```-- |
-- Module:      Math.NumberTheory.Primes.Counting.Impl
-- Copyright:   (c) 2011 Daniel Fischer
-- Licence:     MIT
--
-- Number of primes not exceeding @n@, @&#960;(n)@, and @n@-th prime.
--
{-# LANGUAGE BangPatterns        #-}
{-# LANGUAGE CPP                 #-}
{-# LANGUAGE FlexibleContexts    #-}
{-# LANGUAGE ScopedTypeVariables #-}

{-# OPTIONS_GHC -fspec-constr-count=24 #-}
module Math.NumberTheory.Primes.Counting.Impl
( primeCount
, primeCountMaxArg
, nthPrime
) where

#include "MachDeps.h"

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 Data.Array.Base
import Data.Array.ST
import Data.Bits
import Data.Int
import Unsafe.Coerce

-- | Maximal allowed argument of 'primeCount'. Currently 8e18.
primeCountMaxArg :: Integer
primeCountMaxArg = 8000000000000000000

-- | @'primeCount' n == &#960;(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  = fromIntegral . length . takeWhile (<= n) . map unPrime . primeList . primeSieve \$ max 242 n
| n < 30000 = runST \$ do
ba <- sieveTo n
(s,e) <- getBounds ba
ct <- countFromTo s e ba
return (fromIntegral \$ 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 (fromIntegral 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 (fromIntegral 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], ba `unsafeAt` 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 (fromIntegral cr)
pisr = approxPrimeCount sr
picr = approxPrimeCount cr
diff = pisr - picr
size = fromIntegral (diff + diff `quot` 50) + 30
store <- unsafeNewArray_ (0,size-1) :: ST s (STUArray s Int Int64)
let feed :: Int64 -> Int -> Int -> UArray Int Bool -> [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 <- unsafeThaw uar
eat 0 0 voff (wi-1) ri stu sves
| uar `unsafeAt` ri = do
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 -> STUArray s Int Bool -> [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 <- unsafeThaw ba
eat acc btw (fromInteger vO) wi 0 nstu more
| wi < 0    = return acc
| otherwise = do
let dist = qb - voff - 7
if dist < fromIntegral 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 + fromIntegral new + 1
eat (acc+nbtw) nbtw voff (wi-1) (li+1) stu sves
else do
let (cpl,fds) = dist `quotRem` fromIntegral sieveRange
(b,j) = idxPr (fds+7)
!li = (b `shiftL` 3) .|. j
ctLoop !lac 0 (PS vO ba : more) = do
nstu <- unsafeThaw ba
new <- countFromTo 0 li nstu
let nbtw = btw + lac + 1 + fromIntegral new
eat (acc+nbtw) nbtw (fromIntegral vO) (wi-1) (li+1) nstu more
ctLoop lac s (ps : more) = do
let !new = countAll ps
ctLoop (lac + fromIntegral new) (s-1) more
ctLoop _ _ [] = error "Primes ended"
new <- countFromTo si (sieveBits-1) stu
ctLoop (fromIntegral 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 (fromIntegral plim)
(plo,phi) <- getBounds parr
!pct <- countFromTo plo phi parr
!ar1 <- unsafeNewArray_ (0,end-1)
unsafeWrite ar1 0 lim
unsafeWrite ar1 1 1
!ar2 <- unsafeNewArray_ (0,end-1)
let go :: Int -> Int -> STUArray s Int Int64 -> STUArray s Int Int64 -> ST s Integer
go cap pix old new
| pix == 2  =   coll cap old
| otherwise = do
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 -> STUArray s Int Int64 -> ST s Integer
coll stop ar =
let cgo !acc i
| i < stop  = do
cgo (acc + fromIntegral v*cp6 k) (i+2)
| otherwise = return (acc+fromIntegral pct+2)
in cgo 0 0
go 2 start ar1 ar2
where
(bt,ri) = idxPr plim
!start = 8*bt + ri
!size = fromIntegral \$ (integerSquareRoot lim) `quot` 4
!end = 2*size

treat :: Int -> Int64 -> STUArray s Int Int64 -> STUArray s Int 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
if k < stop
then do
collect stop (acc-v) (ix+2)
else return (acc,ix)
| otherwise = return (acc,ix)
goTreat !wi !ci qi
| qi < end  = do
let !q0 = key `quot` n
!r0 = fromIntegral (q0 `rem` 30030)
!nkey = q0 - fromIntegral (cpDfAr `unsafeAt` r0)
nk0 = q0 + fromIntegral (cpGpAr `unsafeAt` (r0+1) + 1)
!nlim = n*nk0
(wi1,ci1) <- copyTo end nkey old ci new wi
(!acc, !ci2) <- if ckey == nkey
then do
