arithmoi-0.13.2.0: Math/NumberTheory/Primes/Counting/Impl.hs
-- |
-- Module: Math.NumberTheory.Primes.Counting.Impl
-- Copyright: (c) 2011 Daniel Fischer
-- Licence: MIT
-- Maintainer: Daniel Fischer <daniel.is.fischer@googlemail.com>
--
-- Number of primes not exceeding @n@, @π(n)@, and @n@-th prime.
--
{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# OPTIONS_GHC -fspec-constr-count=24 #-}
{-# OPTIONS_GHC -Wno-incomplete-uni-patterns #-}
module Math.NumberTheory.Primes.Counting.Impl
( primeCount
, primeCountMaxArg
, nthPrime
) where
import Math.NumberTheory.Primes.Sieve.Eratosthenes
(PrimeSieve(..), primeList, primeSieve, psieveFrom, sieveTo, sieveBits, sieveRange)
import Math.NumberTheory.Primes.Sieve.Indexing (toPrim, idxPr)
import Math.NumberTheory.Primes.Counting.Approximate (nthPrimeApprox, approxPrimeCount)
import Math.NumberTheory.Primes.Types
import Math.NumberTheory.Roots
import Math.NumberTheory.Utils.FromIntegral
import Control.Monad.ST
import Data.Bit
import Data.Bits
import Data.Int
import qualified Data.Vector.Unboxed as U
import qualified Data.Vector.Unboxed.Mutable as MU
-- | Maximal allowed argument of 'primeCount'. Currently 8e18.
primeCountMaxArg :: Integer
primeCountMaxArg = 8000000000000000000
-- | @'primeCount' n == π(n)@ is the number of (positive) primes not exceeding @n@.
--
-- For efficiency, the calculations are done on 64-bit signed integers, therefore @n@ must
-- not exceed 'primeCountMaxArg'.
--
-- Requires @/O/(n^0.5)@ space, the time complexity is roughly @/O/(n^0.7)@.
-- For small bounds, @'primeCount' n@ simply counts the primes not exceeding @n@,
-- for bounds from @30000@ on, Meissel's algorithm is used in the improved form due to
-- D.H. Lehmer, cf.
-- <http://en.wikipedia.org/wiki/Prime_counting_function#Algorithms_for_evaluating_.CF.80.28x.29>.
primeCount :: Integer -> Integer
primeCount n
| n > primeCountMaxArg = error $ "primeCount: can't handle bound " ++ show n
| n < 2 = 0
| n < 1000 = intToInteger . length . takeWhile (<= n) . map unPrime . primeList . primeSieve $ max 242 n
| n < 30000 = runST $ do
ba <- sieveTo n
let (s, e) = (0, MU.length ba - 1)
ct <- countFromTo s e ba
return (intToInteger $ ct+3)
| otherwise =
let !ub = cop $ fromInteger n
!sr = integerSquareRoot ub
!cr = nxtEnd $ integerCubeRoot ub + 15
nxtEnd k = k - (k `rem` 30) + 31
!phn1 = calc ub cr
!cs = cr+6
!pdf = sieveCount ub cs sr
in phn1 - pdf
-- | @'nthPrime' n@ calculates the @n@-th prime. Numbering of primes is
-- @1@-based, so @'nthPrime' 1 == 2@.
--
-- Requires @/O/((n*log n)^0.5)@ space, the time complexity is roughly @/O/((n*log n)^0.7@.
-- The argument must be strictly positive.
nthPrime :: Int -> Prime Integer
nthPrime 1 = Prime 2
nthPrime 2 = Prime 3
nthPrime 3 = Prime 5
nthPrime 4 = Prime 7
nthPrime 5 = Prime 11
nthPrime 6 = Prime 13
nthPrime n
| n < 1
= error "Prime indexing starts at 1"
| n < 200000
= Prime $ countToNth (n - 3) [primeSieve (p0 + p0 `quot` 32 + 37)]
| p0 > toInteger (maxBound :: Int)
= error $ "nthPrime: index " ++ show n ++ " is too large to handle"
| miss > 0
= Prime $ tooLow n (fromInteger p0) miss
| otherwise
= Prime $ tooHigh n (fromInteger p0) (negate miss)
where
p0 = nthPrimeApprox (toInteger n)
miss = n - fromInteger (primeCount p0)
--------------------------------------------------------------------------------
-- The Works --
--------------------------------------------------------------------------------
-- TODO: do something better in case we guess too high.
