sbv-8.14: Data/SBV/Core/Floating.hs
-----------------------------------------------------------------------------
-- |
-- Module : Data.SBV.Core.Floating
-- Copyright : (c) Levent Erkok
-- License : BSD3
-- Maintainer: erkokl@gmail.com
-- Stability : experimental
--
-- Implementation of floating-point operations mapping to SMT-Lib2 floats
-----------------------------------------------------------------------------
{-# LANGUAGE DefaultSignatures #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE Rank2Types #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE UndecidableInstances #-}
{-# OPTIONS_GHC -Wall -Werror -fno-warn-orphans #-}
module Data.SBV.Core.Floating (
IEEEFloating(..), IEEEFloatConvertible(..)
, sFloatAsSWord32, sDoubleAsSWord64, sFloatingPointAsSWord
, sWord32AsSFloat, sWord64AsSDouble, sWordAsSFloatingPoint
, blastSFloat, blastSDouble, blastSFloatingPoint
, sFloatAsComparableSWord32, sDoubleAsComparableSWord64, sFloatingPointAsComparableSWord
, sComparableSWord32AsSFloat, sComparableSWord64AsSDouble, sComparableSWordAsSFloatingPoint
) where
import Data.Bits (testBit)
import Data.Int (Int8, Int16, Int32, Int64)
import Data.Word (Word8, Word16, Word32, Word64)
import Data.Proxy
import Data.SBV.Core.AlgReals (isExactRational)
import Data.SBV.Core.Sized
import Data.SBV.Core.SizedFloats
import Data.SBV.Core.Data
import Data.SBV.Core.Kind
import Data.SBV.Core.Model
import Data.SBV.Core.Symbolic (addSValOptGoal)
import Data.SBV.Utils.Numeric
import Data.Ratio
import GHC.TypeLits
import LibBF
import Data.SBV.Core.Operations
-- For doctest use only
--
-- $setup
-- >>> :set -XTypeApplications
-- >>> :set -XRankNTypes
-- >>> :set -XScopedTypeVariables
-- >>> import Data.SBV.Provers.Prover (prove)
-- | A class of floating-point (IEEE754) operations, some of
-- which behave differently based on rounding modes. Note that unless
-- the rounding mode is concretely RoundNearestTiesToEven, we will
-- not concretely evaluate these, but rather pass down to the SMT solver.
class (SymVal a, RealFloat a) => IEEEFloating a where
-- | Compute the floating point absolute value.
fpAbs :: SBV a -> SBV a
-- | Compute the unary negation. Note that @0 - x@ is not equivalent to @-x@ for floating-point, since @-0@ and @0@ are different.
fpNeg :: SBV a -> SBV a
-- | Add two floating point values, using the given rounding mode
fpAdd :: SRoundingMode -> SBV a -> SBV a -> SBV a
-- | Subtract two floating point values, using the given rounding mode
fpSub :: SRoundingMode -> SBV a -> SBV a -> SBV a
-- | Multiply two floating point values, using the given rounding mode
fpMul :: SRoundingMode -> SBV a -> SBV a -> SBV a
-- | Divide two floating point values, using the given rounding mode
fpDiv :: SRoundingMode -> SBV a -> SBV a -> SBV a
-- | Fused-multiply-add three floating point values, using the given rounding mode. @fpFMA x y z = x*y+z@ but with only
-- one rounding done for the whole operation; not two. Note that we will never concretely evaluate this function since
-- Haskell lacks an FMA implementation.
fpFMA :: SRoundingMode -> SBV a -> SBV a -> SBV a -> SBV a
-- | Compute the square-root of a float, using the given rounding mode
fpSqrt :: SRoundingMode -> SBV a -> SBV a
-- | Compute the remainder: @x - y * n@, where @n@ is the truncated integer nearest to x/y. The rounding mode
-- is implicitly assumed to be @RoundNearestTiesToEven@.
fpRem :: SBV a -> SBV a -> SBV a
-- | Round to the nearest integral value, using the given rounding mode.
fpRoundToIntegral :: SRoundingMode -> SBV a -> SBV a
-- | Compute the minimum of two floats, respects @infinity@ and @NaN@ values
fpMin :: SBV a -> SBV a -> SBV a
-- | Compute the maximum of two floats, respects @infinity@ and @NaN@ values
fpMax :: SBV a -> SBV a -> SBV a
-- | Are the two given floats exactly the same. That is, @NaN@ will compare equal to itself, @+0@ will /not/ compare
-- equal to @-0@ etc. This is the object level equality, as opposed to the semantic equality. (For the latter, just use '.=='.)
fpIsEqualObject :: SBV a -> SBV a -> SBool
-- | Is the floating-point number a normal value. (i.e., not denormalized.)
fpIsNormal :: SBV a -> SBool
-- | Is the floating-point number a subnormal value. (Also known as denormal.)
fpIsSubnormal :: SBV a -> SBool
-- | Is the floating-point number 0? (Note that both +0 and -0 will satisfy this predicate.)
fpIsZero :: SBV a -> SBool
-- | Is the floating-point number infinity? (Note that both +oo and -oo will satisfy this predicate.)
fpIsInfinite :: SBV a -> SBool
-- | Is the floating-point number a NaN value?
fpIsNaN :: SBV a -> SBool
-- | Is the floating-point number negative? Note that -0 satisfies this predicate but +0 does not.
fpIsNegative :: SBV a -> SBool
-- | Is the floating-point number positive? Note that +0 satisfies this predicate but -0 does not.
fpIsPositive :: SBV a -> SBool
-- | Is the floating point number -0?
fpIsNegativeZero :: SBV a -> SBool
-- | Is the floating point number +0?
fpIsPositiveZero :: SBV a -> SBool
-- | Is the floating-point number a regular floating point, i.e., not NaN, nor +oo, nor -oo. Normals or denormals are allowed.
