6590428819525483e232+  , 5.012888748274991e234+  , 6.917786472619488e236+  , 9.615723196941088e238+  , 1.3462012475717523e241+  , 1.898143759076171e243+  , 2.6953641378881624e245+  , 3.8543707171800725e247+  , 5.5502938327393044e249+  , 8.047926057471992e251+  , 1.1749972043909107e254+  , 1.7272458904546386e256+  , 2.5563239178728654e258+  , 3.808922637630569e260+  , 5.713383956445854e262+  , 8.62720977423324e264+  , 1.3113358856834524e267+  , 2.0063439050956823e269+  , 3.0897696138473508e271+  , 4.789142901463393e273+  , 7.471062926282894e275+  , 1.1729568794264143e278+  , 1.8532718694937346e280+  , 2.946702272495038e282+  , 4.714723635992061e284+  , 7.590705053947218e286+  , 1.2296942187394494e289+  , 2.0044015765453023e291+  , 3.287218585534296e293+  , 5.423910666131589e295+  , 9.003691705778436e297+  , 1.5036165148649988e300+  , 2.526075744973198e302+  , 4.269068009004705e304+  , 7.257415615307998e306+  ]+++-- [NOTE: incompleteGamma.taylorP]+--+-- Incompltete gamma uses several algorithms for different parts of+-- parameter space. Most troublesome is P(a,x) Taylor series+-- [Temme1994,Eq.5.5] which requires to evaluate rather nasty+-- expression:+--+--       x^a             x^a+--  ------------- = -------------+--  exp(x)·Γ(a+1)   exp(x)·a·Γ(a)+--+--  Conditions:+--    | 0.5<x<1.1  = x < 4/3*a+--    | otherwise  = x < a+--+-- For small `a` computation could be performed directly. However for+-- largish values of `a` it's possible some of factor in the+-- expression overflow. Values below take into account ranges for+-- Taylor P approximation:+--+--  · a > 155    - x^a could overflow+--  · a > 1182.5 - exp(x) could overflow+--+-- Usual way to avoid overflow problem is to perform calculations in+-- the log domain. It however doesn't work very well in this case+-- since we encounter catastrophic cancellations and could easily lose+-- up to 6(!) digits for large `a`.+--+-- So we take another approach and use Stirling approximation with+-- correction (logGammaCorrection).+--+--              x^a               / x·e \^a         1+--  ≈ ------------------------- = | --- | · ----------------+--    exp(x)·sqrt(2πa)·(a/e)^a)   \  a  /   exp(x)·sqrt(2πa)+--+-- We're using this approach as soon as logGammaCorrection starts+-- working (a>10) because we don't have implementation for gamma+-- function and exp(logGamma z) results in errors for large a.+--+-- Once we get into region when exp(x) could overflow we rewrite+-- expression above once more:+--+--  / x·e            \^a     1+--  | --- · e^(-x/a) | · ---------+--  \  a             /   sqrt(2πa)+--+-- This approach doesn't work very well but it's still big improvement+-- over calculations in the log domain.
Numeric/Sum.hs view
@@ -1,5 +1,5 @@-{-# LANGUAGE BangPatterns, CPP, DeriveDataTypeable, FlexibleContexts,-    MultiParamTypeClasses, TemplateHaskell, TypeFamilies #-}+{-# LANGUAGE BangPatterns, DeriveDataTypeable, FlexibleContexts,+    MultiParamTypeClasses, TypeFamilies #-} {-# OPTIONS_GHC -fno-warn-name-shadowing #-} -- | -- Module    : Numeric.Sum@@ -53,14 +53,14 @@ import Control.DeepSeq (NFData(..)) import Data.Bits (shiftR) import Data.Data (Typeable, Data)-import Data.Vector.Generic (Vector(..), foldl')-import Data.Vector.Unboxed.Deriving (derivingUnbox)--- Needed for GHC 7.2 & 7.4 to derive Unbox instances+import Data.Semigroup               (Semigroup(..)) -- Needed for GHC <8.4+import Data.Vector.Generic          (Vector(..)) import Data.Vector.Generic.Mutable (MVector(..))  import qualified Data.Foldable as F import qualified Data.Vector as V import qualified Data.Vector.Generic as G+import qualified Data.Vector.Generic.Mutable as GM import qualified Data.Vector.Unboxed as U  -- | A class for summation of floating point numbers.@@ -74,7 +74,7 @@     -- | Sum a collection of values.     --     -- Example:-    -- @foo = 'sum' 'kbn' [1,2,3]@+    -- @foo = 'Numeric.Sum.sum' 'kbn' [1,2,3]@     sum  :: (F.Foldable f) => (s -> Double) -> f Double -> Double     sum  f = f . F.foldl' add zero     {-# INLINE sum #-}@@ -94,11 +94,54 @@ data KahanSum = KahanSum {-# UNPACK #-} !Double {-# UNPACK #-} !Double               deriving (Eq, Show, Typeable, Data) -derivingUnbox "KahanSum"-    [t| KahanSum -> (Double, Double) |]-    [| \ (KahanSum a b) -> (a, b) |]-    [| \ (a, b) -> KahanSum a b |]+instance U.Unbox KahanSum+newtype instance U.MVector s KahanSum = MV_KahanSum (U.MVector s (Double, Double))+instance MVector U.MVector KahanSum where+  {-# INLINE GM.basicLength #-}+  {-# INLINE GM.basicUnsafeSlice #-}+  {-# INLINE basicOverlaps #-}+  {-# INLINE basicUnsafeNew #-}+  {-# INLINE basicInitialize #-}+  {-# INLINE basicUnsafeReplicate #-}+  {-# INLINE basicUnsafeRead #-}+  {-# INLINE basicUnsafeWrite #-}+  {-# INLINE basicClear #-}+  {-# INLINE basicSet #-}+  {-# INLINE GM.basicUnsafeCopy #-}+  {-# INLINE basicUnsafeMove #-}+  {-# INLINE basicUnsafeGrow #-}+  basicLength (MV_KahanSum mvec) = GM.basicLength mvec+  basicUnsafeSlice idx len (MV_KahanSum mvec) = MV_KahanSum (GM.basicUnsafeSlice idx len mvec)+  basicOverlaps (MV_KahanSum mvec) (MV_KahanSum mvec') = basicOverlaps mvec mvec'+  basicUnsafeNew len = MV_KahanSum <$> basicUnsafeNew len+  basicInitialize (MV_KahanSum mvec) = basicInitialize mvec+  basicUnsafeReplicate len val = MV_KahanSum <$> basicUnsafeReplicate len ((\ (KahanSum a b) -> (a, b)) val)+  basicUnsafeRead (MV_KahanSum mvec) idx = (\ (a, b) -> KahanSum a b) <$> basicUnsafeRead mvec idx+  basicUnsafeWrite (MV_KahanSum mvec) idx val = basicUnsafeWrite mvec idx ((\ (KahanSum a b) -> (a, b)) val)+  basicClear (MV_KahanSum mvec) = basicClear mvec+  basicSet (MV_KahanSum mvec) val = basicSet mvec ((\ (KahanSum a b) -> (a, b)) val)+  basicUnsafeCopy (MV_KahanSum mvec) (MV_KahanSum mvec') = GM.basicUnsafeCopy mvec mvec'+  basicUnsafeMove (MV_KahanSum mvec) (MV_KahanSum mvec') = basicUnsafeMove mvec mvec'+  basicUnsafeGrow (MV_KahanSum mvec) len = MV_KahanSum <$> basicUnsafeGrow mvec len +newtype instance U.Vector KahanSum = V_KahanSum (U.Vector (Double, Double))+instance Vector U.Vector KahanSum where+  {-# INLINE basicUnsafeFreeze #-}+  {-# INLINE basicUnsafeThaw #-}+  {-# INLINE G.basicLength #-}+  {-# INLINE G.basicUnsafeSlice #-}+  {-# INLINE basicUnsafeIndexM #-}+  {-# INLINE G.basicUnsafeCopy #-}+  {-# INLINE elemseq #-}+  basicUnsafeFreeze (MV_KahanSum mvec) = V_KahanSum <$> basicUnsafeFreeze mvec+  basicUnsafeThaw (V_KahanSum vec) = MV_KahanSum <$> basicUnsafeThaw vec+  basicLength (V_KahanSum vec) = G.basicLength vec+  basicUnsafeSlice idx len (V_KahanSum vec) = V_KahanSum (G.basicUnsafeSlice idx len vec)+  basicUnsafeIndexM (V_KahanSum vec) idx = (\ (a, b) -> KahanSum a b) <$> basicUnsafeIndexM vec idx+  basicUnsafeCopy (MV_KahanSum mvec) (V_KahanSum vec) = G.basicUnsafeCopy mvec vec+  elemseq (V_KahanSum vec) val = elemseq vec ((\ (KahanSum a b) -> (a, b)) val)++ instance Summation KahanSum where     zero = KahanSum 0 0     add  = kahanAdd@@ -106,6 +149,15 @@ instance NFData KahanSum where     rnf !_ = () +-- | @since 0.3.0.0+instance Monoid KahanSum where+  mempty = zero+  mappend = (<>)++-- | @since 0.3.0.0+instance Semigroup KahanSum where+  s <> KahanSum s' _ = add s s'+ kahanAdd :: KahanSum -> Double -> KahanSum kahanAdd (KahanSum sum c) x = KahanSum sum' c'   where sum' = sum + y@@ -122,11 +174,54 @@ data KBNSum = KBNSum {-# UNPACK #-} !Double {-# UNPACK #-} !Double             deriving (Eq, Show, Typeable, Data) -derivingUnbox "KBNSum"-    [t| KBNSum -> (Double, Double) |]-    [| \ (KBNSum a b) -> (a, b) |]-    [| \ (a, b) -> KBNSum a b |]+instance U.Unbox KBNSum+newtype instance U.MVector s KBNSum = MV_KBNSum (U.MVector s (Double, Double))+instance MVector U.MVector KBNSum where+  {-# INLINE GM.basicLength #-}+  {-# INLINE GM.basicUnsafeSlice #-}+  {-# INLINE basicOverlaps #-}+  {-# INLINE basicUnsafeNew #-}+  {-# INLINE basicInitialize #-}+  {-# INLINE basicUnsafeReplicate #-}+  {-# INLINE basicUnsafeRead #-}+  {-# INLINE basicUnsafeWrite #-}+  {-# INLINE basicClear #-}+  {-# INLINE basicSet #-}+  {-# INLINE GM.basicUnsafeCopy #-}+  {-# INLINE basicUnsafeMove #-}+  {-# INLINE basicUnsafeGrow #-}+  basicLength (MV_KBNSum mvec) = GM.basicLength mvec+  basicUnsafeSlice idx len (MV_KBNSum mvec) = MV_KBNSum (GM.basicUnsafeSlice idx len mvec)+  basicOverlaps (MV_KBNSum mvec) (MV_KBNSum mvec') = basicOverlaps mvec mvec'+  basicUnsafeNew len = MV_KBNSum <$> basicUnsafeNew len+  basicInitialize (MV_KBNSum mvec) = basicInitialize mvec+  basicUnsafeReplicate len val = MV_KBNSum <$> basicUnsafeReplicate len ((\ (KBNSum a b) -> (a, b)) val)+  basicUnsafeRead (MV_KBNSum mvec) idx = (\ (a, b) -> KBNSum a b) <$> basicUnsafeRead mvec idx+  basicUnsafeWrite (MV_KBNSum mvec) idx val = basicUnsafeWrite mvec idx ((\ (KBNSum a b) -> (a, b)) val)+  basicClear (MV_KBNSum mvec) = basicClear mvec+  basicSet (MV_KBNSum mvec) val = basicSet mvec ((\ (KBNSum a b) -> (a, b)) val)+  basicUnsafeCopy (MV_KBNSum mvec) (MV_KBNSum mvec') = GM.basicUnsafeCopy mvec mvec'+  basicUnsafeMove (MV_KBNSum mvec) (MV_KBNSum mvec') = basicUnsafeMove mvec mvec'+  basicUnsafeGrow (MV_KBNSum mvec) len = MV_KBNSum <$> basicUnsafeGrow mvec len +newtype instance U.Vector KBNSum = V_KBNSum (U.Vector (Double, Double))+instance Vector U.Vector KBNSum where+  {-# INLINE basicUnsafeFreeze #-}+  {-# INLINE basicUnsafeThaw #-}+  {-# INLINE G.basicLength #-}+  {-# INLINE G.basicUnsafeSlice #-}+  {-# INLINE basicUnsafeIndexM #-}+  {-# INLINE G.basicUnsafeCopy #-}+  {-# INLINE elemseq #-}+  basicUnsafeFreeze (MV_KBNSum mvec) = V_KBNSum <$> basicUnsafeFreeze mvec+  basicUnsafeThaw (V_KBNSum vec) = MV_KBNSum <$> basicUnsafeThaw vec+  basicLength (V_KBNSum vec) = G.basicLength vec+  basicUnsafeSlice idx len (V_KBNSum vec) = V_KBNSum (G.basicUnsafeSlice idx len vec)+  basicUnsafeIndexM (V_KBNSum vec) idx = (\ (a, b) -> KBNSum a b) <$> basicUnsafeIndexM vec idx+  basicUnsafeCopy (MV_KBNSum mvec) (V_KBNSum vec) = G.basicUnsafeCopy mvec vec+  elemseq (V_KBNSum vec) val = elemseq vec ((\ (KBNSum a b) -> (a, b)) val)++ instance Summation KBNSum where     zero = KBNSum 0 0     add  = kbnAdd@@ -134,6 +229,16 @@ instance NFData KBNSum where     rnf !