-- Not too pressing, since I think a) nthPrimeApprox always underestimates
-- in the range we can handle, and b) it's always "goodEnough"
tooLow :: Int -> Int -> Int -> Integer
tooLow n p0 shortage
| p1 > toInteger (maxBound :: Int)
= error $ "nthPrime: index " ++ show n ++ " is too large to handle"
| goodEnough
= lowSieve p0 shortage
| c1 < n
= lowSieve (fromInteger p1) (n-c1)
| otherwise
= lowSieve p0 shortage -- a third count wouldn't make it faster, I think
where
gap = truncate (log (intToDouble p0 :: Double))
est = toInteger shortage * gap
p1 = toInteger p0 + est
goodEnough = 3*est*est*est < 2*p1*p1 -- a second counting would be more work than sieving
c1 = fromInteger (primeCount p1)
tooHigh :: Int -> Int -> Int -> Integer
tooHigh n p0 surplus
| c < n
= lowSieve b (n-c)
| otherwise
= tooHigh n b (c-n)
where
gap = truncate (log (intToDouble p0 :: Double))
b = p0 - (surplus * gap * 11) `quot` 10
c = fromInteger (primeCount (toInteger b))
lowSieve :: Int -> Int -> Integer
lowSieve a miss = countToNth (miss+rep) psieves
where
strt = a + 1 + (a .&. 1)
psieves@(PS vO ba:_) = psieveFrom (toInteger strt)
rep | o0 < 0 = 0
| otherwise = sum [1 | i <- [0 .. r2], unBit (ba `U.unsafeIndex` i)]
where
o0 = toInteger strt - vO - 9 -- (strt - 2) - v0 - 7
r0 = fromInteger o0 `rem` 30
r1 = r0 `quot` 3
r2 = min 7 (if r1 > 5 then r1-1 else r1)
-- highSieve :: Integer -> Integer -> Integer -> Integer
-- highSieve a surp gap = error "Oh shit"
sieveCount :: Int64 -> Int64 -> Int64 -> Integer
sieveCount ub cr sr = runST (sieveCountST ub cr sr)
sieveCountST :: forall s. Int64 -> Int64 -> Int64 -> ST s Integer
sieveCountST ub cr sr = do
let psieves = psieveFrom (int64ToInteger cr)
pisr = approxPrimeCount sr
picr = approxPrimeCount cr
diff = pisr - picr
size = int64ToInt (diff + diff `quot` 50) + 30
store <- MU.unsafeNew size :: ST s (MU.MVector s Int64)
let feed :: Int64 -> Int -> Int -> U.Vector Bit -> [PrimeSieve] -> ST s Integer
feed voff !wi !ri uar sves
| ri == sieveBits = case sves of
(PS vO ba : more) -> feed (fromInteger vO) wi 0 ba more
_ -> error "prime stream ended prematurely"
| pval > sr = do
stu <- U.unsafeThaw uar
eat 0 0 voff (wi-1) ri stu sves
| unBit (uar `U.unsafeIndex` ri) = do
MU.unsafeWrite store wi (ub `quot` pval)
feed voff (wi+1) (ri+1) uar sves
| otherwise = feed voff wi (ri+1) uar sves
where
pval = voff + toPrim ri
eat :: Integer -> Integer -> Int64 -> Int -> Int -> MU.MVector s Bit -> [PrimeSieve] -> ST s Integer
eat !acc !btw voff !wi !si stu sves
| si == sieveBits =
case sves of
[] -> error "Premature end of prime stream"
(PS vO ba : more) -> do
nstu <- U.unsafeThaw ba
eat acc btw (fromInteger vO) wi 0 nstu more
| wi < 0 = return acc
| otherwise = do
qb <- MU.unsafeRead store wi
let dist = qb - voff - 7
if dist < intToInt64 sieveRange
then do
let (b,j) = idxPr (dist+7)
!li = (b `shiftL` 3) .|. j
new <- if li < si then return 0 else countFromTo si li stu
let nbtw = btw + intToInteger new + 1
eat (acc+nbtw) nbtw voff (wi-1) (li+1) stu sves
else do
let (cpl,fds) = dist `quotRem` intToInt64 sieveRange
(b,j) = idxPr (fds+7)
!li = (b `shiftL` 3) .|. j