fpIsPoint :: SBV a -> SBool
-- Default definitions. Minimal complete definition: None! All should be taken care by defaults
-- Note that we never evaluate FMA concretely, as there's no fma operator in Haskell
fpAbs = lift1 FP_Abs (Just abs) Nothing
fpNeg = lift1 FP_Neg (Just negate) Nothing
fpAdd = lift2 FP_Add (Just (+)) . Just
fpSub = lift2 FP_Sub (Just (-)) . Just
fpMul = lift2 FP_Mul (Just (*)) . Just
fpDiv = lift2 FP_Div (Just (/)) . Just
fpFMA = lift3 FP_FMA Nothing . Just
fpSqrt = lift1 FP_Sqrt (Just sqrt) . Just
fpRem = lift2 FP_Rem (Just fpRemH) Nothing
fpRoundToIntegral = lift1 FP_RoundToIntegral (Just fpRoundToIntegralH) . Just
fpMin = liftMM FP_Min (Just fpMinH) Nothing
fpMax = liftMM FP_Max (Just fpMaxH) Nothing
fpIsEqualObject = lift2B FP_ObjEqual (Just fpIsEqualObjectH) Nothing
fpIsNormal = lift1B FP_IsNormal fpIsNormalizedH
fpIsSubnormal = lift1B FP_IsSubnormal isDenormalized
fpIsZero = lift1B FP_IsZero (== 0)
fpIsInfinite = lift1B FP_IsInfinite isInfinite
fpIsNaN = lift1B FP_IsNaN isNaN
fpIsNegative = lift1B FP_IsNegative (\x -> x < 0 || isNegativeZero x)
fpIsPositive = lift1B FP_IsPositive (\x -> x >= 0 && not (isNegativeZero x))
fpIsNegativeZero x = fpIsZero x .&& fpIsNegative x
fpIsPositiveZero x = fpIsZero x .&& fpIsPositive x
fpIsPoint x = sNot (fpIsNaN x .|| fpIsInfinite x)
-- | SFloat instance
instance IEEEFloating Float
-- | SDouble instance
instance IEEEFloating Double
-- | Capture convertability from/to FloatingPoint representations.
--
-- Conversions to float: 'toSFloat' and 'toSDouble' simply return the
-- nearest representable float from the given type based on the rounding
-- mode provided. Similarly, 'toSFloatingPoint' converts to a generalized
-- floating-point number with specified exponent and significand bith widths.
--
-- Conversions from float: 'fromSFloat', 'fromSDouble', 'fromSFloatingPoint' functions do
-- the reverse conversion. However some care is needed when given values
-- that are not representable in the integral target domain. For instance,
-- converting an 'SFloat' to an 'SInt8' is problematic. The rules are as follows:
--
-- If the input value is a finite point and when rounded in the given rounding mode to an
-- integral value lies within the target bounds, then that result is returned.
-- (This is the regular interpretation of rounding in IEEE754.)
--
-- Otherwise (i.e., if the integral value in the float or double domain) doesn't
-- fit into the target type, then the result is unspecified. Note that if the input
-- is @+oo@, @-oo@, or @NaN@, then the result is unspecified.
--
-- Due to the unspecified nature of conversions, SBV will never constant fold
-- conversions from floats to integral values. That is, you will always get a
-- symbolic value as output. (Conversions from floats to other floats will be
-- constant folded. Conversions from integral values to floats will also be
-- constant folded.)
--
-- Note that unspecified really means unspecified: In particular, SBV makes
-- no guarantees about matching the behavior between what you might get in
-- Haskell, via SMT-Lib, or the C-translation. If the input value is out-of-bounds
-- as defined above, or is @NaN@ or @oo@ or @-oo@, then all bets are off. In particular
-- C and SMTLib are decidedly undefine this case, though that doesn't mean they do the
-- same thing! Same goes for Haskell, which seems to convert via Int64, but we do
-- not model that behavior in SBV as it doesn't seem to be intentional nor well documented.
--
-- You can check for @NaN@, @oo@ and @-oo@, using the predicates 'fpIsNaN', 'fpIsInfinite',
-- and 'fpIsPositive', 'fpIsNegative' predicates, respectively; and do the proper conversion
-- based on your needs. (0 is a good choice, as are min/max bounds of the target type.)
--
-- Currently, SBV provides no predicates to check if a value would lie within range for a
-- particular conversion task, as this depends on the rounding mode and the types involved
-- and can be rather tricky to determine. (See <http://github.com/LeventErkok/sbv/issues/456>
-- for a discussion of the issues involved.) In a future release, we hope to be able to
-- provide underflow and overflow predicates for these conversions as well.
class SymVal a => IEEEFloatConvertible a where
-- | Convert from an IEEE74 single precision float.
fromSFloat :: SRoundingMode -> SFloat -> SBV a
fromSFloat = genericFromFloat
-- | Convert to an IEEE-754 Single-precision float.
--
-- >>> :{
-- roundTrip :: forall a. (Eq a, IEEEFloatConvertible a) => SRoundingMode -> SBV a -> SBool
-- roundTrip m x = fromSFloat m (toSFloat m x) .== x
-- :}
--
-- >>> prove $ roundTrip @Int8
-- Q.E.D.
-- >>> prove $ roundTrip @Word8
-- Q.E.D.
-- >>> prove $ roundTrip @Int16
-- Q.E.D.
-- >>> prove $ roundTrip @Word16
-- Q.E.D.