_ = () +-- | @since 0.3.0.0+instance Monoid KBNSum where+  mempty = zero+  mappend = (<>)++-- | @since 0.3.0.0+instance Semigroup KBNSum where+  s <> KBNSum s' c' = add (add s s') c'+  + kbnAdd :: KBNSum -> Double -> KBNSum kbnAdd (KBNSum sum c) x = KBNSum sum' c'   where c' | abs sum >= abs x = c + ((sum - sum') + x)@@ -157,11 +262,53 @@                      {-# UNPACK #-} !Double             deriving (Eq, Show, Typeable, Data) -derivingUnbox "KB2Sum"-    [t| KB2Sum -> (Double, Double, Double) |]-    [| \ (KB2Sum a b c) -> (a, b, c) |]-    [| \ (a, b, c) -> KB2Sum a b c |]+instance U.Unbox KB2Sum+newtype instance U.MVector s KB2Sum = MV_KB2Sum (U.MVector s (Double, Double, Double))+instance MVector U.MVector KB2Sum where+  {-# INLINE GM.basicLength #-}+  {-# INLINE GM.basicUnsafeSlice #-}+  {-# INLINE basicOverlaps #-}+  {-# INLINE basicUnsafeNew #-}+  {-# INLINE basicInitialize #-}+  {-# INLINE basicUnsafeReplicate #-}+  {-# INLINE basicUnsafeRead #-}+  {-# INLINE basicUnsafeWrite #-}+  {-# INLINE basicClear #-}+  {-# INLINE basicSet #-}+  {-# INLINE GM.basicUnsafeCopy #-}+  {-# INLINE basicUnsafeMove #-}+  {-# INLINE basicUnsafeGrow #-}+  basicLength (MV_KB2Sum mvec) = GM.basicLength mvec+  basicUnsafeSlice idx len (MV_KB2Sum mvec) = MV_KB2Sum (GM.basicUnsafeSlice idx len mvec)+  basicOverlaps (MV_KB2Sum mvec) (MV_KB2Sum mvec') = basicOverlaps mvec mvec'+  basicUnsafeNew len = MV_KB2Sum <$> basicUnsafeNew len+  basicInitialize (MV_KB2Sum mvec) = basicInitialize mvec+  basicUnsafeReplicate len val = MV_KB2Sum <$> basicUnsafeReplicate len ((\ (KB2Sum a b c) -> (a, b, c)) val)+  basicUnsafeRead (MV_KB2Sum mvec) idx = (\ (a, b, c) -> KB2Sum a b c) <$> basicUnsafeRead mvec idx+  basicUnsafeWrite (MV_KB2Sum mvec) idx val = basicUnsafeWrite mvec idx ((\ (KB2Sum a b c) -> (a, b, c)) val)+  basicClear (MV_KB2Sum mvec) = basicClear mvec+  basicSet (MV_KB2Sum mvec) val = basicSet mvec ((\ (KB2Sum a b c) -> (a, b, c)) val)+  basicUnsafeCopy (MV_KB2Sum mvec) (MV_KB2Sum mvec') = GM.basicUnsafeCopy mvec mvec'+  basicUnsafeMove (MV_KB2Sum mvec) (MV_KB2Sum mvec') = basicUnsafeMove mvec mvec'+  basicUnsafeGrow (MV_KB2Sum mvec) len = MV_KB2Sum <$> basicUnsafeGrow mvec len +newtype instance U.Vector KB2Sum = V_KB2Sum (U.Vector (Double, Double, Double))+instance Vector U.Vector KB2Sum where+  {-# INLINE basicUnsafeFreeze #-}+  {-# INLINE basicUnsafeThaw #-}+  {-# INLINE G.basicLength #-}+  {-# INLINE G.basicUnsafeSlice #-}+  {-# INLINE basicUnsafeIndexM #-}+  {-# INLINE G.basicUnsafeCopy #-}+  {-# INLINE elemseq #-}+  basicUnsafeFreeze (MV_KB2Sum mvec) = V_KB2Sum <$> basicUnsafeFreeze mvec+  basicUnsafeThaw (V_KB2Sum vec) = MV_KB2Sum <$> basicUnsafeThaw vec+  basicLength (V_KB2Sum vec) = G.basicLength vec+  basicUnsafeSlice idx len (V_KB2Sum vec) = V_KB2Sum (G.basicUnsafeSlice idx len vec)+  basicUnsafeIndexM (V_KB2Sum vec) idx = (\ (a, b, c) -> KB2Sum a b c) <$> basicUnsafeIndexM vec idx+  basicUnsafeCopy (MV_KB2Sum mvec) (V_KB2Sum vec) = G.basicUnsafeCopy mvec vec+  elemseq (V_KB2Sum vec) val = elemseq vec ((\ (KB2Sum a b c) -> (a, b, c)) val)+ instance Summation KB2Sum where     zero = KB2Sum 0 0 0     add  = kb2Add@@ -169,6 +316,16 @@ instance NFData KB2Sum where     rnf !_ = () +-- | @since 0.3.0.0+instance Monoid KB2Sum where+  mempty = zero+  mappend = (<>)++-- | @since 0.3.0.0+instance Semigroup KB2Sum where+  s <> KB2Sum s' c' cc' = add (add (add s s') c') cc'++ kb2Add :: KB2Sum -> Double -> KB2Sum kb2Add (KB2Sum sum c cc) x = KB2Sum sum' c' cc'   where sum'                 = sum + x@@ -185,7 +342,7 @@ -- | /O(n)/ Sum a vector of values. sumVector :: (Vector v Double, Summation s) =>              (s -> Double) -> v Double -> Double-sumVector f = f . foldl' add zero+sumVector f = f . G.foldl' add zero {-# INLINE sumVector #-}  -- | /O(n)/ Sum a vector of values using pairwise summation.@@ -223,7 +380,7 @@ --           where s'    = 'add' s x -- @ ----- In most instances, you can simply use the much more general 'sum'+-- In most instances, you can simply use the much more general 'Numeric.Sum.sum' -- function instead of writing a summation function by hand. -- -- @@@ -231,7 +388,7 @@ -- import Prelude hiding (sum) -- -- -- betterSumList :: [Double] -> Double--- betterSumList xs = 'sum' 'kbn' xs+-- betterSumList xs = 'Numeric.Sum.sum' 'kbn' xs -- @  -- Note well the use of 'seq' in the example above to force the
README.markdown view
@@ -1,7 +1,21 @@-# math-functions: efficient, special purpose mathematical functions+# math-functions: collection of tools for numeric computations -This package provides a number of special-purpose mathematical-functions used in statistical and numerical computing.+[![Build Status](https://travis-ci.org/Shimuuar/math-functions.png?branch=master)](https://travis-ci.org/Shimuuar/math-functions)+[![Build status](https://ci.appveyor.com/api/projects/status/6xexxj9g6rnbg2q4/branch/master?svg=true)](https://ci.appveyor.com/project/Shimuuar/math-functions/branch/master)++This package provides collection of various tools for numeric+computations. Namely:++ - Number pure haskell implementations of special function which are used in+   statistical and numerical computing.++ - Compensated summation (Kahan summation) which allows to++ - Root finding for functions of single real variable++ - Series summation++ - Functions for comparing IEEE754 numbers  Where possible, we give citations and computational complexity estimates for the algorithms used.
+ bench/bench.hs view
@@ -0,0 +1,126 @@+{-# LANGUAGE NumDecimals #-}+import Data.Default.Class+import qualified Data.Vector.Unboxed as U+import Text.Printf+import Test.Tasty.Bench+import System.Random (randomIO)++import qualified Numeric.Sum as Sum+import Numeric.SpecFunctions+import Numeric.Polynomial+import Numeric.RootFinding++++-- Uniformly sample logGamma performance between 10^-6 to 10^6+benchmarkLogGamma :: (Double -> Double) -> [Benchmark]+benchmarkLogGamma logG =+  [ bench (printf "%.3g" x) $ nf logG x+  | x <- [ m * 10**n | n <- [ -8 .. 8 ]+                     , m <- [ 10**(i / tics) | i <- [0 .. tics-1] ]+         ]+  ]+  where tics = 3+{-# INLINE benchmarkLogGamma #-}+++-- Power of polynomial to be evaluated (In other words length of coefficients vector)+coef_size :: [Int]+coef_size = [ 1,2,3,4,5,6,7,8,9+            , 10,    30+            , 100,   300+            , 1000,  3000+            , 10000, 30000+            ]+{-# INLINE coef_size #-}++-- Precalculated coefficients+coef_list :: [U.Vector Double]+coef_list = [ U.replicate n 1.2 | n <- coef_size]+{-# NOINLINE coef_list #-}++++main :: IO ()+main = do+  v <- U.replicateM 1e6 randomIO :: IO (U.Vector Double)+  defaultMain+    [ bgroup "logGamma" $+      benchmarkLogGamma logGamma+    , bgroup "incompleteGamma" $+        [ bench (show p) $ nf (incompleteGamma p) p+        | p <- [ 0.1+               , 1,   3+               , 10,  30+               , 100, 300+               , 999, 1000+               ]+        ]+    , bgroup "factorial"+      [ bench (show n) $ nf factorial n+      | n <- [ 0, 1, 3, 6, 9, 11, 15+             , 20, 30, 40, 50, 60, 70, 80, 90, 100+             ]+      ]+    , bgroup "incompleteBeta"+      [ bench (show (p,q,x)) $ nf (incompleteBeta p q) x+      | (p,q,x) <- [ (10,      10,      0.5)+                   , (101,     101,     0.5)+                   , (1010,    1010,    0.5)+                   , (10100,   10100,   0.5)+                   , (100100,  100100,  0.5)+                   , (1001000, 1001000, 0.5)+                   , (10010000,10010000,0.5)+                   ]+      ]+    , bgroup "log1p"+        [ bench (show x) $ nf log1p x+        | x <- [ -0.9+               , -0.5+               , -0.1+               ,  0.1+               ,  0.5+               ,  1+               ,  10+               ,  100+               ] :: [Double]+        ]+    , bgroup "sinc" $+          bench "sin" (nf sin (0.55 :: Double))+        : [ bench (show x) $ nf sinc x+          | x <- [0, 1e-6, 1e-3,  0.5]+          ]+    , bgroup "erf & Co"+      [ bgroup "erf"+        [ bench (show x) $ nf erf x+        | x <- [0, 1.1, 100, 1000]+        ]+      , bgroup "erfc"+        [ bench (show x) $ nf erfc x+        | x <- [0, 1.1, 100, 1000]+        ]+      , bgroup "invErfc"+        [ bench (show x) $ nf erfc x+        | x <- [1e-9, 1e-6, 1e-3, 0.1, 1]+        ]+      ]+    , bgroup "expm1"+      [ bench (show x) $ nf expm1 (x :: Double)+      | x <- [-0.1, 0, 1, 19]+      ]+    , bgroup "poly"+        $  [ bench ("vector_"++show (U.length coefs)) $ nf (\x -> evaluatePolynomial x coefs) (1 :: Double)+           | coefs <- coef_list ]+        ++ [ bench ("unpacked_"++show n) $ nf (\x -> evaluatePolynomialL x (map fromIntegral [1..n])) (1 :: Double)+           | n <- coef_size ]+    , bgroup "RootFinding"+      [ bench "ridders sin" $ nf (ridders       def (0,pi/2))     (\x -> sin x - 0.525)+      , bench "newton sin"  $ nf (newtonRaphson def (0,1.2,pi/2)) (\x -> (sin x - 0.525,cos x))+      ]+    , bgroup "Sum"+      [ bench "naive"    $ whnf U.sum v+      , bench "kahan"    $ whnf (Sum.sumVector Sum.kahan) v+      , bench "kbn"      $ whnf (Sum.sumVector Sum.kbn) v+      , bench "kb2"      $ whnf (Sum.sumVector Sum.kb2) v+      ]+    ]
− benchmark/Summation.hs 
@@ -1,15 +0,0 @@-import Criterion.Main-import Numeric.Sum as Sum-import System.Random.MWC-import qualified Data.Vector.Unboxed as U--main = do-  gen <- createSystemRandom-  v <- uniformVector gen 10000000 :: IO (U.Vector Double)-  defaultMain [-      bench "naive" $ whnf U.sum v-    , bench "pairwise" $ whnf pairwiseSum v-    , bench "kahan" $ whnf (sumVector kahan) v-    , bench "kbn" $ whnf (sumVector kbn) v-    , bench "kb2" $ whnf (sumVector kb2) v-    ]
− benchmark/bench.hs 
@@ -1,89 +0,0 @@-import Criterion.Main-import qualified Data.Vector.Unboxed as U-import Numeric.SpecFunctions-import Numeric.Polynomial-import Text.Printf---- Uniformly sample logGamma performance between 10^-6 to 10^6-benchmarkLogGamma logG =-  [ bench (printf "%.3g" x) $ nf logG x-  | x <- [ m * 10**n | n <- [ -8 .. 8 ]-                     , m <- [ 10**(i / tics) | i <- [0 .. tics-1] ]-         ]-  ]-  where tics = 3-{-# INLINE benchmarkLogGamma #-}----- Power of polynomial to be evaluated (In other words length of coefficients vector)-coef_size :: [Int]-coef_size = [ 1,2,3,4,5,6,7,8,9-            , 10,    30-            , 100,   300-            , 1000,  3000-            , 10000, 30000-            ]-{-# INLINE coef_size #-}---- Precalculated coefficients-coef_list :: [U.Vector Double]-coef_list = [ U.replicate n 1.2 | n <- coef_size]-{-# NOINLINE coef_list #-}----main :: IO ()-main = defaultMain-  [ bgroup "logGamma" $-    benchmarkLogGamma logGamma-  , bgroup "logGammaL" $-    benchmarkLogGamma logGammaL-  , bgroup "incompleteGamma" $-      [ bench (show p) $ nf (incompleteGamma p) p-      | p <- [ 0.1-             , 1,   3-             , 10,  30-             , 100, 300-             , 999, 1000-             ]-      ]-  , bgroup "factorial"-    [ bench (show n) $ nf factorial n-    | n <- [ 0, 1, 3, 6, 9, 11, 15-           , 20, 30, 40, 50, 60, 70, 80, 90, 100-           ]-    ]-  , bgroup "incompleteBeta"-    [ bench (show (p,q,x)) $ nf (incompleteBeta p q) x-    | (p,q,x) <- [ (10,      10,      0.5)-                 , (101,     101,     0.5)-                 , (1010,    1010,    0.5)-                 , (10100,   10100,   0.5)-                 , (100100,  100100,  0.5)-                 , (1001000, 1001000, 0.5)-                 , (10010000,10010000,0.5)-                 ]-    ]-  , bgroup "log1p"-      [ bench (show x) $ nf log1p x-      | x <- [ -0.9-             , -0.5-             , -0.1-             ,  0.1-             ,  0.5-             ,  1-             ,  10-             ,  100-             ]-      ]-  , bgroup "sinc" $-        bench "sin" (nf sin (0.55 :: Double))-      : [ bench (show x) $ nf sinc x-        | x <- [0, 1e-6, 1e-3,  0.5]-        ]-  , bgroup "poly"-      $  [ bench ("vector_"++show (U.length coefs)) $ nf (\x -> evaluatePolynomial x coefs) (1 :: Double)-         | coefs <- coef_list ]-      ++ [ bench ("unpacked_"++show n) $ nf (\x -> evaluatePolynomialL x (map fromIntegral [1..n])) (1 :: Double)-         | n <- coef_size ]-  ]
changelog.md view