return (ov-val,ci1+2)
else return (-val,ci1)
(!tot, !nqi) <- collect nlim acc (qi+2)
unsafeWrite new wi1 nkey
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 -> STUArray s Int Int64 -> ST s Int
locate p low high arr = do
let go lo hi
| lo < hi     = do
let !md = (lo+hi) `quot` 2
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 -> STUArray s Int Int64 -> Int
-> STUArray s Int Int64 -> Int -> ST s (Int,Int)
copyTo end lim old oi new ni = do
let go ri wi
| ri < end  = do
if ok < lim
then do
unsafeWrite new wi ok
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 -> STUArray s Int Int64 -> Int -> STUArray s Int Int64 -> Int -> ST s Int
copyRem end old oi new ni = do
let go ri wi
| ri < end    = do
unsafeRead old ri >>= unsafeWrite new wi
go (ri+1) (wi+1)
| otherwise   = return wi
go oi ni

{-# INLINE cp6 #-}
cp6 :: Int64 -> Integer
cp6 k =
case k `quotRem` 30030 of
(q,r) -> 5760*fromIntegral q +
fromIntegral (cpCtAr `unsafeAt` fromIntegral r)

cop :: Int64 -> Int64
cop m = m - fromIntegral (cpDfAr `unsafeAt` fromIntegral (m `rem` 30030))

--------------------------------------------------------------------------------
--                           Ugly helper arrays                               --
--------------------------------------------------------------------------------

cpCtAr :: UArray Int Int16
cpCtAr = runSTUArray \$ do
ar <- newArray (0,30029) 1
let zilch s i
| i < 30030 = unsafeWrite ar i 0 >> zilch s (i+s)
| otherwise = return ()
accumulate ct i
| i < 30030 = do
let !ct' = ct+v
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
accumulate 1 2

cpDfAr :: UArray Int Int8
cpDfAr = runSTUArray \$ do
ar <- newArray (0,30029) 0
let note s i
| i < 30029 = unsafeWrite ar i 1 >> note s (i+s)
| otherwise = return ()
accumulate d i
| i < 30029 = do
if v == 0
then accumulate 2 (i+2)
else do 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
accumulate 2 3

cpGpAr :: UArray Int Int8
cpGpAr = runSTUArray \$ do
ar <- newArray (0,30030) 0
unsafeWrite ar 30030 1
let note s i
| i < 30029 = unsafeWrite ar i 1 >> note s (i+s)
| otherwise = return ()
accumulate d i
| i < 1     = return ar
| otherwise = do
if v == 0
then accumulate 2 (i-2)
else do 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
accumulate 2 30027

-------------------------------------------------------------------------------
-- Prime counting

#if SIZEOF_HSWORD == 8

#define WSHFT 6
#define TOPB 32
#define TOPM 0xFFFFFFFF

#else

#define WSHFT 5
#define TOPB 16
#define TOPM 0xFFFF

#endif

-- 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) = go n 0
where
wa :: UArray Int Word
wa = unsafeCoerce bs

go !k i
| i == snd (bounds wa)
= countToNth k more
| otherwise
= let w = unsafeAt wa i
bc = popCount w
in if bc < k
then go (k-bc) (i+1)
else let j = bc - k
px = top w j bc
in v0 + toPrim (px + (i `shiftL` WSHFT))

-- count all set bits in a chunk, do it wordwise for speed.
countAll :: PrimeSieve -> Int
countAll (PS _ bs) = go 0 0
where
wa :: UArray Int Word
wa = unsafeCoerce bs

go !ct i
| i == snd (bounds wa)
= ct
| otherwise
= go (ct + popCount (unsafeAt wa i)) (i+1)

-- Find the j-th highest of bc set bits in the Word w.
top :: Word -> Int -> Int -> Int
top w j bc = go 0 TOPB TOPM bn w
where
!bn = bc-j
go !_ _ !_ !_ 0 = error "Too few bits set"
go bs 0 _ _ wd = if wd .&. 1 == 0 then error "Too few bits, shift 0" else bs
go bs a msk ix wd =
case popCount (wd .&. msk) of
lc | lc < ix  -> go (bs+a) a msk (ix-lc) (wd `unsafeShiftR` a)
| otherwise ->
let !na = a `shiftR` 1
in go bs na (msk `unsafeShiftR` na) ix wd

-- count set bits between two indices (inclusive)
-- start and end must both be valid indices and start <= end
countFromTo :: Int -> Int -> STUArray s Int Bool -> ST s Int
countFromTo start end ba = do
wa <- (castSTUArray :: STUArray s Int Bool -> ST s (STUArray s Int Word)) ba
let !sb = start `shiftR` WSHFT
!eb = end `shiftR` WSHFT
count !acc i
| i == eb = do