-- >>> prove $ roundTrip @Int32
-- Falsifiable. Counter-example:
-- s0 = RoundTowardNegative :: RoundingMode
-- s1 = -44297675 :: Int32
--
-- Note how we get a failure on `Int32`. The counter-example value is not representable exactly as a single precision float:
--
-- >>> toRational (-44297675 :: Float)
-- (-44297676) % 1
--
-- Note how the numerator is different, it is off by 1. This is hardly surprising, since floats become sparser as
-- the magnitude increases to be able to cover all the integer values representable.
toSFloat :: SRoundingMode -> SBV a -> SFloat
-- default definition if we have an integral like
default toSFloat :: Integral a => SRoundingMode -> SBV a -> SFloat
toSFloat = genericToFloat (onlyWhenRNE (Just . fromRational . fromIntegral))
-- | Convert from an IEEE74 double precision float.
fromSDouble :: SRoundingMode -> SDouble -> SBV a
fromSDouble = genericFromFloat
-- | Convert to an IEEE-754 Double-precision float.
--
-- >>> :{
-- roundTrip :: forall a. (Eq a, IEEEFloatConvertible a) => SRoundingMode -> SBV a -> SBool
-- roundTrip m x = fromSDouble m (toSDouble m x) .== x
-- :}
--
-- >>> prove $ roundTrip @Int8
-- Q.E.D.
-- >>> prove $ roundTrip @Word8
-- Q.E.D.
-- >>> prove $ roundTrip @Int16
-- Q.E.D.
-- >>> prove $ roundTrip @Word16
-- Q.E.D.
-- >>> prove $ roundTrip @Int32
-- Q.E.D.
-- >>> prove $ roundTrip @Word32
-- Q.E.D.
-- >>> prove $ roundTrip @Int64
-- Falsifiable. Counter-example:
-- s0 = RoundNearestTiesToAway :: RoundingMode
-- s1 = -3871901667041025751 :: Int64
--
-- Just like in the `SFloat` case, once we reach 64-bits, we no longer can exactly represent the
-- integer value for all possible values:
--
-- >>> toRational (-3871901667041025751 :: Double)
-- (-3871901667041025536) % 1
--
-- In this case the numerator is off by 15.
toSDouble :: SRoundingMode -> SBV a -> SDouble
-- default definition if we have an integral like
default toSDouble :: Integral a => SRoundingMode -> SBV a -> SDouble
toSDouble = genericToFloat (onlyWhenRNE (Just . fromRational . fromIntegral))
-- | Convert from an arbitrary floating point.
fromSFloatingPoint :: ValidFloat eb sb => SRoundingMode -> SFloatingPoint eb sb -> SBV a
fromSFloatingPoint = genericFromFloat
-- | Convert to an arbitrary floating point.
toSFloatingPoint :: ValidFloat eb sb => SRoundingMode -> SBV a -> SFloatingPoint eb sb
-- -- default definition if we have an integral like
default toSFloatingPoint :: (Integral a, ValidFloat eb sb) => SRoundingMode -> SBV a -> SFloatingPoint eb sb
toSFloatingPoint = genericToFloat (const (Just . fromRational . fromIntegral))
-- Run the function if the conversion is in RNE. Otherwise return Nothing.
onlyWhenRNE :: (a -> Maybe b) -> RoundingMode -> a -> Maybe b
onlyWhenRNE f RoundNearestTiesToEven v = f v
onlyWhenRNE _ _ _ = Nothing
-- | A generic from-float converter. Note that this function does no constant folding since
-- it's behavior is undefined when the input float is out-of-bounds or not a point.
genericFromFloat :: forall a r. (IEEEFloating a, IEEEFloatConvertible r)
=> SRoundingMode -- Rounding mode
-> SBV a -- Input float/double
-> SBV r
genericFromFloat rm f = SBV (SVal kTo (Right (cache r)))
where kFrom = kindOf f
kTo = kindOf (Proxy @r)
r st = do msv <- sbvToSV st rm
xsv <- sbvToSV st f
newExpr st kTo (SBVApp (IEEEFP (FP_Cast kFrom kTo msv)) [xsv])
-- | A generic to-float converter, which will constant-fold as necessary, but only in the sRNE mode for regular floats.
genericToFloat :: forall a r. (IEEEFloatConvertible a, IEEEFloating r)
=> (RoundingMode -> a -> Maybe r) -- How to convert concretely, if possible
-> SRoundingMode -- Rounding mode
-> SBV a -- Input convertible
-> SBV r
genericToFloat converter rm i
| Just w <- unliteral i, Just crm <- unliteral rm, Just result <- converter crm w
= literal result
| True
= SBV (SVal kTo (Right (cache r)))
where kFrom = kindOf i
kTo = kindOf (Proxy @r)
r st = do msv <- sbvToSV st rm
xsv <- sbvToSV st i
newExpr st kTo (SBVApp (IEEEFP (FP_Cast kFrom kTo msv)) [xsv])
instance IEEEFloatConvertible Int8
instance IEEEFloatConvertible Int16
instance IEEEFloatConvertible Int32
instance IEEEFloatConvertible Int64
instance IEEEFloatConvertible Word8
instance IEEEFloatConvertible Word16
instance IEEEFloatConvertible Word32
instance IEEEFloatConvertible Word64
instance IEEEFloatConvertible Integer
-- For float and double, skip the conversion if the same and do the constant folding, unlike all others.