@@ -1,5 +1,136 @@-Changes in 0.1.7.0+## Changes in 0.3.4.4 +  * Support for future versions of GHC. Monoid and Monad instances now have+    canonical form+++## Changes in 0.3.4.3+ +  * Support for `QuickCheck >= 2.14`. Test no longer fail++  * Support for GHC<8.0 dropped+  ++## Changes in 0.3.4.2++  * Fixed crash in `invIncompleteBeta` (#68) for some inputs initial approximation+    was computed incorrectly.+++## Changes in 0.3.4.1++  * Precision of `incompleteGamma` improved.+++## Changes in 0.3.4.0++  * Dependency on `vector-th-unbox` is dropped. All instances are written by+    hand now.+++## Changes in 0.3.3.0++  * New implementation for `logGamma`. Now it's precise within 2 ulps at full+    range including zeros at 1 and 2.++  * Bug with precision loss of `invErf` for parameters near zero is fixed.++  * Fixed loss of precision in bundled `erf` near zero. (Affect primarily GHCJS)++  * `factorial` for now uses lookup table++  * `logFactorial` is optimized a bit (less number of terms is used)++  * `m_sqrt_eps` constant added.++  * Module `Numeric.SpecFunctions.Internal` is exposed.++  * Many improvements for test suite. Tables of expected function values are+    generated with mpmath, error estimates improved. Test suite itself is+    migrated to `tasty` from `test-framework`.+++## Changes in 0.3.2.1++  * Fixes build on windows for GHC<8.0+++## Changes in 0.3.2.0++  * GHCJS is now supported++  * Flag `system-expm1` is set to true by default. Only affects GHC<8.0++## Changes in 0.3.1.0++  * Exported data types for iteration steps in root finding++  * Defaults for root finding algorithm are documented+++## Changes in 0.3.0.2++  * Fix license field in cabal file+++## Changes in 0.3.0.0++  * `Semigroup` and `Monoid` instances added for data types from `Numeric.Sum`++  * API for finding roots of real functions reworked. 1) All algorithm+    parameters are now tweakable. 2) Functions for getting list of iterations+    added.++  * `Foldable` and `Traversable` instances for `Root` were added.++## Changes in 0.2.1.0++  * `log1p` and `expm1` are simply reexported from `GHC.Float`. They're methods+     of `Floating` type class.++  * On windows `expm1` is implemented in pure haskell for older GHCs.+++## Changes in 0.2.0.0++  * Bug fixes and documentation tweaks+++## Changes in 0.2.0.0++  * `logGamma` now uses Lancsoz approximation and same as `logGammaL`.  Old+     implementation of `logGamma` moved to `Numeric.SpecFunctions.Extra.logGammaAS245`.++  * Precision of `logGamma` for z<1 improved.++  * New much more precise implementation for `incompleteGamma`++  * Dependency on `erf` package dropped. `erf` and `erfc` just do direct calls+    to C.++  * `Numeric.SpecFunctions.expm1` added++  * `Numeric.SpecFunctions.log1pmx` added.++  * `logGammaCorrection` exported in `Numeric.SpecFunctions.Extra`.++  * Module `Numeric.Series` added for working with infinite sequences, series+    summation and evaluation of continued fractions.++  * Module `statistics: Statistics.Math.RootFinding` copied to+    `Numeric.RootFinding`. Instances for `binary` and `aeson` dropped.++  * Root-finding using Newton-Raphson added++  * `Numeric.MathFunctions.Comparison.ulpDelta` added. It calculates signed+    distance between two doubles.++  * Other bug fixes.++++## Changes in 0.1.7.0+   * Module `statistics: Statistics.Function.Comparison` moved to     `Numeric.MathFunctions.Comparison`. Old implementation if `within` compared     negative numbers incorrectly.@@ -11,7 +142,7 @@   * Precision of `logFactorial` is slightly improved.  -Changes in 0.1.6.0+## Changes in 0.1.6.0    * `logChoose` added for calculation of logarithm of binomial coefficient @@ -20,17 +151,17 @@   * `sinc` added  -Changes in 0.1.5.3+## Changes in 0.1.5.3    * Fix for test suite on 32bit platform  -Changes in 0.1.5+## Changes in 0.1.5    * Numeric.Sum: new module adds accurate floating point summation.  -Changes in 0.1.4+## Changes in 0.1.4    * logFactorial type is genberalized. It accepts any `Integral` type @@ -38,7 +169,7 @@     are store in lists added  -Changes in 0.1.3+## Changes in 0.1.3    * Error function and its inverse added.
math-functions.cabal view
@@ -1,50 +1,100 @@ name:           math-functions-version:        0.1.7.0-cabal-version:  >= 1.8-license:        BSD3+version:        0.3.4.4+cabal-version:  >= 1.10+license:        BSD2 license-file:   LICENSE author:         Bryan O'Sullivan <bos@serpentine.com>,-                Aleksey Khudyakov <alexey.skladnoy@gmail.com>-maintainer:     Bryan O'Sullivan <bos@serpentine.com>-homepage:       https://github.com/bos/math-functions-bug-reports:    https://github.com/bos/math-functions/issues+                Alexey Khudyakov <alexey.skladnoy@gmail.com>+maintainer:     Alexey Khudyakov <alexey.skladnoy@gmail.com>+homepage:       https://github.com/haskell/math-functions+bug-reports:    https://github.com/haskell/math-functions/issues category:       Math, Numeric build-type:     Simple-synopsis:       Special functions and Chebyshev polynomials+synopsis:       Collection of tools for numeric computations description:-  This library provides implementations of special mathematical-  functions and Chebyshev polynomials.  These functions are often-  useful in statistical and numerical computing. +  This library contain collection of various utilities for numerical+  computing. So far there're special mathematical functions,+  compensated summation algorithm, summation of series, root finding+  for real functions, polynomial summation and Chebyshev+  polynomials. ++tested-with:+    GHC ==8.0.2+     || ==8.2.2+     || ==8.4.4+     || ==8.6.5+     || ==8.8.4+     || ==8.10.7+     || ==9.0.2+     || ==9.2.7+     || ==9.4.5+     || ==9.6.2++ extra-source-files:-  changelog.md-  README.markdown-  benchmark/*.hs-  tests/*.hs-  tests/Tests/*.hs-  tests/Tests/SpecFunctions/gen.py-  doc/sinc.hs+                   changelog.md+                   README.markdown+                   tests/Tests/SpecFunctions/gen.py+                   tests/tables/generate.py+                   tests/tables/shell.nix+                   tests/tables/*.dat+                   tests/tables/inputs/*.dat+                   doc/sinc.hs +flag system-expm1+     description: Use expm1 provided by GHC. On GHCJS we don't have one so we+                  have to use hand-coded one.+     default:     True+     manual:      True++flag system-erf+     description: Use erf and erfc provided by system. On GHCJS+                  version provided by library is used regardless of+                  flag for that lack of libc.+     default:     True+     manual:      True+ library-  ghc-options:          -Wall-  build-depends:        base >=3 && <5,-                        deepseq,-                        erf >= 2,-                        vector >= 0.7,-                        primitive,-                        vector-th-unbox+  default-language: Haskell2010+  other-extensions:+    BangPatterns+    CPP+    DeriveDataTypeable+    FlexibleContexts+    MultiParamTypeClasses+    ScopedTypeVariables+    TypeFamilies+    DeriveGeneric++  ghc-options:          -Wall -O2+  build-depends:        base                >= 4.9 && < 5+                      , deepseq+                      , data-default-class  >= 0.1.2.0+                      , vector              >= 0.11+                      , primitive+  if flag(system-expm1) && !impl(ghcjs)+    cpp-options: -DUSE_SYSTEM_EXPM1+  if flag(system-erf) && !impl(ghcjs)+    cpp-options: -DUSE_SYSTEM_ERF   exposed-modules:     Numeric.MathFunctions.Constants     Numeric.MathFunctions.Comparison     Numeric.Polynomial     Numeric.Polynomial.Chebyshev+    Numeric.RootFinding     Numeric.SpecFunctions     Numeric.SpecFunctions.Extra+    Numeric.SpecFunctions.Internal+    Numeric.Series     Numeric.Sum   other-modules:-    Numeric.SpecFunctions.Internal+    Numeric.SpecFunctions.Compat -test-suite tests+test-suite math-function-tests+  default-language: Haskell2010+  other-extensions: ViewPatterns+   type:           exitcode-stdio-1.0   ghc-options:    -Wall -threaded   if arch(i386)@@ -56,27 +106,50 @@     Tests.Helpers     Tests.Chebyshev     Tests.Comparison+    Tests.RootFinding     Tests.SpecFunctions     Tests.SpecFunctions.Tables     Tests.Sum-  build-depends:-    math-functions,-    base >=3 && <5,-    deepseq,-    primitive,-    vector >= 0.7,-    vector-th-unbox,-    erf,-    HUnit      >= 1.2,-    QuickCheck >= 2.7,-    test-framework,-    test-framework-hunit,-    test-framework-quickcheck2+  if flag(system-erf)   && !impl(ghcjs)+    cpp-options: -DUSE_SYSTEM_ERF+  build-depends:    base >=4.5 && <5+                  , math-functions+                  , data-default-class  >= 0.1.2.0+                  , deepseq+                  , primitive+                  , vector >= 0.7+                  , vector-th-unbox+                  , erf+                  , QuickCheck       >= 2.7+                  , tasty            >= 1.2+                  , tasty-hunit      >= 0.10+                  , tasty-quickcheck >= 0.10 +benchmark math-functions-bench+  type:             exitcode-stdio-1.0+  if impl(ghcjs)+     buildable: False+  default-language: Haskell2010+  other-extensions:+    BangPatterns+    CPP+    DeriveDataTypeable+    FlexibleContexts+    MultiParamTypeClasses+    ScopedTypeVariables+    TemplateHaskell+    TypeFamilies+    DeriveGeneric+  ghc-options:          -Wall -O2+  hs-source-dirs:       bench+  Main-is:              bench.hs+  build-depends:        base                >= 4.5 && < 5+                      , math-functions+                      , data-default-class+                      , vector+                      , random+                      , tasty-bench         >=0.3.4+ source-repository head   type:     git   location: https://github.com/bos/math-functions--source-repository head-  type:     mercurial-  location: https://bitbucket.org/bos/math-functions
tests/Tests/Chebyshev.hs view
@@ -1,12 +1,10 @@ {-# OPTIONS_GHC -fno-warn-type-defaults #-} -module Tests.Chebyshev (-  tests-  ) where+module Tests.Chebyshev ( tests ) where  import Data.Vector.Unboxed                  (fromList)-import Test.Framework-import Test.Framework.Providers.QuickCheck2+import Test.Tasty+import Test.Tasty.QuickCheck                (testProperty) import Test.QuickCheck                      (Arbitrary(..),counterexample,Property)  import Tests.Helpers@@ -14,7 +12,7 @@ import Numeric.MathFunctions.Comparison  -tests :: Test+tests :: TestTree tests = testGroup "Chebyshev polynomials"   [ testProperty "Chebyshev 0" $ \a0 (Ch x) ->       testCheb [a0] x
tests/Tests/Comparison.hs view
@@ -2,15 +2,15 @@ -- Tests for approximate comparison module Tests.Comparison (tests) where -import Test.Framework-import Test.Framework.Providers.QuickCheck2+import Test.Tasty+import Test.Tasty.QuickCheck  import Tests.Helpers  import Numeric.MathFunctions.Comparison import Numeric.MathFunctions.Constants (m_epsilon) -tests :: Test+tests :: TestTree tests = testGroup "Comparison"   [ testProperty "addUlps 0"       $ \x   -> x == addUlps 0 x   , testProperty "addUlps sym"     $ \i x -> x == (addUlps (-i) . addUlps i) x
tests/Tests/Helpers.hs view