ctLoop !lac 0 (PS vO ba : more) = do
nstu <- U.unsafeThaw ba
new <- countFromTo 0 li nstu
let nbtw = btw + lac + 1 + intToInteger new
eat (acc+nbtw) nbtw (integerToInt64 vO) (wi-1) (li+1) nstu more
ctLoop lac s (ps : more) = do
let !new = countAll ps
ctLoop (lac + intToInteger new) (s-1) more
ctLoop _ _ [] = error "Primes ended"
new <- countFromTo si (sieveBits-1) stu
ctLoop (intToInteger new) (cpl-1) sves
case psieves of
(PS vO ba : more) -> feed (fromInteger vO) 0 0 ba more
_ -> error "No primes sieved"
calc :: Int64 -> Int64 -> Integer
calc lim plim = runST (calcST lim plim)
calcST :: forall s. Int64 -> Int64 -> ST s Integer
calcST lim plim = do
!parr <- sieveTo (int64ToInteger plim)
let (plo, phi) = (0, MU.length parr - 1)
!pct <- countFromTo plo phi parr
!ar1 <- MU.unsafeNew end
MU.unsafeWrite ar1 0 lim
MU.unsafeWrite ar1 1 1
!ar2 <- MU.unsafeNew end
let go :: Int -> Int -> MU.MVector s Int64 -> MU.MVector s Int64 -> ST s Integer
go cap pix old new
| pix == 2 = coll cap old
| otherwise = do
Bit isp <- MU.unsafeRead parr pix
if isp
then do
let !n = fromInteger (toPrim pix)
!ncap <- treat cap n old new
go ncap (pix-1) new old
else go cap (pix-1) old new
coll :: Int -> MU.MVector s Int64 -> ST s Integer
coll stop ar =
let cgo !acc i
| i < stop = do
!k <- MU.unsafeRead ar i
!v <- MU.unsafeRead ar (i+1)
cgo (acc + int64ToInteger v*cp6 k) (i+2)
| otherwise = return (acc+intToInteger pct+2)
in cgo 0 0
go 2 start ar1 ar2
where
(bt,ri) = idxPr plim
!start = 8*bt + ri
!size = int64ToInt $ integerSquareRoot lim `quot` 4
!end = 2*size
treat :: Int -> Int64 -> MU.MVector s Int64 -> MU.MVector s Int64 -> ST s Int
treat end n old new = do
qi0 <- locate n 0 (end `quot` 2 - 1) old
let collect stop !acc ix
| ix < end = do
!k <- MU.unsafeRead old ix
if k < stop
then do
v <- MU.unsafeRead old (ix+1)
collect stop (acc-v) (ix+2)
else return (acc,ix)
| otherwise = return (acc,ix)
goTreat !wi !ci qi
| qi < end = do
!key <- MU.unsafeRead old qi
!val <- MU.unsafeRead old (qi+1)
let !q0 = key `quot` n
!r0 = int64ToInt (q0 `rem` 30030)
!nkey = q0 - int8ToInt64 (cpDfAr `U.unsafeIndex` r0)
nk0 = q0 + int8ToInt64 (cpGpAr `U.unsafeIndex` (r0+1) + 1)
!nlim = n*nk0
(wi1,ci1) <- copyTo end nkey old ci new wi
ckey <- MU.unsafeRead old ci1
(!acc, !ci2) <- if ckey == nkey
then do
!ov <- MU.unsafeRead old (ci1+1)
return (ov-val,ci1+2)
else return (-val,ci1)
(!tot, !nqi) <- collect nlim acc (qi+2)
MU.unsafeWrite new wi1 nkey
MU.unsafeWrite new (wi1+1) tot
goTreat (wi1+2) ci2 nqi
| otherwise = copyRem end old ci new wi
goTreat 0 0 qi0
--------------------------------------------------------------------------------
-- Auxiliaries --
--------------------------------------------------------------------------------
locate :: Int64 -> Int -> Int -> MU.MVector s Int64 -> ST s Int
locate p low high arr = do
let go lo hi
| lo < hi = do
let !md = (lo+hi) `quot` 2
v <- MU.unsafeRead arr (2*md)
case compare p v of
LT -> go lo md
EQ -> return (2*md)
GT -> go (md+1) hi
| otherwise = return (2*lo)
go low high
{-# INLINE copyTo #-}
copyTo :: Int -> Int64 -> MU.MVector s Int64 -> Int