instance IEEEFloatConvertible Float where
toSFloat _ f = f
toSDouble = genericToFloat (onlyWhenRNE (Just . fp2fp))
toSFloatingPoint rm f = toSFloatingPoint rm $ toSDouble rm f
fromSFloat _ f = f
fromSDouble rm f
| Just RoundNearestTiesToEven <- unliteral rm
, Just fv <- unliteral f
= literal (fp2fp fv)
| True
= genericFromFloat rm f
instance IEEEFloatConvertible Double where
toSFloat = genericToFloat (onlyWhenRNE (Just . fp2fp))
toSDouble _ d = d
toSFloatingPoint rm sd
| Just d <- unliteral sd, Just brm <- rmToRM rm
= literal $ FloatingPoint $ FP ei si $ fst (bfRoundFloat (mkBFOpts ei si brm) (bfFromDouble d))
| True
= res
where (k, ei, si) = case kindOf res of
kr@(KFP eb sb) -> (kr, eb, sb)
kr -> error $ "Unexpected kind in toSFloatingPoint: " ++ show (kr, rm, sd)
res = SBV $ SVal k $ Right $ cache r
r st = do msv <- sbvToSV st rm
xsv <- sbvToSV st sd
newExpr st k (SBVApp (IEEEFP (FP_Cast KDouble k msv)) [xsv])
fromSDouble _ d = d
fromSFloat rm d
| Just RoundNearestTiesToEven <- unliteral rm
, Just dv <- unliteral d
= literal (fp2fp dv)
| True
= genericFromFloat rm d
convertWhenExactRational :: Fractional a => AlgReal -> Maybe a
convertWhenExactRational r
| isExactRational r = Just (fromRational (toRational r))
| True = Nothing
-- For AlgReal; be careful to only process exact rationals concretely
instance IEEEFloatConvertible AlgReal where
toSFloat = genericToFloat (onlyWhenRNE convertWhenExactRational)
toSDouble = genericToFloat (onlyWhenRNE convertWhenExactRational)
toSFloatingPoint = genericToFloat (const convertWhenExactRational)
-- Arbitrary floats can handle all rounding modes in concrete mode
instance ValidFloat eb sb => IEEEFloatConvertible (FloatingPoint eb sb) where
toSFloat rm i
| Just (FloatingPoint (FP _ _ v)) <- unliteral i, Just brm <- rmToRM rm
= literal $ fp2fp $ fst (bfToDouble brm (fst (bfRoundFloat (mkBFOpts ei si brm) v)))
| True
= genericToFloat (\_ _ -> Nothing) rm i
where ei = intOfProxy (Proxy @eb)
si = intOfProxy (Proxy @sb)
fromSFloat rm i
| Just f <- unliteral i, Just brm <- rmToRM rm
= literal $ FloatingPoint $ FP ei si $ fst (bfRoundFloat (mkBFOpts ei si brm) (bfFromDouble (fp2fp f :: Double)))
| True
= genericFromFloat rm i
where ei = intOfProxy (Proxy @eb)
si = intOfProxy (Proxy @sb)
toSDouble rm i
| Just (FloatingPoint (FP _ _ v)) <- unliteral i, Just brm <- rmToRM rm
= literal $ fst (bfToDouble brm (fst (bfRoundFloat (mkBFOpts ei si brm) v)))
| True
= genericToFloat (\_ _ -> Nothing) rm i
where ei = intOfProxy (Proxy @eb)
si = intOfProxy (Proxy @sb)
fromSDouble rm i
| Just f <- unliteral i, Just brm <- rmToRM rm
= literal $ FloatingPoint $ FP ei si $ fst (bfRoundFloat (mkBFOpts ei si brm) (bfFromDouble f))
| True
= genericFromFloat rm i
where ei = intOfProxy (Proxy @eb)
si = intOfProxy (Proxy @sb)
toSFloatingPoint rm i
| Just (FloatingPoint (FP _ _ v)) <- unliteral i, Just brm <- rmToRM rm
= literal $ FloatingPoint $ FP ei si $ fst (bfRoundFloat (mkBFOpts ei si brm) v)
| True
= genericToFloat (\_ _ -> Nothing) rm i
where ei = intOfProxy (Proxy @eb)
si = intOfProxy (Proxy @sb)
-- From and To are the same when the source is an arbitrary float!
fromSFloatingPoint = toSFloatingPoint
-- | Concretely evaluate one arg function, if rounding mode is RoundNearestTiesToEven and we have enough concrete data
concEval1 :: SymVal a => Maybe (a -> a) -> Maybe SRoundingMode -> SBV a -> Maybe (SBV a)
concEval1 mbOp mbRm a = do op <- mbOp
v <- unliteral a
case unliteral =<< mbRm of
Nothing -> (Just . literal) (op v)
Just RoundNearestTiesToEven -> (Just . literal) (op v)
_ -> Nothing
-- | Concretely evaluate two arg function, if rounding mode is RoundNearestTiesToEven and we have enough concrete data
concEval2 :: SymVal a => Maybe (a -> a -> a) -> Maybe SRoundingMode -> SBV a -> SBV a -> Maybe (SBV a)
concEval2 mbOp mbRm a b = do op <- mbOp
v1 <- unliteral a
v2 <- unliteral b
case unliteral =<< mbRm of
Nothing -> (Just . literal) (v1 `op` v2)
Just RoundNearestTiesToEven -> (Just . literal) (v1 `op` v2)
_ -> Nothing
-- | Concretely evaluate a bool producing two arg function, if rounding mode is RoundNearestTiesToEven and we have enough concrete data
concEval2B :: SymVal a => Maybe (a -> a -> Bool) -> Maybe SRoundingMode -> SBV a -> SBV a -> Maybe SBool
concEval2B mbOp mbRm a b = do op <- mbOp
v1 <- unliteral a
v2 <- unliteral b
case unliteral =<< mbRm of
Nothing -> (Just . literal) (v1 `op` v2)
Just RoundNearestTiesToEven -> (Just . literal) (v1 `op` v2)
_ -> Nothing
-- | Concretely evaluate two arg function, if rounding mode is RoundNearestTiesToEven and we have enough concrete data
concEval3 :: SymVal a => Maybe (a -> a -> a -> a) -> Maybe SRoundingMode -> SBV a -> SBV a -> SBV a -> Maybe (SBV a)
concEval3 mbOp mbRm a b c = do op <- mbOp
v1 <- unliteral a
v2 <- unliteral b