@@ -15,9 +15,8 @@ import Data.Complex import Data.Typeable -import qualified Test.HUnit      as HU-import Test.Framework-import Test.Framework.Providers.HUnit+import Test.Tasty+import Test.Tasty.HUnit  import Numeric.MathFunctions.Comparison @@ -71,8 +70,8 @@ -- HUnit helpers ---------------------------------------------------------------- -testAssertion :: String -> Bool -> Test-testAssertion str cont = testCase str $ HU.assertBool str cont+testAssertion :: String -> Bool -> TestTree+testAssertion str cont = testCase str $ assertBool str cont -testEquality :: (Show a, Eq a) => String -> a -> a -> Test-testEquality msg a b = testCase msg $ HU.assertEqual msg a b+testEquality :: (Show a, Eq a) => String -> a -> a -> TestTree+testEquality msg a b = testCase msg $ assertEqual msg a b
+ tests/Tests/RootFinding.hs view
@@ -0,0 +1,44 @@+-- |+module Tests.RootFinding ( tests ) where++import Data.Default.Class+import Test.Tasty+import Test.Tasty.HUnit++import Numeric.RootFinding+import Tests.Helpers+++tests :: TestTree+tests = testGroup "Root finding"+  [ testGroup "Ridders"+    [ testAssertion "sin x - 0.525 [exact]"     $ testRiddersSin0_525 (AbsTol 0)+    , testAssertion "sin x - 0.525 [abs 1e-12]" $ testRiddersSin0_525 (AbsTol 1e-12)+    , testAssertion "sin x - 0.525 [abs 1e-6]"  $ testRiddersSin0_525 (AbsTol 1e-6)+    , testAssertion "sin x - 0.525 [rel 1e-12]" $ testRiddersSin0_525 (RelTol 1e-12)+    , testAssertion "sin x - 0.525 [rel 1e-6]"  $ testRiddersSin0_525 (RelTol 1e-6)+    ]+  , testGroup "Newton-Raphson"+    [ testAssertion "sin x - 0.525 [rel 1e-12]" $ testNewtonSin0_525 (RelTol 1e-12)+    , testAssertion "sin x - 0.525 [rel 1e-6]"  $ testNewtonSin0_525 (RelTol 1e-6)+    , testAssertion "sin x - 0.525 [abs 1e-12]" $ testNewtonSin0_525 (AbsTol 1e-12)+    , testAssertion "sin x - 0.525 [abs 1e-6]"  $ testNewtonSin0_525 (AbsTol 1e-6)+    , testAssertion "1/x - 0.5     [0]"         $+        let Root r = newtonRaphson def{newtonTol=RelTol 0} (1,1000,1000)+                       (\x -> (1/x - 0.5, -1/(x*x)))+        in  r == 2+    ]+  ]+  where+    -- Exact root for equation: sin x - 0.525 = 0+    exactRoot = 0.5527151130967832+    --+    testRiddersSin0_525 tol+      = withinTolerance tol r exactRoot+      where+        Root r = ridders def{riddersTol = tol} (0, pi/2) (\x -> sin x - 0.525)+    --+    testNewtonSin0_525 tol+      = withinTolerance tol r exactRoot+      where+        Root r = newtonRaphson def{newtonTol=tol} (0, pi/4, pi/2) (\x -> (sin x - 0.525, cos x))
tests/Tests/SpecFunctions.hs view
@@ -1,106 +1,373 @@+{-# LANGUAGE BangPatterns #-}+{-# LANGUAGE CPP          #-} {-# LANGUAGE ViewPatterns #-} -- | Tests for Statistics.Math module Tests.SpecFunctions (   tests   ) where +import Control.Monad+import Data.List+import Data.Maybe import qualified Data.Vector as V import           Data.Vector   ((!))+import qualified Data.Vector.Unboxed as U  import Test.QuickCheck  hiding (choose,within)-import Test.Framework-import Test.Framework.Providers.QuickCheck2+import Test.Tasty+import Test.Tasty.QuickCheck   (testProperty)+import Test.Tasty.HUnit  import Tests.Helpers import Tests.SpecFunctions.Tables import Numeric.SpecFunctions-import Numeric.MathFunctions.Comparison (within)+import Numeric.SpecFunctions.Internal   (factorialTable)+import Numeric.MathFunctions.Comparison (within,ulpDistance)+import Numeric.MathFunctions.Constants  (m_epsilon,m_tiny) +erfTol,erfcTol,erfcLargeTol :: Int+-- Pure haskell implementation is not very good+#if !defined(USE_SYSTEM_ERF) || defined(__GHCJS__)+erfTol       = 4+erfcTol      = 4+erfcLargeTol = 64+-- Macos's erf is slightly worse that GNU one+#elif defined(darwin_HOST_OS)+erfTol       = 2+erfcTol      = 2+erfcLargeTol = 2+-- Windows' one is not very good too+#elif defined(mingw32_HOST_OS)+erfTol       = 2+erfcTol      = 2+erfcLargeTol = 4+#else+erfTol       = 1+erfcTol      = 2+erfcLargeTol = 2+#endif -tests :: Test+isGHCJS :: Bool+#if defined(__GHCJS__)+isGHCJS = True+#else+isGHCJS = False+#endif++isWindows :: Bool+#if defined(mingw32_HOST_OS)+isWindows = True+#else+isWindows = False+#endif+++tests :: TestTree tests = testGroup "Special functions"-  [ testProperty "Gamma(x+1) = x*Gamma(x) [logGamma]"  $ gammaReccurence logGamma  3e-8-  , testProperty "Gamma(x+1) = x*Gamma(x) [logGammaL]" $ gammaReccurence logGammaL 2e-13-  , testProperty "gamma(1,x) = 1 - exp(-x)"      $ incompleteGammaAt1Check-  , testProperty "0 <= gamma <= 1"               $ incompleteGammaInRange-  , testProperty "0 <= I[B] <= 1"            $ incompleteBetaInRange-  -- XXX FIXME DISABLED due to failures-  -- , testProperty "invIncompleteGamma = gamma^-1" $ invIGammaIsInverse-  -- , testProperty "invIncompleteBeta  = B^-1" $ invIBetaIsInverse-  -- , testProperty "gamma - increases"             $-  --     \s x y -> s > 0 && x > 0 && y > 0 ==> monotonicallyIncreases (incompleteGamma s) x y-  , testProperty "invErfc = erfc^-1"         $ invErfcIsInverse-  , testProperty "invErf  = erf^-1"          $ invErfIsInverse-    -- Unit tests-  , testAssertion "Factorial is expected to be precise at 1e-15 level"-      $ and [ eq 1e-15 (factorial (fromIntegral n :: Int))-                       (fromIntegral (factorial' n))-            |n <- [0..170]]-  , testAssertion "Log factorial is expected to be precise at 1e-15 level"-      $ and [ eq 1e-15 (logFactorial (fromIntegral n :: Int))-                       (log $ fromIntegral $ factorial' n)-            | n <- [2..170]]-  , testAssertion "logGamma is expected to be precise at 1e-9 level [integer points]"-      $ and [ eq 1e-9 (logGamma (fromIntegral n))-                      (logFactorial (n-1))-            | n <- [3..10000::Int]]-  , testAssertion "logGamma is expected to be precise at 1e-9 level [fractional points]"-      $ and [ eq 1e-9 (logGamma x) lg | (x,lg) <- tableLogGamma ]-  , testAssertion "logGammaL is expected to be precise at 1e-15 level"-      $ and [ eq 1e-15 (logGammaL (fromIntegral n))-                       (logFactorial (n-1))-            | n <- [3..10000::Int]]-    -- FIXME: Too low!-  , testAssertion "logGammaL is expected to be precise at 1e-10 level [fractional points]"-      $ and [ eq 1e-10 (logGammaL x) lg | (x,lg) <- tableLogGamma ]-    -- FIXME: loss of precision when logBeta p q ≈ 0.-    --        Relative error doesn't work properly in this case.-  , testAssertion "logBeta is expected to be precise at 1e-6 level"-      $ and [ eq 1e-6 (logBeta p q)-                      (logGammaL p + logGammaL q - logGammaL (p+q))-            | p <- [0.1,0.2 .. 0.9] ++ [2 .. 20]-            , q <- [0.1,0.2 .. 0.9] ++ [2 .. 20]-            ]-  , testAssertion "digamma is expected to be precise at 1e-14 [integers]"-      $ digammaTestIntegers 1e-14-    -- Relative precision is lost when digamma(x) ≈ 0-  , testAssertion "digamma is expected to be precise at 1e-12"-      $ and [ eq 1e-12 r (digamma x) | (x,r) <- tableDigamma ]-    -- FIXME: Why 1e-8? Is it due to poor precision of logBeta?-  , testAssertion "incompleteBeta is expected to be precise at 1e-8 level"-      $ and [ eq 1e-8 (incompleteBeta p q x) ib | (p,q,x,ib) <- tableIncompleteBeta ]-  , testAssertion "incompleteBeta with p > 3000 and q > 3000"-      $ and [ eq 1e-11 (incompleteBeta p q x) ib | (x,p,q,ib) <--                 [ (0.495,  3001,  3001, 0.2192546757957825068677527085659175689142653854877723)-                 , (0.501,  3001,  3001, 0.5615652382981522803424365187631195161665429270531389)-                 , (0.531,  3500,  3200, 0.9209758089734407825580172472327758548870610822321278)-                 , (0.501, 13500, 13200, 0.0656209987264794057358373443387716674955276089622780)-                 ]-            ]-  , testAssertion "choose is expected to precise at 1e-12 level"-      $ and [ eq 1e-12 (choose (fromIntegral n) (fromIntegral k)) (fromIntegral $ choose' n k)-            | n <- [0..1000], k <- [0..n]]-  , testAssertion "logChoose == log . choose"-      $ and [ let n' = fromIntegral n-                  k' = fromIntegral k-              in within 2 (logChoose n' k') (log $ choose n' k')-            | n <- [0..1000], k <- [0..n]]+  [ testGroup "erf"+    [ -- implementation from numerical recipes loses precision for+      -- large arguments+      testCase "erfc table" $+        forTable "tests/tables/erfc.dat" $ \[x, exact] ->+          checkTabularPure erfcTol (show x) exact (erfc x)+    , testCase "erfc table [large]" $+        forTable "tests/tables/erfc-large.dat" $ \[x, exact] ->+          checkTabularPure erfcLargeTol (show x) exact (erfc x)+      --+    , testCase "erf table" $+        forTable "tests/tables/erf.dat" $ \[x, exact] -> do+          checkTabularPure erfTol (show x) exact (erf x)+    , testProperty "id = erfc . invErfc" invErfcIsInverse+    , testProperty "id = invErfc . erfc" invErfcIsInverse2+    , testProperty "invErf  = erf^-1"    invErfIsInverse+    ]+  --+  , testGroup "log1p & Co"+    [ testCase "expm1 table" $+        forTable "tests/tables/expm1.dat" $ \[x, exact] ->+          checkTabularPure 2 (show x) exact (expm1 x)+    , testCase "log1p table" $+        forTable "tests/tables/log1p.dat" $ \[x, exact] ->+          checkTabularPure 1 (show x) exact (log1p x)+    ]+  ----------------+  , testGroup "gamma function"+    [ testCase "logGamma table [fractional points" $+        forTable "tests/tables/loggamma.dat" $ \[x, exact] -> do+          checkTabularPure 2 (show x) exact (logGamma x)+    , testProperty "Gamma(x+1) = x*Gamma(x)" $ gammaReccurence+    , testCase     "logGamma is expected to be precise at 1e-15 level" $+        forM_ [3..10000::Int] $ \n -> do+          let exact = logFactorial (n-1)+              val   = logGamma (fromIntegral n)+          checkTabular 8 (show n) exact val+    ]+  ----------------+  , testGroup "incomplete gamma"+    [ testCase "incompleteGamma table" $+        forTable "tests/tables/igamma.dat" $ \[a,x,exact] -> do+          let err | a < 10    = 16+                  | a <= 101  = case () of+                      _| isGHCJS   -> 64+                       | isWindows -> 64+                       | otherwise -> 32+                  | a == 201  = 200+                  | otherwise = 32+          checkTabularPure err (show (a,x)) exact (incompleteGamma a x)+    , testProperty "incomplete gamma - increases" $+        \(abs -> s) (abs -> x) (abs -> y) -> s > 0 ==> monotonicallyIncreases (incompleteGamma s) x y+    , testProperty "0 <= gamma <= 1"               incompleteGammaInRange+    , testProperty "gamma(1,x) = 1 - exp(-x)"      incompleteGammaAt1Check+    , testProperty "invIncompleteGamma = gamma^-1" invIGammaIsInverse+    ]+  ----------------+  , testGroup "beta function"+    [ testCase "logBeta table" $+        forTable "tests/tables/logbeta.dat" $ \[p,q,exact] ->+          let errEst+                -- For Stirling approx. errors are very good+                | b > 10          = 2+                -- Partial Stirling approx+                | a > 10 = case () of+                    _| b >= 1    -> 4+                     | otherwise -> 2 * est+                -- sum of logGamma+                | otherwise = case () of+                    _| a <= 1 && b <= 1 -> 8+                     | a >= 1 && b >= 1 -> 8+                     | otherwise        -> 2 * est+                where+                  a = max p q+                  b = min p q+                  --+                  est = ceiling+                      $ abs (logGamma a) + abs (logGamma b) + abs (logGamma (a + b))+                      / abs (logBeta a b)+          in checkTabularPure errEst (show (p,q)) exact (logBeta p q)+    , testCase "logBeta factorial" betaFactorial+    , testProperty "beta(1,p) = 1/p"   beta1p+    -- , testProperty "beta recurrence"   betaRecurrence+    ]+  ----------------+  , testGroup "incomplete beta"+    [ testCase "incompleteBeta table" $+        forM_ tableIncompleteBeta $ \(p,q,x,exact) ->+          checkTabular 64 (show (x,p,q)) (incompleteBeta p q x) exact+    , testCase "incompleteBeta table with p > 3000 and q > 3000" $+        forM_ tableIncompleteBetaP3000 $ \(x,p,q,exact) ->+          checkTabular 7000 (show (x,p,q)) (incompleteBeta p q x) exact+    --+    , testProperty "0 <= I[B] <= 1" incompleteBetaInRange+    , testProperty "ibeta symmetry" incompleteBetaSymmetry+    , testCase "Regression #68" $ do+        let a = 1+            b = 0.3+            p = 0.3+            x = invIncompleteBeta a b p+        assertBool "Inversion OK" $ incompleteBeta a b x `ulpDistance` p < 4+    -- XXX FIXME DISABLED due to failures+    -- , testProperty "invIncompleteBeta  = B^-1" $ invIBetaIsInverse+    ]+  ----------------+  , testGroup "digamma"+    [ testAssertion "digamma is expected to be precise at 1e-14 [integers]"+        $ digammaTestIntegers 1e-14+      -- Relative precision is lost when digamma(x) ≈ 0+    , testCase "digamma is expected to be precise at 1e-12" $+      forTable "tests/tables/digamma.dat" $ \[x, exact] ->+        checkTabularPure 2048+          (show x) (digamma x) exact+    ]+  ----------------+  , testGroup "factorial"+    [ testCase "Factorial table" $+      forM_ [0 .. 170] $ \n -> do+        checkTabular 1+          (show n)+          (fromIntegral (factorial' n))+          (factorial (fromIntegral n :: Int))+      --+    , testCase "Log factorial from integer" $+      forM_ [2 .. 170] $ \n -> do+        checkTabular 1+          (show n)+          (log $ fromIntegral $ factorial' n)+          (logFactorial (fromIntegral n :: Int))+    , testAssertion "Factorial table is OK"+    $ U.length factorialTable == 171+    , testCase "Log factorial table" $+      forTable "tests/tables/factorial.dat" $ \[i,exact] ->+        checkTabularPure 3+          (show i) (logFactorial (round i :: Int)) exact+    ]+  ----------------+  , testGroup "combinatorics"+    [ testCase "choose table" $+      forM_ [0 .. 1000] $ \n ->+        forM_ [0 .. n]  $ \k -> do+          checkTabular (if isWindows then 3072 else 2048)+            (show (n,k))+            (fromIntegral $ choose' n k)+            (choose (fromInteger n) (fromInteger k))+    --+    , testCase "logChoose == log . choose" $+      forM_ [0 .. 1000] $ \n ->+        forM_ [0 .. n]  $ \k -> do+          checkTabular 2+            (show (n,k))+            (log $ choose n k)+            (logChoose n k)+    ]     ----------------------------------------------------------------     -- Self tests-  , testProperty "Self-test: 0 <= range01 <= 1" $ \x -> let f = range01 x in f <= 1 && f >= 0+  , testGroup "self-test"+    [ testProperty "Self-test: 0 <= range01 <= 1" $ \x -> let f = range01 x in f <= 1 && f >= 0+    ]   ]  ----------------------------------------------------------------+-- efr tests+----------------------------------------------------------------++roundtrip_erfc_invErfc,+  roundtrip_invErfc_erfc,+  roundtrip_erf_invErf+  :: (Double,Double)+#if !defined(USE_SYSTEM_ERF) || defined(__GHCJS__)+roundtrip_erfc_invErfc = (2,8)+roundtrip_invErfc_erfc = (8,4)+roundtrip_erf_invErf   = (128,128)+#elif defined(darwin_HOST_OS)+roundtrip_erfc_invErfc = (4,4)+roundtrip_invErfc_erfc = (4,4)+roundtrip_erf_invErf   = (2,2)+#elif defined(mingw32_HOST_OS)+roundtrip_erfc_invErfc = (4,4)+roundtrip_invErfc_erfc = (4,4)+roundtrip_erf_invErf   = (4,4)+#else+roundtrip_erfc_invErfc = (2,2)+roundtrip_invErfc_erfc = (2,2)+roundtrip_erf_invErf   = (1,1)+#endif++-- id ≈ erfc . invErfc+invErfcIsInverse :: Double -> Property+invErfcIsInverse ((*2) . range01 -> x)+  = (not $ isInfinite x) ==>+  ( counterexample ("x        = " ++ show x )+  $ counterexample ("y        = " ++ show y )+  $ counterexample ("x'       = " ++ show x')+  $ counterexample ("calc.err = " ++ show (delta, delta-e'))+  $ counterexample ("ulps     = " ++ show (ulpDistance x x'))+  $ ulpDistance x x' <= round delta+  )+  where+    (e,e') = roundtrip_erfc_invErfc+    delta  = e' + e * abs ( y / x  *  2 / sqrt pi * exp( -y*y ))+    y  = invErfc x+    x' = erfc y++-- id ≈ invErfc . erfc+invErfcIsInverse2 :: Double -> Property+invErfcIsInverse2 x+  = (not $ isInfinite x') ==>+    (y > m_tiny)          ==>+    (x /= 0) ==>+    counterexample ("x        = " ++ show x )+  $ counterexample ("y        = " ++ show y )+  $ counterexample ("x'       = " ++ show x')+  $ counterexample ("calc.err = " ++ show delta)+  $ counterexample ("ulps     = " ++ show (ulpDistance x x'))+  $ ulpDistance x x' <= delta+  where+    (e,e') = roundtrip_invErfc_erfc+    delta  = round+           $ e' + e * abs (y / x  /  (2 / sqrt pi * exp( -x*x )))+    y  = erfc x+    x' = invErfc y++-- id ≈ erf . invErf+invErfIsInverse :: Double -> Property+invErfIsInverse a+  = (x /= 0) ==>+    counterexample ("x        = " ++ show x )+  $ counterexample ("y        = " ++ show y )+  $ counterexample ("x'       = " ++ show x')+  $ counterexample ("calc.err = " ++ show delta)+  $ counterexample ("ulps     = " ++ show (ulpDistance x x'))+  $ ulpDistance x x' <= delta+  where+    (e,e') = roundtrip_erf_invErf+    delta  = round+           $ e + e' * abs (y / x  *  2 / sqrt pi * exp ( -y * y ))+    x  | a < 0     = - range01 a+       | otherwise =   range01 a+    y  = invErf x+    x' = erf y++---------------------------------------------------------------- -- QC tests ---------------------------------------------------------------- +-- B(p,q) = (x - 1)!(y-1)! / (x + y - 1)!+betaFactorial :: IO ()+betaFactorial = do+  forM_ prod $ \(p,q,facP,facQ,facProd) -> do+    let exact = fromIntegral (facQ * facP)+              / fromIntegral facProd+    checkTabular 16 (show (p,q))+      (logBeta (fromIntegral p) (fromIntegral q))+      (log exact)+  where+    prod    = [ (p,q,facP,facQ, factorial' (p + q - 1))+              | (p,facP) <- facList+              , (q,facQ) <- facList+              , p + q < 170+              , not (p == 1 && q== 1)+              ]+    facList = [(p,factorial' (p-1)) | p <- [1 .. 170]]++-- B(1,p) = 1/p+beta1p :: Double -> Property+beta1p (abs -> p)+  = p > 2 ==>+    counterexample ("p    = " ++ show p)+  $ counterexample ("logB = " ++ show lb)+  $ counterexample ("err  = " ++ show d)+  $ d <= 24+  where+    lb = logBeta 1 p+    d  = ulpDistance lb (- log p)++{-+-- B(p+1,q) = B(p,q) · p/(p+q)+betaRecurrence :: Double -> Double -> Property+betaRecurrence (abs -> p) (abs -> q)+  = p > 0  &&  q > 0  ==>+    counterexample ("p          = " ++ show p)+  $ counterexample ("q          = " ++ show q)+  $ counterexample ("log B(p,q) = " ++ show (logBeta p q))+  $ counterexample ("log B(p+1,q) = " ++ show (logBeta (p+1) q))+  $ counterexample ("err        = " ++ show d)+  $ d <= 128+  where+    logB  = logBeta p q + log (p / (p + q))+    logB' = logBeta (p + 1) q+    d     = ulpDistance logB logB'+-}+ -- Γ(x+1) = x·Γ(x)-gammaReccurence :: (Double -> Double) -> Double -> Double -> Property-gammaReccurence logG ε x =-  (x > 0 && x < 100)  ==>  (abs (g2 - g1 - log x) < ε)+gammaReccurence :: Double -> Property+gammaReccurence x+  = x > 0  ==>  err < errEst     where-      g1 = logG x-      g2 = logG (x+1)+      g1     = logGamma x+      g2     = logGamma (x+1)+      err    = abs (g2 - g1 - log x)+      -- logGamma apparently is not as precise for small x. See #59 for details+      errEst = max 1e-14+             $ 2 * m_epsilon * sum (map abs [ g1 , g2 , log x ])  -- γ(s,x) is in [0,1] range incompleteGammaInRange :: Double -> Double -> Property@@ -109,55 +376,52 @@  -- γ(1,x) = 1 - exp(-x) -- Since Γ(1) = 1 normalization doesn't make any difference-incompleteGammaAt1Check :: Double -> Property+incompleteGammaAt1Check :: Double -> Bool incompleteGammaAt1Check (abs -> x) =-  x > 0 ==> (incompleteGamma 1 x + exp(-x)) ≈ 1-  where-    (≈) = eq 1e-13+  ulpDistance (incompleteGamma 1 x) (-expm1(-x)) < 16  -- invIncompleteGamma is inverse of incompleteGamma invIGammaIsInverse :: Double -> Double -> Property invIGammaIsInverse (abs -> a) (range01 -> p) =-  a > 0 && p > 0 && p < 1  ==> ( counterexample ("a  = " ++ show a )-                               $ counterexample ("p  = " ++ show p )-                               $ counterexample ("x  = " ++ show x )-                               $ counterexample ("p' = " ++ show p')-                               $ counterexample ("Δp = " ++ show (p - p'))-                               $ abs (p - p') <= 1e-12-                               )+  a > m_tiny && p > m_tiny && p < 1 && x > m_tiny  ==>+    ( counterexample ("a    = " ++ show a )+    $ counterexample ("p    = " ++ show p )+    $ counterexample ("x    = " ++ show x )+    $ counterexample ("p'   = " ++ show p')+    $ counterexample ("err  = " ++ show (ulpDistance p p'))+    $ counterexample ("est  = " ++ show est)+    $ ulpDistance p p' <= est+    )   where     x  = invIncompleteGamma a p+    f' = exp ( log x * (a-1) - x - logGamma a)     p' = incompleteGamma    a x---- invErfc is inverse of erfc-invErfcIsInverse :: Double -> Property-invErfcIsInverse ((*2) . range01 -> p)-  = counterexample ("p  = " ++ show p )-  $ counterexample ("x  = " ++ show x )-  $ counterexample ("p' = " ++ show p')-  $ abs (p - p') <= 1e-14-  where-    x  = invErfc p-    p' = erfc x---- invErf is inverse of erf-invErfIsInverse :: Double -> Property-invErfIsInverse a-  = counterexample ("p  = " ++ show p )-  $ counterexample ("x  = " ++ show x )-  $ counterexample ("p' = " ++ show p')-  $ abs (p - p') <= 1e-14-  where-    x  = invErf p-    p' = erf x-    p  | a < 0     = - range01 a-       | otherwise =   range01 a+    -- FIXME: Test should be rechecked when #42 is fixed+    (e,e') = (32,32)+    est    = round+           $ e' + e * abs (x / p * f') --- B(s,x) is in [0,1] range+-- I(x;p,q) is in [0,1] range incompleteBetaInRange :: Double -> Double -> Double -> Property incompleteBetaInRange (abs -> p) (abs -> q) (range01 -> x) =   p > 0 && q > 0  ==> let i = incompleteBeta p q x in i >= 0 && i <= 1 +-- I(0.5; p,p) = 0.5+incompleteBetaSymmetry :: Double -> Property+incompleteBetaSymmetry (abs -> p) =+    p > 0 ==>+    counterexample ("p   = " ++ show p)+  $ counterexample ("ib  = " ++ show ib)+  $ counterexample ("err = " ++ show d)+  $ counterexample ("est = " ++ show est)+  $ d <= est+  where+    est | p < 1     = 80+        | p < 10    = 200+        | otherwise = round $ 6 * p+    d  = ulpDistance ib 0.5+    ib = incompleteBeta p p 0.5+ -- invIncompleteBeta is inverse of incompleteBeta invIBetaIsInverse :: Double -> Double -> Double -> Property invIBetaIsInverse (abs -> p) (abs -> q) (range01 -> x) =@@ -210,3 +474,33 @@ -- Truncate double to [0,1] range01 :: Double -> Double range01 = abs . (snd :: (Integer, Double) -> Double) . properFraction+++readTable :: FilePath -> IO [[Double]]+readTable+  = fmap (fmap (fmap read . words) . lines)+  . readFile++forTable :: FilePath -> ([Double] -> Maybe String) -> IO ()+forTable path fun = do+  rows <- readTable path+  case mapMaybe fun rows of+    []   -> return ()+    errs -> assertFailure $ intercalate "---\n" errs++checkTabular :: Int -> String -> Double -> Double -> IO ()+checkTabular prec x exact val =+  case checkTabularPure prec x exact val of+    Nothing -> return ()+    Just s  -> assertFailure s++checkTabularPure :: Int -> String -> Double -> Double -> Maybe String+checkTabularPure prec x exact val+  | within prec exact val = Nothing+  | otherwise             = Just $ unlines+      [ " x         = " ++ x+      , " expected  = " ++ show exact+      , " got       = " ++ show val+      , " ulps diff = " ++ show (ulpDistance exact val)+      , " err.est.  = " ++ show prec+      ]