-> MU.MVector s Int64 -> Int -> ST s (Int,Int)
copyTo end lim old oi new ni = do
let go ri wi
| ri < end = do
ok <- MU.unsafeRead old ri
if ok < lim
then do
!ov <- MU.unsafeRead old (ri+1)
MU.unsafeWrite new wi ok
MU.unsafeWrite new (wi+1) ov
go (ri+2) (wi+2)
else return (wi,ri)
| otherwise = return (wi,ri)
go oi ni
{-# INLINE copyRem #-}
copyRem :: Int -> MU.MVector s Int64 -> Int -> MU.MVector s Int64 -> Int -> ST s Int
copyRem end old oi new ni = do
let len = end - oi
MU.copy (MU.slice ni len new) (MU.slice oi len old)
pure $ ni + len
{-# INLINE cp6 #-}
cp6 :: Int64 -> Integer
cp6 k =
case k `quotRem` 30030 of
(q,r) -> 5760*int64ToInteger q +
int16ToInteger (cpCtAr `U.unsafeIndex` int64ToInt r)
cop :: Int64 -> Int64
cop m = m - int8ToInt64 (cpDfAr `U.unsafeIndex` int64ToInt (m `rem` 30030))
--------------------------------------------------------------------------------
-- Ugly helper arrays --
--------------------------------------------------------------------------------
cpCtAr :: U.Vector Int16
cpCtAr = runST $ do
ar <- MU.replicate 30030 1
let zilch s i
| i < 30030 = MU.unsafeWrite ar i 0 >> zilch s (i+s)
| otherwise = return ()
accumulate ct i
| i < 30030 = do
v <- MU.unsafeRead ar i
let !ct' = ct+v
MU.unsafeWrite ar i ct'
accumulate ct' (i+1)
| otherwise = return ar
zilch 2 0
zilch 6 3
zilch 10 5
zilch 14 7
zilch 22 11
zilch 26 13
vec <- accumulate 1 2
U.unsafeFreeze vec
cpDfAr :: U.Vector Int8
cpDfAr = runST $ do
ar <- MU.replicate 30030 0
let note s i
| i < 30029 = MU.unsafeWrite ar i 1 >> note s (i+s)
| otherwise = return ()
accumulate d i
| i < 30029 = do
v <- MU.unsafeRead ar i
if v == 0
then accumulate 2 (i+2)
else do MU.unsafeWrite ar i d
accumulate (d+1) (i+1)
| otherwise = return ar
note 2 0
note 6 3
note 10 5
note 14 7
note 22 11
note 26 13
vec <- accumulate 2 3
U.unsafeFreeze vec
cpGpAr :: U.Vector Int8
cpGpAr = runST $ do
ar <- MU.replicate 30031 0
MU.unsafeWrite ar 30030 1
let note s i
| i < 30029 = MU.unsafeWrite ar i 1 >> note s (i+s)
| otherwise = return ()
accumulate d i
| i < 1 = return ar
| otherwise = do
v <- MU.unsafeRead ar i
if v == 0
then accumulate 2 (i-2)
else do MU.unsafeWrite ar i d
accumulate (d+1) (i-1)
note 2 0
note 6 3
note 10 5
note 14 7
note 22 11
note 26 13
vec <- accumulate 2 30027
U.unsafeFreeze vec
-------------------------------------------------------------------------------
-- Prime counting
-- find the n-th set bit in a list of PrimeSieves,
-- aka find the (n+3)-rd prime
countToNth :: Int -> [PrimeSieve] -> Integer
countToNth !_ [] = error "countToNth: Prime stream ended prematurely"
countToNth !n (PS v0 bs : more) = case nthBitIndex (Bit True) n bs of
Just i -> v0 + toPrim i
Nothing -> countToNth (n - countBits bs) more
-- count all set bits in a chunk, do it wordwise for speed.
countAll :: PrimeSieve -> Int
countAll (PS _ bs) = countBits bs
-- count set bits between two indices (inclusive)
-- start and end must both be valid indices and start <= end
countFromTo :: Int -> Int -> MU.MVector s Bit -> ST s Int
countFromTo start end =
fmap countBits . U.unsafeFreeze . MU.slice start (end - start + 1)