v3 <- unliteral c
case unliteral =<< mbRm of
Nothing -> (Just . literal) (op v1 v2 v3)
Just RoundNearestTiesToEven -> (Just . literal) (op v1 v2 v3)
_ -> Nothing
-- | Add the converted rounding mode if given as an argument
addRM :: State -> Maybe SRoundingMode -> [SV] -> IO [SV]
addRM _ Nothing as = return as
addRM st (Just rm) as = do svm <- sbvToSV st rm
return (svm : as)
-- | Lift a 1 arg FP-op
lift1 :: SymVal a => FPOp -> Maybe (a -> a) -> Maybe SRoundingMode -> SBV a -> SBV a
lift1 w mbOp mbRm a
| Just cv <- concEval1 mbOp mbRm a
= cv
| True
= SBV $ SVal k $ Right $ cache r
where k = kindOf a
r st = do sva <- sbvToSV st a
args <- addRM st mbRm [sva]
newExpr st k (SBVApp (IEEEFP w) args)
-- | Lift an FP predicate
lift1B :: SymVal a => FPOp -> (a -> Bool) -> SBV a -> SBool
lift1B w f a
| Just v <- unliteral a = literal $ f v
| True = SBV $ SVal KBool $ Right $ cache r
where r st = do sva <- sbvToSV st a
newExpr st KBool (SBVApp (IEEEFP w) [sva])
-- | Lift a 2 arg FP-op
lift2 :: SymVal a => FPOp -> Maybe (a -> a -> a) -> Maybe SRoundingMode -> SBV a -> SBV a -> SBV a
lift2 w mbOp mbRm a b
| Just cv <- concEval2 mbOp mbRm a b
= cv
| True
= SBV $ SVal k $ Right $ cache r
where k = kindOf a
r st = do sva <- sbvToSV st a
svb <- sbvToSV st b
args <- addRM st mbRm [sva, svb]
newExpr st k (SBVApp (IEEEFP w) args)
-- | Lift min/max: Note that we protect against constant folding if args are alternating sign 0's, since
-- SMTLib is deliberately nondeterministic in this case
liftMM :: (SymVal a, RealFloat a) => FPOp -> Maybe (a -> a -> a) -> Maybe SRoundingMode -> SBV a -> SBV a -> SBV a
liftMM w mbOp mbRm a b
| Just v1 <- unliteral a
, Just v2 <- unliteral b
, not ((isN0 v1 && isP0 v2) || (isP0 v1 && isN0 v2)) -- If not +0/-0 or -0/+0
, Just cv <- concEval2 mbOp mbRm a b
= cv
| True
= SBV $ SVal k $ Right $ cache r
where isN0 = isNegativeZero
isP0 x = x == 0 && not (isN0 x)
k = kindOf a
r st = do sva <- sbvToSV st a
svb <- sbvToSV st b
args <- addRM st mbRm [sva, svb]
newExpr st k (SBVApp (IEEEFP w) args)
-- | Lift a 2 arg FP-op, producing bool
lift2B :: SymVal a => FPOp -> Maybe (a -> a -> Bool) -> Maybe SRoundingMode -> SBV a -> SBV a -> SBool
lift2B w mbOp mbRm a b
| Just cv <- concEval2B mbOp mbRm a b
= cv
| True
= SBV $ SVal KBool $ Right $ cache r
where r st = do sva <- sbvToSV st a
svb <- sbvToSV st b
args <- addRM st mbRm [sva, svb]
newExpr st KBool (SBVApp (IEEEFP w) args)
-- | Lift a 3 arg FP-op
lift3 :: SymVal a => FPOp -> Maybe (a -> a -> a -> a) -> Maybe SRoundingMode -> SBV a -> SBV a -> SBV a -> SBV a
lift3 w mbOp mbRm a b c
| Just cv <- concEval3 mbOp mbRm a b c
= cv
| True
= SBV $ SVal k $ Right $ cache r
where k = kindOf a
r st = do sva <- sbvToSV st a
svb <- sbvToSV st b
svc <- sbvToSV st c
args <- addRM st mbRm [sva, svb, svc]
newExpr st k (SBVApp (IEEEFP w) args)
-- | Convert an 'SFloat' to an 'SWord32', preserving the bit-correspondence. Note that since the
-- representation for @NaN@s are not unique, this function will return a symbolic value when given a
-- concrete @NaN@.
--
-- Implementation note: Since there's no corresponding function in SMTLib for conversion to
-- bit-representation due to partiality, we use a translation trick by allocating a new word variable,
-- converting it to float, and requiring it to be equivalent to the input. In code-generation mode, we simply map
-- it to a simple conversion.
sFloatAsSWord32 :: SFloat -> SWord32
sFloatAsSWord32 (SBV v) = SBV $ svFloatAsSWord32 v
-- | Convert an 'SDouble' to an 'SWord64', preserving the bit-correspondence. Note that since the
-- representation for @NaN@s are not unique, this function will return a symbolic value when given a
-- concrete @NaN@.
--
-- See the implementation note for 'sFloatAsSWord32', as it applies here as well.
sDoubleAsSWord64 :: SDouble -> SWord64
sDoubleAsSWord64 (SBV v) = SBV $ svDoubleAsSWord64 v
-- | Extract the sign\/exponent\/mantissa of a single-precision float. The output will have
-- 8 bits in the second argument for exponent, and 23 in the third for the mantissa.
blastSFloat :: SFloat -> (SBool, [SBool], [SBool])
blastSFloat = extract . sFloatAsSWord32
where extract x = (sTestBit x 31, sExtractBits x [30, 29 .. 23], sExtractBits x [22, 21 .. 0])
-- | Extract the sign\/exponent\/mantissa of a single-precision float. The output will have
-- 11 bits in the second argument for exponent, and 52 in the third for the mantissa.
blastSDouble :: SDouble -> (SBool, [SBool], [SBool])
blastSDouble = extract . sDoubleAsSWord64
where extract x = (sTestBit x 63, sExtractBits x [62, 61 .. 52], sExtractBits x [51, 50 .. 0])
-- | Extract the sign\/exponent\/mantissa of an arbitrary precision float. The output will have
-- @eb@ bits in the second argument for exponent, and @sb-1@ bits in the third for mantissa.