tests/Tests/SpecFunctions/Tables.hs view
@@ -1,38 +1,5 @@ module Tests.SpecFunctions.Tables where -tableLogGamma :: [(Double,Double)]-tableLogGamma =-  [(0.000001250000000, 13.592366285131767256)-  , (0.000068200000000, 9.5930266308318756785)-  , (0.000246000000000, 8.3100370767447948595)-  , (0.000880000000000, 7.0350813373524845318)-  , (0.003120000000000, 5.7681293583655666168)-  , (0.026700000000000, 3.6082588918892972707)-  , (0.077700000000000, 2.5148371858768232556)-  , (0.234000000000000, 1.3579557559432757774)-  , (0.860000000000000, 0.098146578027685588141)-  , (1.340000000000000, -0.11404757557207759189)-  , (1.890000000000000, -0.042511642297870140539)-  , (2.450000000000000, 0.25014296569217620014)-  , (3.650000000000000, 1.3701041997380685178)-  , (4.560000000000000, 2.5375143317949575561)-  , (6.660000000000000, 5.9515377269550207018)-  , (8.250000000000000, 9.0331869196051215454)-  , (11.300000000000001, 15.814180681373947834)-  , (25.600000000000001, 56.711261598328121636)-  , (50.399999999999999, 146.12815158702164808)-  , (123.299999999999997, 468.85500075897556371)-  , (487.399999999999977, 2526.9846647543727158)-  , (853.399999999999977, 4903.9359135978220365)-  , (2923.300000000000182, 20402.93198938705973)-  , (8764.299999999999272, 70798.268343590112636)-  , (12630.000000000000000, 106641.7726498250704)-  , (34500.000000000000000, 325976.34838781820145)-  , (82340.000000000000000, 849629.79603036714252)-  , (234800.000000000000000, 2668846.4390507955104)-  , (834300.000000000000000, 10540830.912557533011)-  , (1230000.000000000000000, 16017699.322315014899)-  ] tableIncompleteBeta :: [(Double,Double,Double,Double)] tableIncompleteBeta =   [(2.000000000000000, 3.000000000000000, 0.030000000000000, 0.0051864299999999988189)@@ -45,106 +12,12 @@   , (13.100000000000000, 9.800000000000001, 0.420000000000000, 0.071321857831804780226)   , (13.100000000000000, 9.800000000000001, 0.920000000000000, 0.99999578339197070509)   ]-tableDigamma :: [(Double,Double)]-tableDigamma =-  [(10.0261172557341425, 2.2544954834170942704)-  , (0.9070101446062873, -0.74152778337908598072)-  , (3.4679213262860156, 1.0925031389314479036)-  , (28.5703089405901878, 3.3347652650101657912)-  , (5.9700184459319399, 1.7006665338476731897)-  , (20.5303177686997920, 2.9973508205248808878)-  , (5.6622605630542511, 1.6429280447671743559)-  , (4.4741465342999014, 1.3824198603491071324)-  , (21.4416006516504787, 3.0418326144933285349)-  , (47.6946291432301663, 3.8542988022858128971)-  , (11.2357450115053670, 2.37393979612347783)-  , (0.3352840110772935, -3.1124447967622668187)-  , (2.5037441860153118, 0.70499097759044615508)-  , (0.5241560861477529, -1.8489960634174653631)-  , (0.1972018552655726, -5.3635382066874592866)-  , (0.8289440927562556, -0.90024805153750442344)-  , (2.0717397641759350, 0.4680412969073853291)-  , (9.1173553049782452, 2.1543380160183831507)-  , (1.1815938184339669, -0.31262126373727594508)-  , (7.3600347508772019, 1.9265946441432049152)-  , (19.7457045917841398, 2.9574003365402390386)-  , (4.1956416643620571, 1.3101672771843546617)-  , (7.3868205159465790, 1.9304848277860633399)-  , (1.2786090750546355, -0.19373178842778399078)-  , (10.6498308581562604, 2.3178608134278069208)-  , (10.6750266252851169, 2.3203381265132185796)-  , (10.6883248506773985, 2.3216431742802625671)-  , (14.3373372205836365, 2.6275879484098640937)-  , (3.3932538441985769, 1.0672611106295626371)-  , (11.4168205413938768, 2.3906538776946248959)-  , (3.2500957742991048, 1.0170253699094919941)-  , (2.7573211981404855, 0.82209952378707851217)-  , (21.8943170241258827, 3.063216323919045081)-  , (16.7950471612825254, 2.7910180230044043803)-  , (9.2578640399661225, 2.1704940538770385317)-  , (5.3213868642873896, 1.5748408574979930741)-  , (9.4381079039564071, 2.1908443398518979706)-  , (13.1568457441413429, 2.538458049596743038)-  , (10.6478950333943825, 2.3176702242110884811)-  , (6.4894496431749733, 1.7911554320176725774)-  , (20.3998669454332315, 2.9908182167188113176)-  , (3.6989463639934752, 1.1668268193484248041)-  , (3.4716258279958572, 1.093739186127963281)-  , (24.7013029455164919, 3.1864775907749920414)-  , (1.1608524325026863, -0.33982067949719851896)-  , (1.9482800424522431, 0.3888762195060542215)-  , (30.4956621109554185, 3.4010990755913685923)-  , (16.3105956379859052, 2.7608468922073350349)-  , (10.6908820268137070, 2.3218939328714371939)-  , (3.4369121607821915, 1.082096765647714065)-  , (2.2914619096171260, 0.5953971130541900747)-  , (24.1273989930028883, 3.1624816269998849982)-  , (14.9455957898231535, 2.6705890837495616097)-  , (32.2002179941400826, 3.4563650137673369578)-  , (1.7232417075599473, 0.22682264125689588496)-  , (9.9662376350778192, 2.248195612105357899)-  , (10.9702870318273966, 2.348920912357223223)-  , (18.8934063317711676, 2.912115343761407793)-  , (8.6720493874148570, 2.1013420151521415846)-  , (20.4905634096258815, 2.9953645521238549954)-  , (1.4654265058258678, 0.0036653372399428492921)-  , (15.4401781010745509, 2.7042406258657996077)-  , (13.6688064138713390, 2.5780909087521290957)-  , (2.4073661551765566, 0.65668881914974130964)-  , (0.8108729056729371, -0.94026521559981879328)-  , (29.5024809785193902, 3.367430902728568487)-  , (7.5321882978878660, 1.9513375601887514854)-  , (3.3716588961200955, 1.0598414578703589939)-  , (2.9310065630306474, 0.89516303667430119351)-  , (7.2023118361897769, 1.9033764996201536501)-  , (3.1362387322050900, 0.97520764792577085966)-  , (6.5709053027851487, 1.8046329737306385788)-  , (3.7348491113356177, 1.1779005641199544741)-  , (1.2328105814385013, -0.24823346907893503732)-  , (7.9098387372709587, 2.0035651569967258823)-  , (2.8590898311999715, 0.86554629114604864082)-  , (2.1964374279534344, 0.54225028515290207842)-  , (3.8933394033155189, 1.2253803767351847398)-  , (10.7410508007627694, 2.3268008547643748152)-  , (2.4921048837305193, 0.69927782909414781809)-  , (2.2101710538553756, 0.55010424351998354897)-  , (14.0357118427322334, 2.6055587167248708269)-  , (4.1320729121597584, 1.2929216807716104043)-  , (0.2766365979680845, -3.8108738889017752527)-  , (27.9448247140513644, 3.3122329205038494315)-  , (9.3081256750537182, 2.1762105230057038341)-  , (1.4222181352589696, -0.038843893649701873028)-  , (1.5107587188614726, 0.046499571962236106726)-  , (3.3467578222470555, 1.0512176183500512305)-  , (12.2373583939228876, 2.4630788434421742039)-  , (0.9385094944630431, -0.68317598609698348966)-  , (5.8655552400886410, 1.6814385243672138603)-  , (17.1377048621110468, 2.8118219246156086477)-  , (4.0502102843199079, 1.2702685434611069581)-  , (2.2041235084734976, 0.54665320805956585382)-  , (0.9498749870396368, -0.66283138696545962354)-  , (5.5020466797149687, 1.6115010556650317675)-  , (1.8741725410778542, 0.33826100356492333487)-  , (14.1730624058772161, 2.6156503142962224118)-  , (1.0704026637921555, -0.46701211139417769802)++-- (x, p, q, exact)+tableIncompleteBetaP3000 :: [(Double,Double,Double,Double)]+tableIncompleteBetaP3000 =+  [ (0.495,  3001,  3001, 0.2192546757957825068677527085659175689142653854877723)+  , (0.501,  3001,  3001, 0.5615652382981522803424365187631195161665429270531389)+  , (0.531,  3500,  3200, 0.9209758089734407825580172472327758548870610822321278)+  , (0.501, 13500, 13200, 0.0656209987264794057358373443387716674955276089622780)   ]
− tests/Tests/SpecFunctions_flymake.hs 
@@ -1,206 +0,0 @@-{-# LANGUAGE ViewPatterns #-}--- | Tests for Statistics.Math-module Tests.SpecFunctions (-  tests-  ) where--import qualified Data.Vector as V-import           Data.Vector   ((!))--import Test.QuickCheck  hiding (choose)-import Test.Framework-import Test.Framework.Providers.QuickCheck2--import Tests.Helpers-import Tests.SpecFunctions.Tables-import Numeric.SpecFunctions---tests :: Test-tests = testGroup "Special functions"-  [ testProperty "Gamma(x+1) = x*Gamma(x) [logGamma]"  $ gammaReccurence logGamma  3e-8-  , testProperty "Gamma(x+1) = x*Gamma(x) [logGammaL]" $ gammaReccurence logGammaL 2e-13-  , testProperty "gamma(1,x) = 1 - exp(-x)"      $ incompleteGammaAt1Check-  , testProperty "0 <= gamma <= 1"               $ incompleteGammaInRange-  , testProperty "0 <= I[B] <= 1"            $ incompleteBetaInRange-  -- XXX FIXME DISABLED due to failures-  -- , testProperty "invIncompleteGamma = gamma^-1" $ invIGammaIsInverse-  -- , testProperty "invIncompleteBeta  = B^-1" $ invIBetaIsInverse-  -- , testProperty "gamma - increases"             $-  --     \s x y -> s > 0 && x > 0 && y > 0 ==> monotonicallyIncreases (incompleteGamma s) x y-  , testProperty "invErfc = erfc^-1"         $ invErfcIsInverse-  , testProperty "invErf  = erf^-1"          $ invErfIsInverse-    -- Unit tests-  , testAssertion "Factorial is expected to be precise at 1e-15 level"-      $ and [ eq 1e-15 (factorial (fromIntegral n :: Int))-                       (fromIntegral (factorial' n))-            |n <- [0..170]]-  , testAssertion "Log factorial is expected to be precise at 1e-15 level"-      $ and [ eq 1e-15 (logFactorial (fromIntegral n :: Int))-                       (log $ fromIntegral $ factorial' n)-            | n <- [2..170]]-  , testAssertion "logGamma is expected to be precise at 1e-9 level [integer points]"-      $ and [ eq 1e-9 (logGamma (fromIntegral n))-                      (logFactorial (n-1))-            | n <- [3..10000::Int]]-  , testAssertion "logGamma is expected to be precise at 1e-9 level [fractional points]"-      $ and [ eq 1e-9 (logGamma x) lg | (x,lg) <- tableLogGamma ]-  , testAssertion "logGammaL is expected to be precise at 1e-15 level"-      $ and [ eq 1e-15 (logGammaL (fromIntegral n))-                       (logFactorial (n-1))-            | n <- [3..10000::Int]]-    -- FIXME: Too low!-  , testAssertion "logGammaL is expected to be precise at 1e-10 level [fractional points]"-      $ and [ eq 1e-10 (logGammaL x) lg | (x,lg) <- tableLogGamma ]-    -- FIXME: loss of precision when logBeta p q ≈ 0.-    --        Relative error doesn't work properly in this case.-  , testAssertion "logBeta is expected to be precise at 1e-6 level"-      $ and [ eq 1e-6 (logBeta p q)-                      (logGammaL p + logGammaL q - logGammaL (p+q))-            | p <- [0.1,0.2 .. 