-----------------------------------------------------------------------------
-- |
-- Module : Data.SBV.Utils.Numeric
-- Copyright : (c) Levent Erkok
-- License : BSD3
-- Maintainer: erkokl@gmail.com
-- Stability : experimental
--
-- Various number related utilities
-----------------------------------------------------------------------------
module Data.SBV.Utils.Numeric where
-- | The SMT-Lib (in particular Z3) implementation for min/max for floats does not agree with
-- Haskell's; and also it does not agree with what the hardware does. Sigh.. See:
--
--
-- So, we codify here what the Z3 (SMTLib) is implementing for fpMax.
-- The discrepancy with Haskell is that the NaN propagation doesn't work in Haskell
-- The discrepancy with x86 is that given +0/-0, x86 returns the second argument; SMTLib is non-deterministic
fpMaxH :: RealFloat a => a -> a -> a
fpMaxH x y
| isNaN x = y
| isNaN y = x
| (isN0 x && isP0 y) || (isN0 y && isP0 x) = error "fpMaxH: Called with alternating-sign 0's. Not supported"
| x > y = x
| True = y
where isN0 = isNegativeZero
isP0 a = a == 0 && not (isN0 a)
-- | SMTLib compliant definition for 'Data.SBV.fpMin'. See the comments for 'Data.SBV.fpMax'.
fpMinH :: RealFloat a => a -> a -> a
fpMinH x y
| isNaN x = y
| isNaN y = x
| (isN0 x && isP0 y) || (isN0 y && isP0 x) = error "fpMinH: Called with alternating-sign 0's. Not supported"
| x < y = x
| True = y
where isN0 = isNegativeZero
isP0 a = a == 0 && not (isN0 a)
-- | Convert double to float and back. Essentially @fromRational . toRational@
-- except careful on NaN, Infinities, and -0.
fp2fp :: (RealFloat a, RealFloat b) => a -> b
fp2fp x
| isNaN x = 0 / 0
| isInfinite x && x < 0 = -1 / 0
| isInfinite x = 1 / 0
| isNegativeZero x = negate 0
| True = fromRational (toRational x)
-- | Compute the "floating-point" remainder function, the float/double value that
-- remains from the division of @x@ and @y@. There are strict rules around 0's, Infinities,
-- and NaN's as coded below, See , towards the
-- end of section 4.c.
fpRemH :: RealFloat a => a -> a -> a
fpRemH x y
| isInfinite x || isNaN x = 0 / 0
| y == 0 || isNaN y = 0 / 0
| isInfinite y = x
| True = pSign (x - fromRational (fromInteger d * ry))
where rx, ry, rd :: Rational
rx = toRational x
ry = toRational y
rd = rx / ry
d :: Integer
d = round rd
-- If the result is 0, make sure we preserve the sign of x
pSign r
| r == 0 = if x < 0 || isNegativeZero x then -0.0 else 0.0
| True = r
-- | Convert a float to the nearest integral representable in that type
fpRoundToIntegralH :: RealFloat a => a -> a
fpRoundToIntegralH x
| isNaN x = x
| x == 0 = x
| isInfinite x = x
| i == 0 = if x < 0 || isNegativeZero x then -0.0 else 0.0
| True = fromInteger i
where i :: Integer
i = round x
-- | Check that two floats are the exact same values, i.e., +0/-0 does not
-- compare equal, and NaN's compare equal to themselves.
fpIsEqualObjectH :: RealFloat a => a -> a -> Bool
fpIsEqualObjectH a b
| isNaN a = isNaN b
| isNegativeZero a = isNegativeZero b
| isNegativeZero b = isNegativeZero a
| True = a == b
-- | Ordering for floats, avoiding the +0/-0/NaN issues. Note that this is
-- essentially used for indexing into a map, so we need to be total. Thus,
-- the order we pick is:
-- NaN -oo -0 +0 +oo
-- The placement of NaN here is questionable, but immaterial.
fpCompareObjectH :: RealFloat a => a -> a -> Ordering
fpCompareObjectH a b
| a `fpIsEqualObjectH` b = EQ
| isNaN a = LT
| isNaN b = GT
| isNegativeZero a, b == 0 = LT
| isNegativeZero b, a == 0 = GT
| True = a `compare` b
-- | Check if a number is "normal." Note that +0/-0 is not considered a normal-number
-- and also this is not simply the negation of isDenormalized!
fpIsNormalizedH :: RealFloat a => a -> Bool
fpIsNormalizedH x = not (isDenormalized x || isInfinite x || isNaN x || x == 0)