blastSFloatingPoint :: forall eb sb. (ValidFloat eb sb, KnownNat (eb + sb), BVIsNonZero (eb + sb))
=> SFloatingPoint eb sb -> (SBool, [SBool], [SBool])
blastSFloatingPoint = extract . sFloatingPointAsSWord
where ei = intOfProxy (Proxy @eb)
si = intOfProxy (Proxy @sb)
extract x = (sTestBit x (ei + si - 1), sExtractBits x [ei + si - 2, ei + si - 3 .. si - 1], sExtractBits x [si - 2, si - 3 .. 0])
-- | Reinterpret the bits in a 32-bit word as a single-precision floating point number
sWord32AsSFloat :: SWord32 -> SFloat
sWord32AsSFloat fVal
| Just f <- unliteral fVal = literal $ wordToFloat f
| True = SBV (SVal KFloat (Right (cache y)))
where y st = do xsv <- sbvToSV st fVal
newExpr st KFloat (SBVApp (IEEEFP (FP_Reinterpret (kindOf fVal) KFloat)) [xsv])
-- | Reinterpret the bits in a 32-bit word as a single-precision floating point number
sWord64AsSDouble :: SWord64 -> SDouble
sWord64AsSDouble dVal
| Just d <- unliteral dVal = literal $ wordToDouble d
| True = SBV (SVal KDouble (Right (cache y)))
where y st = do xsv <- sbvToSV st dVal
newExpr st KDouble (SBVApp (IEEEFP (FP_Reinterpret (kindOf dVal) KDouble)) [xsv])
-- | Convert a float to a comparable 'SWord32'. The trick is to ignore the
-- sign of -0, and if it's a negative value flip all the bits, and otherwise
-- only flip the sign bit. This is known as the lexicographic ordering on floats
-- and it works as long as you do not have a @NaN@.
sFloatAsComparableSWord32 :: SFloat -> SWord32
sFloatAsComparableSWord32 f = ite (fpIsNegativeZero f) (sFloatAsComparableSWord32 0) (fromBitsBE $ sNot sb : ite sb (map sNot rest) rest)
where (sb : rest) = blastBE $ sFloatAsSWord32 f
-- | Inverse transformation to 'sFloatAsComparableSWord32'. Note that this isn't a perfect inverse, since @-0@ maps to @0@ and back to @0@.
-- Otherwise, it's faithful:
--
-- >>> prove $ \x -> let f = sComparableSWord32AsSFloat x in fpIsNaN f .|| fpIsNegativeZero f .|| sFloatAsComparableSWord32 f .== x
-- Q.E.D.
-- >>> prove $ \x -> fpIsNegativeZero x .|| sComparableSWord32AsSFloat (sFloatAsComparableSWord32 x) `fpIsEqualObject` x
-- Q.E.D.
sComparableSWord32AsSFloat :: SWord32 -> SFloat
sComparableSWord32AsSFloat w = sWord32AsSFloat $ ite sb (fromBitsBE $ sFalse : rest) (fromBitsBE $ map sNot allBits)
where allBits@(sb : rest) = blastBE w
-- | Convert a double to a comparable 'SWord64'. The trick is to ignore the
-- sign of -0, and if it's a negative value flip all the bits, and otherwise
-- only flip the sign bit. This is known as the lexicographic ordering on doubles
-- and it works as long as you do not have a @NaN@.
sDoubleAsComparableSWord64 :: SDouble -> SWord64
sDoubleAsComparableSWord64 d = ite (fpIsNegativeZero d) (sDoubleAsComparableSWord64 0) (fromBitsBE $ sNot sb : ite sb (map sNot rest) rest)
where (sb : rest) = blastBE $ sDoubleAsSWord64 d
-- | Inverse transformation to 'sDoubleAsComparableSWord64'. Note that this isn't a perfect inverse, since @-0@ maps to @0@ and back to @0@.
-- Otherwise, it's faithful:
--
-- >>> prove $ \x -> let d = sComparableSWord64AsSDouble x in fpIsNaN d .|| fpIsNegativeZero d .|| sDoubleAsComparableSWord64 d .== x
-- Q.E.D.
-- >>> prove $ \x -> fpIsNegativeZero x .|| sComparableSWord64AsSDouble (sDoubleAsComparableSWord64 x) `fpIsEqualObject` x
-- Q.E.D.
sComparableSWord64AsSDouble :: SWord64 -> SDouble
sComparableSWord64AsSDouble w = sWord64AsSDouble $ ite sb (fromBitsBE $ sFalse : rest) (fromBitsBE $ map sNot allBits)
where allBits@(sb : rest) = blastBE w
-- | 'Float' instance for 'Metric' goes through the lexicographic ordering on 'Word32'.
-- It implicitly makes sure that the value is not @NaN@.
instance Metric Float where
type MetricSpace Float = Word32
toMetricSpace = sFloatAsComparableSWord32
fromMetricSpace = sComparableSWord32AsSFloat
msMinimize nm o = do constrain $ sNot $ fpIsNaN o
addSValOptGoal $ unSBV `fmap` Minimize nm (toMetricSpace o)
msMaximize nm o = do constrain $ sNot $ fpIsNaN o
addSValOptGoal $ unSBV `fmap` Maximize nm (toMetricSpace o)
-- | 'Double' instance for 'Metric' goes through the lexicographic ordering on 'Word64'.