0.9] ++ [2 .. 20]-            , q <- [0.1,0.2 .. 0.9] ++ [2 .. 20]-            ]-  , testAssertion "digamma is expected to be precise at 1e-14 [integers]"-      $ digammaTestIntegers 1e-14-    -- Relative precision is lost when digamma(x) ≈ 0-  , testAssertion "digamma is expected to be precise at 1e-12"-      $ and [ eq 1e-12 r (digamma x) | (x,r) <- tableDigamma ]-    -- FIXME: Why 1e-8? Is it due to poor precision of logBeta?-  , testAssertion "incompleteBeta is expected to be precise at 1e-8 level"-      $ and [ eq 1e-8 (incompleteBeta p q x) ib | (p,q,x,ib) <- tableIncompleteBeta ]-  , testAssertion "incompleteBeta with p > 3000 and q > 3000"-      $ and [ eq 1e-11 (incompleteBeta p q x) ib | (x,p,q,ib) <--                 [ (0.495,  3001,  3001, 0.2192546757957825068677527085659175689142653854877723)-                 , (0.501,  3001,  3001, 0.5615652382981522803424365187631195161665429270531389)-                 , (0.531,  3500,  3200, 0.9209758089734407825580172472327758548870610822321278)-                 , (0.501, 13500, 13200, 0.0656209987264794057358373443387716674955276089622780)-                 ]-            ]-  , testAssertion "choose is expected to precise at 1e-12 level"-      $ and [ eq 1e-12 (choose (fromIntegral n) (fromIntegral k)) (fromIntegral $ choose' n k)-            | n <- [0..300], k <- [0..n]]-    -----------------------------------------------------------------    -- Self tests-  , testProperty "Self-test: 0 <= range01 <= 1" $ \x -> let f = range01 x in f <= 1 && f >= 0-  ]--------------------------------------------------------------------- QC tests--------------------------------------------------------------------- Γ(x+1) = x·Γ(x)-gammaReccurence :: (Double -> Double) -> Double -> Double -> Property-gammaReccurence logG ε x =-  (x > 0 && x < 100)  ==>  (abs (g2 - g1 - log x) < ε)-    where-      g1 = logG x-      g2 = logG (x+1)---- γ(s,x) is in [0,1] range-incompleteGammaInRange :: Double -> Double -> Property-incompleteGammaInRange (abs -> s) (abs -> x) =-  x >= 0 && s > 0  ==> let i = incompleteGamma s x in i >= 0 && i <= 1---- γ(1,x) = 1 - exp(-x)--- Since Γ(1) = 1 normalization doesn't make any difference-incompleteGammaAt1Check :: Double -> Property-incompleteGammaAt1Check (abs -> x) =-  x > 0 ==> (incompleteGamma 1 x + exp(-x)) ≈ 1-  where-    (≈) = eq 1e-13---- invIncompleteGamma is inverse of incompleteGamma-invIGammaIsInverse :: Double -> Double -> Property-invIGammaIsInverse (abs -> a) (range01 -> p) =-  a > 0 && p > 0 && p < 1  ==> ( printTestCase ("a  = " ++ show a )-                               $ printTestCase ("p  = " ++ show p )-                               $ printTestCase ("x  = " ++ show x )-                               $ printTestCase ("p' = " ++ show p')-                               $ printTestCase ("Δp = " ++ show (p - p'))-                               $ abs (p - p') <= 1e-12-                               )-  where-    x  = invIncompleteGamma a p-    p' = incompleteGamma    a x---- invErfc is inverse of erfc-invErfcIsInverse :: Double -> Property-invErfcIsInverse ((*2) . range01 -> p)-  = printTestCase ("p  = " ++ show p )-  $ printTestCase ("x  = " ++ show x )-  $ printTestCase ("p' = " ++ show p')-  $ abs (p - p') <= 1e-14-  where-    x  = invErfc p-    p' = erfc x---- invErf is inverse of erf-invErfIsInverse :: Double -> Property-invErfIsInverse a-  = printTestCase ("p  = " ++ show p )-  $ printTestCase ("x  = " ++ show x )-  $ printTestCase ("p' = " ++ show p')-  $ abs (p - p') <= 1e-14-  where-    x  = invErf p-    p' = erf x-    p  | a < 0     = - range01 a-       | otherwise =   range01 a---- B(s,x) is in [0,1] range-incompleteBetaInRange :: Double -> Double -> Double -> Property-incompleteBetaInRange (abs -> p) (abs -> q) (range01 -> x) =-  p > 0 && q > 0  ==> let i = incompleteBeta p q x in i >= 0 && i <= 1---- invIncompleteBeta is inverse of incompleteBeta-invIBetaIsInverse :: Double -> Double -> Double -> Property-invIBetaIsInverse (abs -> p) (abs -> q) (range01 -> x) =-  p > 0 && q > 0  ==> ( printTestCase ("p   = " ++ show p )-                      $ printTestCase ("q   = " ++ show q )-                      $ printTestCase ("x   = " ++ show x )-                      $ printTestCase ("x'  = " ++ show x')-                      $ printTestCase ("a   = " ++ show a)-                      $ printTestCase ("err = " ++ (show $ abs $ (x - x') / x))-                      $ abs (x - x') <= 1e-12-                      )-  where-    x' = incompleteBeta    p q a-    a  = invIncompleteBeta p q x---- Table for digamma function:------ Uses equality ψ(n) = H_{n-1} - γ where---   H_{n} = Σ 1/k, k = [1 .. n]     - harmonic number---   γ     = 0.57721566490153286060  - Euler-Mascheroni number-digammaTestIntegers :: Double -> Bool-digammaTestIntegers eps-  = all (uncurry $ eq eps) $ take 3000 digammaInt-  where-    ok approx exact = approx-    -- Harmonic numbers starting from 0-    harmN = scanl (\a n -> a + 1/n) 0 [1::Rational .. ]-    gam   = 0.57721566490153286060-    -- Digamma values-    digammaInt = zipWith (\i h -> (digamma i, realToFrac h - gam)) [1..] harmN---------------------------------------------------------------------- Unit tests--------------------------------------------------------------------- Lookup table for fact factorial calculation. It has fixed size--- which is bad but it's OK for this particular case-factorial_table :: V.Vector Integer-factorial_table = V.generate 2000 (\n -> product [1..fromIntegral n])---- Exact implementation of factorial-factorial' :: Integer -> Integer-factorial' n = factorial_table ! fromIntegral n---- Exact albeit slow implementation of choose-choose' :: Integer -> Integer -> Integer-choose' n k = factorial' n `div` (factorial' k * factorial' (n-k))---- Truncate double to [0,1]-range01 :: Double -> Double-range01 = abs . (snd :: (Integer, Double) -> Double) . properFraction
tests/Tests/Sum.hs view
@@ -4,54 +4,68 @@  import Control.Applicative ((<$>)) import Numeric.Sum as Sum+import Numeric.MathFunctions.Comparison import Prelude hiding (sum)-import Test.Framework (Test, testGroup)-import Test.Framework.Providers.QuickCheck2 (testProperty)+import Test.Tasty (TestTree, testGroup)+import Test.Tasty.QuickCheck import Test.QuickCheck (Arbitrary(..)) import qualified Prelude -t_sum :: ([Double] -> Double) -> [Double] -> Bool-t_sum f xs = f xs == trueSum xs+-- Test that summation result is same as exact sum. That should pass+-- if we're effectively working with quad precision+t_sum :: ([Double] -> Double) -> [Double] -> Property+t_sum f xs+  = counterexample ("APPROX = " ++ show approx)+  $ counterexample ("EXACT  = " ++ show exact)+  $ counterexample ("DELTA  = " ++ show (approx - exact))+  $ counterexample ("ULPS   = " ++ show (ulpDistance approx exact))+  $ approx == exact+  where+    approx = f xs+    exact  = trueSum xs -t_sum_error :: ([Double] -> Double) -> [Double] -> Bool-t_sum_error f xs = abs (ts - f xs) <= abs (ts - Prelude.sum xs)-  where ts = trueSum xs+-- Test that summation has smaller error than naive summation or no+-- worse than given number of ulps. If we're close enough to exact+-- answer naive may get ahead+t_sum_error :: ([Double] -> Double) -> [Double] -> Property+t_sum_error f xs+  = counterexample ("APPROX  = " ++ show approx)+  $ counterexample ("NAIVE   = " ++ show naive)+  $ counterexample ("EXACT   = " ++ show exact)+  $ counterexample ("A-EXACT = " ++ show (approx - exact))+  $ counterexample ("N-EXACT = " ++ show (naive  - exact))+  $ counterexample ("ULPS[A] = " ++ show (ulpDistance approx exact))+  $ counterexample ("ULPS[N] = " ++ show (ulpDistance naive  exact))+  $ abs (exact - approx) <= abs (exact - naive)+  where+    naive  = Prelude.sum xs+    approx = f xs+    exact  = trueSum xs -t_sum_shifted :: ([Double] -> Double) -> [Double] -> Bool+t_sum_shifted :: ([Double] -> Double) -> [Double] -> Property t_sum_shifted f = t_sum_error f . zipWith (+) badvec  trueSum :: (Fractional b, Real a) => [a] -> b trueSum xs = fromRational . Prelude.sum . map toRational $ xs  badvec :: [Double]-badvec = cycle [1,1e16,-1e16]--tests :: Test-tests = testGroup "Summation" [-    testGroup "ID" [-      -- plain summation loses precision quickly-      -- testProperty "t_sum" $ t_sum (sum id)--      -- tautological tests:-      -- testProperty "t_sum_error" $ t_sum_error (sum id)-      -- testProperty "t_sum_shifted" $ t_sum_shifted (sum id)-    ]-  , testGroup "Kahan" [-      -- tests that cannot pass:-      -- testProperty "t_sum" $ t_sum (sum kahan)-      -- testProperty "t_sum_error" $ t_sum_error (sum kahan)+badvec = cycle [1, 1e14, -1e14] -      -- kahan summation only beats normal summation with large values+tests :: TestTree+tests = testGroup "Summation"+  [ testGroup "Kahan" [+      -- Kahan summation only beats naive summation when truly+      -- catastrophic cancellation occurs       testProperty "t_sum_shifted" $ t_sum_shifted (sum kahan)     ]   , testGroup "KBN" [-      testProperty "t_sum" $ t_sum (sum kbn)-    , testProperty "t_sum_error" $ t_sum_error (sum kbn)+      testProperty "t_sum"         $ t_sum         (sum kbn)+    , testProperty "t_sum_error"   $ t_sum_error   (sum kbn)     , testProperty "t_sum_shifted" $ t_sum_shifted (sum kbn)     ]   , testGroup "KB2" [-      testProperty "t_sum" $ t_sum (sum kb2)-    , testProperty "t_sum_error" $ t_sum_error (sum kb2)+      testProperty "t_sum"         $ t_sum         (sum kb2)+    , testProperty "t_sum_error"   $ t_sum_error   (sum kb2)     , testProperty "t_sum_shifted" $ t_sum_shifted (sum kb2)     ]   ]
+ tests/tables/digamma.dat view
@@ -0,0 +1,100 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+ tests/tables/generate.py view
@@ -0,0 +1,102 @@+#!/usr/bin/env python3+"""+"""+import itertools+import re+import mpmath+import sys++# We want to require quad precision (it never hurts to have some+# buffer for errors+mpmath.mp.dps = 30++def skip_comments(gen):+    "Skip comments and empty lines"+    for s in gen:+        s = s.strip()+        if not (s == '' or s[0] == '#'):+            yield s++def tokenize_stream(gen):+    "Extremeli ugly code which splits code on separator"+    nm  = None+    acc = None+    for s in gen:+        r = re.match(r'^([a-zA-Z]+(, *[a-zA-Z]+)*) *= *', s)+        if r:+            if acc is not None:+                for _ in nm:+                    yield(acc)+            nm  = r.groups()[0].split(',')+            acc = []+        elif acc is None:+            raise Exception("Invalid format")+        else:+            acc.append(float(s))+    for _ in nm:+        yield(acc)++++def load_inputs(path):+    "Load inputs for function from tsv file"+    with open(path) as f:+        for s in skip_comments(f):+            yield tuple(map(float, s.split()))+++def load_inputs_cartesian(path):+    "Load inputs for several variables where we want to generate all pair"+    with open(path) as f:+        for x in itertools.product(*tokenize_stream(skip_comments(f))):+            yield x++def fmt(f, inputs, out):+    # Here we set up output. We hope that refcounting will collect fds+    # promptly+    if out is None:+        out = sys.stdout+    else:+        out = open(out, 'w')+    for xs in inputs:+        param = ["%.18g" % x for x in xs]+        sOut  = mpmath.nstr( f(*xs), mpmath.mp.dps )+        print( '\t'.join( param + [sOut]),+               file=out)       +++## ================================================================++fmt( mpmath.erf,+     load_inputs('inputs/erf.dat'),+     'erf.dat')+fmt( mpmath.erf,+     load_inputs('inputs/erf.dat'),+     'erf.dat')+fmt( mpmath.erfc,+     load_inputs('inputs/erfc.dat'),+     'erfc.dat')+fmt( mpmath.erfc,+     load_inputs('inputs/erfc-large.dat'),+     'erfc-large.dat')+fmt( mpmath.loggamma,+     load_inputs('inputs/loggamma.dat'),+     'loggamma.dat')+fmt( mpmath.digamma,+     load_inputs('inputs/digamma.dat'),+     'digamma.dat')+fmt( mpmath.expm1,+     load_inputs('inputs/expm1.dat'),+     'expm1.dat')+fmt( mpmath.log1p,+     load_inputs('inputs/log1p.dat'),+     'log1p.dat')+fmt( lambda x: mpmath.log(mpmath.factorial(x)),+     map(lambda x: (x,), range(0, 20000)),+     'factorial.dat')+fmt( lambda a, x: mpmath.gammainc(z=a, a=0, b=x, regularized=True),+     load_inputs_cartesian('inputs/igamma.dat'),+     'igamma.dat')+fmt( lambda p, q: mpmath.log(mpmath.beta(p,q)),+     load_inputs_cartesian('inputs/logbeta.dat'),+     'logbeta.dat')