-- It implicitly makes sure that the value is not @NaN@.
instance Metric Double where
type MetricSpace Double = Word64
toMetricSpace = sDoubleAsComparableSWord64
fromMetricSpace = sComparableSWord64AsSDouble
msMinimize nm o = do constrain $ sNot $ fpIsNaN o
addSValOptGoal $ unSBV `fmap` Minimize nm (toMetricSpace o)
msMaximize nm o = do constrain $ sNot $ fpIsNaN o
addSValOptGoal $ unSBV `fmap` Maximize nm (toMetricSpace o)
-- | Real instance for FloatingPoint. NB. The methods haven't been subjected to much testing, so beware of any floating-point snafus here.
instance ValidFloat eb sb => Real (FloatingPoint eb sb) where
toRational (FloatingPoint (FP _ _ r)) = case bfToRep r of
BFNaN -> toRational (0/0 :: Double)
BFRep s n -> case n of
Zero -> 0 % 1
Inf -> (if s == Neg then -1 else 1) % 0
Num x y -> -- The value here is x * 2^y
let v :: Integer
v = 2 ^ abs (fromIntegral y :: Integer)
sgn = if s == Neg then ((-1) *) else id
in if y > 0
then sgn $ x * v % 1
else sgn $ x % v
-- | RealFrac instance for FloatingPoint. NB. The methods haven't been subjected to much testing, so beware of any floating-point snafus here.
instance ValidFloat eb sb => RealFrac (FloatingPoint eb sb) where
properFraction (FloatingPoint f) = (a, FloatingPoint b)
where (a, b) = properFraction f
-- | RealFloat instance for FloatingPoint. NB. The methods haven't been subjected to much testing, so beware of any floating-point snafus here.
instance ValidFloat eb sb => RealFloat (FloatingPoint eb sb) where
floatRadix (FloatingPoint f) = floatRadix f
floatDigits (FloatingPoint f) = floatDigits f
floatRange (FloatingPoint f) = floatRange f
isNaN (FloatingPoint f) = isNaN f
isInfinite (FloatingPoint f) = isInfinite f
isDenormalized (FloatingPoint f) = isDenormalized f
isNegativeZero (FloatingPoint f) = isNegativeZero f
isIEEE (FloatingPoint f) = isIEEE f
decodeFloat (FloatingPoint f) = decodeFloat f
encodeFloat m n = res
where res = FloatingPoint $ fpEncodeFloat ei si m n
ei = intOfProxy (Proxy @eb)
si = intOfProxy (Proxy @sb)
-- | Convert a float to the word containing the corresponding bit pattern
sFloatingPointAsSWord :: forall eb sb. (ValidFloat eb sb, KnownNat (eb + sb), BVIsNonZero (eb + sb)) => SFloatingPoint eb sb -> SWord (eb + sb)
sFloatingPointAsSWord (SBV v) = SBV (svFloatingPointAsSWord v)
-- | Convert a float to the correct size word, that can be used in lexicographic ordering. Used in optimization.
sFloatingPointAsComparableSWord :: forall eb sb. (ValidFloat eb sb, KnownNat (eb + sb), BVIsNonZero (eb + sb)) => SFloatingPoint eb sb -> SWord (eb + sb)
sFloatingPointAsComparableSWord f = ite (fpIsNegativeZero f) posZero (fromBitsBE $ sNot sb : ite sb (map sNot rest) rest)
where posZero = sFloatingPointAsComparableSWord (0 :: SFloatingPoint eb sb)
(sb : rest) = blastBE (sFloatingPointAsSWord f :: SWord (eb + sb))
-- | Inverse transformation to 'sFloatingPointAsComparableSWord'. Note that this isn't a perfect inverse, since @-0@ maps to @0@ and back to @0@.
-- Otherwise, it's faithful:
--
-- >>> prove $ \x -> let d = sComparableSWordAsSFloatingPoint x in fpIsNaN d .|| fpIsNegativeZero d .|| sFloatingPointAsComparableSWord (d :: SFPHalf) .== x
-- Q.E.D.
-- >>> prove $ \x -> fpIsNegativeZero x .|| sComparableSWordAsSFloatingPoint (sFloatingPointAsComparableSWord x) `fpIsEqualObject` (x :: SFPHalf)
-- Q.E.D.
sComparableSWordAsSFloatingPoint :: forall eb sb. (KnownNat (eb + sb), BVIsNonZero (eb + sb), ValidFloat eb sb) => SWord (eb + sb) -> SFloatingPoint eb sb
sComparableSWordAsSFloatingPoint w = sWordAsSFloatingPoint $ ite signBit (fromBitsBE $ sFalse : rest) (fromBitsBE $ map sNot allBits)
where allBits@(signBit : rest) = blastBE w
-- | Convert a word to an arbitrary float, by reinterpreting the bits of the word as the corresponding bits of the float.