+ tests/tables/igamma.dat view
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+ tests/tables/inputs/digamma.dat view
@@ -0,0 +1,100 @@+0.1972018552655726+0.2766365979680845+0.3352840110772935+0.5241560861477529+0.8108729056729371+0.8289440927562556+0.9070101446062873+0.9385094944630431+0.9498749870396368+1.0704026637921555+1.1608524325026863+1.1815938184339669+1.2328105814385013+1.2786090750546355+1.4222181352589696+1.4654265058258678+1.5107587188614726+1.7232417075599473+1.8741725410778542+1.9482800424522431+2.0717397641759350+2.1964374279534344+2.2041235084734976+2.2101710538553756+2.2914619096171260+2.4073661551765566+2.4921048837305193+2.5037441860153118+2.7573211981404855+2.8590898311999715+2.9310065630306474+3.1362387322050900+3.2500957742991048+3.3467578222470555+3.3716588961200955+3.3932538441985769+3.4369121607821915+3.4679213262860156+3.4716258279958572+3.6989463639934752+3.7348491113356177+3.8933394033155189+4.0502102843199079+4.1320729121597584+4.1956416643620571+4.4741465342999014+5.3213868642873896+5.5020466797149687+5.6622605630542511+5.8655552400886410+5.9700184459319399+6.4894496431749733+6.5709053027851487+7.2023118361897769+7.3600347508772019+7.3868205159465790+7.5321882978878660+7.9098387372709587+8.6720493874148570+9.1173553049782452+9.2578640399661225+9.3081256750537182+9.4381079039564071+9.9662376350778192+10.0261172557341425+10.6478950333943825+10.6498308581562604+10.6750266252851169+10.6883248506773985+10.6908820268137070+10.7410508007627694+10.9702870318273966+11.2357450115053670+11.4168205413938768+12.2373583939228876+13.1568457441413429+13.6688064138713390+14.0357118427322334+14.1730624058772161+14.3373372205836365+14.9455957898231535+15.4401781010745509+16.3105956379859052+16.7950471612825254+17.1377048621110468+18.8934063317711676+19.7457045917841398+20.3998669454332315+20.4905634096258815+20.5303177686997920+21.4416006516504787+21.8943170241258827+24.1273989930028883+24.7013029455164919+27.9448247140513644+28.5703089405901878+29.5024809785193902+30.4956621109554185+32.2002179941400826+47.6946291432301663
+ tests/tables/inputs/erf.dat view
@@ -0,0 +1,68 @@+-3.000000+-2.500000+-2.400000+-2.300000+-2.200000+-2.100000+-2.000000+-1.900000+-1.800000+-1.700000+-1.600000+-1.500000+-1.400000+-1.300000+-1.200000+-1.100000+-1.000000+-0.900000+-0.800000+-0.700000+-0.600000+-0.500000+-0.400000+-0.300000+-0.200000+-0.100000+-0.080000+-0.060000+-0.040000+-0.020000+0.000000+1e-9+1e-8+1e-7+1e-6+1e-5+1e-4+1e-3+0.020000+0.040000+0.060000+0.080000+0.100000+0.200000+0.300000+0.400000+0.500000+0.600000+0.700000+0.800000+0.900000+1.000000+1.100000+1.200000+1.300000+1.400000+1.500000+1.600000+1.700000+1.800000+1.900000+2.000000+2.100000+2.200000+2.300000+2.400000+2.500000+3.000000
+ tests/tables/inputs/erfc-large.dat view
@@ -0,0 +1,9 @@+2.100000+2.200000+2.300000+2.400000+2.500000+3.000000+3.500000+11.000000+23.000000
+ tests/tables/inputs/erfc.dat view
@@ -0,0 +1,56 @@+-3.000000+-2.500000+-2.400000+-2.300000+-2.200000+-2.100000+-2.000000+-1.900000+-1.800000+-1.700000+-1.600000+-1.500000+-1.400000+-1.300000+-1.200000+-1.100000+-1.000000+-0.900000+-0.800000+-0.700000+-0.600000+-0.500000+-0.400000+-0.300000+-0.200000+-0.100000+-0.080000+-0.060000+-0.040000+-0.020000+0.000000+0.020000+0.040000+0.060000+0.080000+0.100000+0.200000+0.300000+0.400000+0.500000+0.600000+0.700000+0.800000+0.900000+1.000000+1.100000+1.200000+1.300000+1.400000+1.500000+1.600000+1.700000+1.800000+1.900000+2.000000+2.0009765625
+ tests/tables/inputs/expm1.dat view
@@ -0,0 +1,13 @@+-1e-1+-1e-2+-1e-3+-1e-4+-1e-5+-1e-6+0+1e-6+1e-5+1e-4+1e-3+1e-2+1e-1
+ tests/tables/inputs/igamma.dat view
@@ -0,0 +1,31 @@+a =+1e-4+1e-3+0.01+0.1+0.2+0.3+0.4+0.5+0.6+2+3+6+12+18+101+201+1000+3003++x =+1e-6+1e-5+1e-3+1e-2+1e-1+1+10+100+1000+3301
+ tests/tables/inputs/log1p.dat view
@@ -0,0 +1,13 @@+-1e-1+-1e-2+-1e-3+-1e-4+-1e-5+-1e-6+0+1e-6+1e-5+1e-4+1e-3+1e-2+1e-1
+ tests/tables/inputs/logbeta.dat view
@@ -0,0 +1,29 @@+p,q =+0.1+0.2+0.30000000000000004+0.4+0.5+0.6+0.7000000000000001+0.8+0.9+2.0+3.0+4.0+5.0+6.0+7.0+8.0+9.0+10.0+11.0+12.0+13.0+14.0+15.0+16.0+17.0+18.0+19.0+20.0
+ tests/tables/inputs/loggamma.dat view
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+ tests/tables/log1p.dat view
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+ tests/tables/logbeta.dat view
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+ tests/tables/loggamma.dat view
@@ -0,0 +1,43 @@+1.2500000000000001e-06	13.5923662851317682435796016512+6.82000000000000043e-05	9.59302663083187580782892179804+0.000246000000000000016	8.31003707674479645971159589555+0.000880000000000000031	7.03508133735248536292633538786+0.00311999999999999994	5.76812935836556732325877954057+0.0267000000000000015	3.60825889188929751923626846281+0.0777000000000000052	2.51483718587682339914549034673+0.234000000000000014	1.35795575594327598998191258078+0.859999999999999987	0.0981465780276855927795120251869+1.34000000000000008	-0.114047575572077578623579479668+1.8899999999999999	-0.0425116422978701401405661729629+2.45000000000000018	0.250142965692176235761147104829+3	0.693147180559945309417232121458+3.64999999999999991	1.37010419973806862861634409045+4	1.79175946922805500081247735838+4.55999999999999961	2.53751433179495797698708606823+5	3.1780538303479456196469416013+6	4.78749174278204599424770093452+6.66000000000000014	5.95153772695502089846022734702+7	6.5792512120101009950601782929+8	8.52516136106541430016553103635+8.25	9.03318691960512285327455704258+10	12.8018274800814696112077178746+11.3000000000000007	15.8141806813739487483883484684+25.6000000000000014	56.7112615983281240988282625152+33	81.557959456115037178502968666+50.3999999999999986	146.128151587021650625708161324+77	256.221135550009525456082846319+123.299999999999997	468.855000758975584910395998022+222	975.612353993036060800198673231+487.399999999999977	2526.98466475437276484523752546+555	2949.78690775573651020653356334+853.399999999999977	4903.93591359782221336869490715+888	5138.13145428259548143327765439+2923.30000000000018	20402.9319893870607273552436914+3333	23699.9195344962473524769527089+8764.29999999999927	70798.268343590118940583038718+12630	106641.772649825079076675672666+34500	325976.34838781821769124632363+82340	849629.79603036717685116333246+234800	2668846.43905079586519344403509+834300	10540830.9125575340577530715314+1230000	16017699.3223150155655909214342
+ tests/tables/shell.nix view
@@ -0,0 +1,10 @@+let+  pkgs = import <nixpkgs> {};+  pyp  = pkgs.python3.withPackages (py: with py;+    [ ipython+      mpmath+    ]);+in+pkgs.mkShell {+  buildInputs = [ pyp ];+}
tests/tests.hs view
@@ -1,13 +1,16 @@-import Test.Framework       (defaultMain)-import qualified Tests.SpecFunctions+import Test.Tasty       (defaultMain,testGroup) import qualified Tests.Chebyshev-import qualified Tests.Sum import qualified Tests.Comparison+import qualified Tests.RootFinding+import qualified Tests.SpecFunctions+import qualified Tests.Sum  main :: IO ()-main = defaultMain [ Tests.SpecFunctions.tests-                   -- FIXME: tests for chebyshev polynomials fail intermittently-                   -- , Tests.Chebyshev.tests-                   , Tests.Sum.tests-                   , Tests.Comparison.tests-                   ]+main = defaultMain $ testGroup "math-functions"+  [ Tests.SpecFunctions.tests+  -- FIXME: tests for chebyshev polynomials fail intermittently+  -- , Tests.Chebyshev.tests+  , Tests.Sum.tests+  , Tests.Comparison.tests+  , Tests.RootFinding.tests+  ]
− tests/view.hs 
@@ -1,102 +0,0 @@-{-# LANGUAGE OverloadedStrings #-}-import Control.Applicative-import Control.Monad-import Numeric.SpecFunctions-import Numeric.MathFunctions.Constants-import CPython.Sugar-import CPython.MPMath-import qualified CPython as Py--import HEP.ROOT.Plot----------------------------------------------------------------------viewBetaDelta = runPy $ do-  addToPythonPath "."-  m  <- loadMPMath-  mpmSetDps m 100-  xs <- forM pqBeta $ \(p,q) -> do x <- fromMPNum =<< mpmLog m =<< mpmBeta m (MPDouble p) (MPDouble q)-                                   return (p,q, relErr x (logBeta p q))-  draws $ do-    -- let xs = [ (p,q, logBeta p q `relErr` (logGammaL p + logGammaL q - logGammaL (q+p)))-    --          | (p,q) <- pqBeta-    --          ]-    add $ Graph2D xs---pqBeta = [ (p,q)-         | p <- logRange 50 0.3 0.6-         , q <- logRange 50 5 6-         ]-  where-----viewIBeta x = runPy $ do-  addToPythonPath "."-  m <- loadMPMath-  mpmSetDps m 30-  ---  let n  = 40-  let pq =  (,)-        <$> logRange n 100 1000-        <*> logRange n 100 1000-  ---  xs <- forM pq $ \(p,q) -> do-          i <- fromMPNum =<< mpmIncompleteBeta m (MPDouble p) (MPDouble q) (MPDouble x)-          return (p,q, incompleteBeta p q x `relErr` i)-  ---  draws $ do-    add $ Graph2D xs---go = runPy $ do-  addToPythonPath "."-  m <- loadMPMath-  mpmSetDps m 16-  ---  print =<< fromMPNum =<< mpmIncompleteBeta m (MPDouble 10) (MPDouble 10) (MPDouble 0.4)-  print $ incompleteBeta 10 10 0.4-----viewLancrox = runPy $ do-  addToPythonPath "."-  m <- loadMPMath-  mpmSetDps m 50-  ---  let xs = logRange 10000 (1e-8) (1e-1)-  pl <- forM xs $ \x -> do y0 <- fromMPNum =<< mpmLog m =<< mpmGamma m (MPDouble x)-                           return (x, y0)-  draws $ do-    add $ Graph $ [ (x, abs $ y `relErr` logGammaL x) | (x,y) <- pl ]-    set $ lineColor RED-    ---    add $ Graph $ [ (x, abs $ y `relErr` logGamma x) | (x,y) <- pl ]-    set $ lineColor BLUE-    ---    set $ xaxis $ logScale ON-    -- set $ yaxis $ logScale ON-    ---    add $ HLine m_epsilon-    add $ HLine $ negate m_epsilon---------------------------------------------------------------------relErr :: Double -> Double -> Double-relErr 0 0 = 0-relErr x y = (x - y) / max (abs x) (abs y)----logRange :: Int -> Double -> Double -> [Double]-logRange n a b-  = [ a * r^i | i <- [0 .. n] ]-  where-    r = (b / a) ** (1 / fromIntegral n)-