sWordAsSFloatingPoint :: forall eb sb. (KnownNat (eb + sb), BVIsNonZero (eb + sb), ValidFloat eb sb) => SWord (eb + sb) -> SFloatingPoint eb sb
sWordAsSFloatingPoint sw
| Just (f :: WordN (eb + sb)) <- unliteral sw
= let ext i = f `testBit` i
exts = map ext
(s, ebits, sigbits) = (ext (ei + si - 1), exts [ei + si - 2, ei + si - 3 .. si - 1], exts [si - 2, si - 3 .. 0])
cvt :: [Bool] -> Integer
cvt = foldr (\b sofar -> 2 * sofar + if b then 1 else 0) 0 . reverse
eIntV = cvt ebits
sIntV = cvt sigbits
fp = fpFromRawRep s (eIntV, ei) (sIntV, si)
in literal $ FloatingPoint fp
| True
= SBV (SVal kTo (Right (cache y)))
where ei = intOfProxy (Proxy @eb)
si = intOfProxy (Proxy @sb)
kTo = KFP ei si
y st = do xsv <- sbvToSV st sw
newExpr st kTo (SBVApp (IEEEFP (FP_Reinterpret (kindOf sw) kTo)) [xsv])
instance (BVIsNonZero (eb + sb), KnownNat (eb + sb), ValidFloat eb sb) => Metric (FloatingPoint eb sb) where
type MetricSpace (FloatingPoint eb sb) = WordN (eb + sb)
toMetricSpace = sFloatingPointAsComparableSWord
fromMetricSpace = sComparableSWordAsSFloatingPoint
msMinimize nm o = do constrain $ sNot $ fpIsNaN o
addSValOptGoal $ unSBV `fmap` Minimize nm (toMetricSpace o)
msMaximize nm o = do constrain $ sNot $ fpIsNaN o
addSValOptGoal $ unSBV `fmap` Maximize nm (toMetricSpace o)
-- Map SBV's rounding modes to LibBF's
rmToRM :: SRoundingMode -> Maybe RoundMode
rmToRM srm = cvt <$> unliteral srm
where cvt RoundNearestTiesToEven = NearEven
cvt RoundNearestTiesToAway = NearAway
cvt RoundTowardPositive = ToPosInf
cvt RoundTowardNegative = ToNegInf
cvt RoundTowardZero = ToZero
-- | Lift a 1 arg Big-float op
lift1FP :: forall eb sb. ValidFloat eb sb =>
(BFOpts -> BigFloat -> (BigFloat, Status))
-> (Maybe SRoundingMode -> SFloatingPoint eb sb -> SFloatingPoint eb sb)
-> SRoundingMode
-> SFloatingPoint eb sb
-> SFloatingPoint eb sb
lift1FP bfOp mkDef rm a
| Just (FloatingPoint (FP _ _ v)) <- unliteral a
, Just brm <- rmToRM rm
= literal $ FloatingPoint (FP ei si (fst (bfOp (mkBFOpts ei si brm) v)))
| True
= mkDef (Just rm) a
where ei = intOfProxy (Proxy @eb)
si = intOfProxy (Proxy @sb)
-- | Lift a 2 arg Big-float op
lift2FP :: forall eb sb. ValidFloat eb sb =>
(BFOpts -> BigFloat -> BigFloat -> (BigFloat, Status))
-> (Maybe SRoundingMode -> SFloatingPoint eb sb -> SFloatingPoint eb sb -> SFloatingPoint eb sb)
-> SRoundingMode
-> SFloatingPoint eb sb
-> SFloatingPoint eb sb
-> SFloatingPoint eb sb
lift2FP bfOp mkDef rm a b
| Just (FloatingPoint (FP _ _ v1)) <- unliteral a
, Just (FloatingPoint (FP _ _ v2)) <- unliteral b
, Just brm <- rmToRM rm
= literal $ FloatingPoint (FP ei si (fst (bfOp (mkBFOpts ei si brm) v1 v2)))
| True
= mkDef (Just rm) a b
where ei = intOfProxy (Proxy @eb)
si = intOfProxy (Proxy @sb)
-- | Lift a 3 arg Big-float op
lift3FP :: forall eb sb. ValidFloat eb sb =>
(BFOpts -> BigFloat -> BigFloat -> BigFloat -> (BigFloat, Status))
-> (Maybe SRoundingMode -> SFloatingPoint eb sb -> SFloatingPoint eb sb -> SFloatingPoint eb sb -> SFloatingPoint eb sb)
-> SRoundingMode
-> SFloatingPoint eb sb
-> SFloatingPoint eb sb
-> SFloatingPoint eb sb
-> SFloatingPoint eb sb
lift3FP bfOp mkDef rm a b c
| Just (FloatingPoint (FP _ _ v1)) <- unliteral a
, Just (FloatingPoint (FP _ _ v2)) <- unliteral b
, Just (FloatingPoint (FP _ _ v3)) <- unliteral c
, Just brm <- rmToRM rm
= literal $ FloatingPoint (FP ei si (fst (bfOp (mkBFOpts ei si brm) v1 v2 v3)))
| True
= mkDef (Just rm) a b c
where ei = intOfProxy (Proxy @eb)
si = intOfProxy (Proxy @sb)
-- Sized-floats have a special instance, since it can handle arbitrary rounding modes when it matters.
instance ValidFloat eb sb => IEEEFloating (FloatingPoint eb sb) where
fpAdd = lift2FP bfAdd (lift2 FP_Add (Just (+)))
fpSub = lift2FP bfSub (lift2 FP_Sub (Just (-)))
fpMul = lift2FP bfMul (lift2 FP_Mul (Just (*)))
fpDiv = lift2FP bfDiv (lift2 FP_Div (Just (/)))
fpFMA = lift3FP bfFMA (lift3 FP_FMA Nothing)
fpSqrt = lift1FP bfSqrt (lift1 FP_Sqrt (Just sqrt))
fpRoundToIntegral rm a
| Just (FloatingPoint (FP ei si v)) <- unliteral a
, Just brm <- rmToRM rm
= literal $ FloatingPoint (FP ei si (fst (bfRoundInt brm v)))
| True
= lift1 FP_RoundToIntegral (Just fpRoundToIntegralH) (Just rm) a
-- All other operations are agnostic to the rounding mode, hence the defaults are sufficient:
--
-- fpAbs :: SBV a -> SBV a
-- fpNeg :: SBV a -> SBV a
-- fpRem :: SBV a -> SBV a -> SBV a
-- fpMin :: SBV a -> SBV a -> SBV a
-- fpMax :: SBV a -> SBV a -> SBV a
-- fpIsEqualObject :: SBV a -> SBV a -> SBool
-- fpIsNormal :: SBV a -> SBool
-- fpIsSubnormal :: SBV a -> SBool
-- fpIsZero :: SBV a -> SBool
-- fpIsInfinite :: SBV a -> SBool
-- fpIsNaN :: SBV a -> SBool
-- fpIsNegative :: SBV a -> SBool
-- fpIsPositive :: SBV a -> SBool
-- fpIsNegativeZero :: SBV a -> SBool
-- fpIsPositiveZero :: SBV a -> SBool
-- fpIsPoint :: SBV a -> SBool
{-# ANN module ("HLint: ignore Reduce duplication" :: String) #-}