-----------------------------------------------------------------------------
-- |
-- Module    : Data.SBV.Utils.Numeric
-- Copyright : (c) Levent Erkok
-- License   : BSD3
-- Maintainer: erkokl@gmail.com
-- Stability : experimental
--
-- Various number related utilities
-----------------------------------------------------------------------------

{-# OPTIONS_GHC -Wall -Werror #-}

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:
--      <http://ghc.haskell.org/trac/ghc/ticket/10378>
--      <http://github.com/Z3Prover/z3/issues/68>
-- 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 :: a -> a -> a
fpMaxH a
x a
y
   | a -> Bool
forall a. RealFloat a => a -> Bool
isNaN a
x                                  = a
y
   | a -> Bool
forall a. RealFloat a => a -> Bool
isNaN a
y                                  = a
x
   | (a -> Bool
isN0 a
x Bool -> Bool -> Bool
&& a -> Bool
isP0 a
y) Bool -> Bool -> Bool
|| (a -> Bool
isN0 a
y Bool -> Bool -> Bool
&& a -> Bool
isP0 a
x) = [Char] -> a
forall a. HasCallStack => [Char] -> a
error [Char]
"fpMaxH: Called with alternating-sign 0's. Not supported"
   | a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
> a
y                                    = a
x
   | Bool
True                                     = a
y
   where isN0 :: a -> Bool
isN0   = a -> Bool
forall a. RealFloat a => a -> Bool
isNegativeZero
         isP0 :: a -> Bool
isP0 a
a = a
a a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0 Bool -> Bool -> Bool
&& Bool -> Bool
not (a -> Bool
isN0 a
a)

-- | SMTLib compliant definition for 'Data.SBV.fpMin'. See the comments for 'Data.SBV.fpMax'.
fpMinH :: RealFloat a => a -> a -> a
fpMinH :: a -> a -> a
fpMinH a
x a
y
   | a -> Bool
forall a. RealFloat a => a -> Bool
isNaN a
x                                  = a
y
   | a -> Bool
forall a. RealFloat a => a -> Bool
isNaN a
y                                  = a
x
   | (a -> Bool
isN0 a
x Bool -> Bool -> Bool
&& a -> Bool
isP0 a
y) Bool -> Bool -> Bool
|| (a -> Bool
isN0 a
y Bool -> Bool -> Bool
&& a -> Bool
isP0 a
x) = [Char] -> a
forall a. HasCallStack => [Char] -> a
error [Char]
"fpMinH: Called with alternating-sign 0's. Not supported"
   | a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< a
y                                    = a
x
   | Bool
True                                     = a
y
   where isN0 :: a -> Bool
isN0   = a -> Bool
forall a. RealFloat a => a -> Bool
isNegativeZero
         isP0 :: a -> Bool
isP0 a
a = a
a a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0 Bool -> Bool -> Bool
&& Bool -> Bool
not (a -> Bool
isN0 a
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 :: a -> b
fp2fp a
x
 | a -> Bool
forall a. RealFloat a => a -> Bool
isNaN a
x               =  b
0 b -> b -> b
forall a. Fractional a => a -> a -> a
/ b
0
 | a -> Bool
forall a. RealFloat a => a -> Bool
isInfinite a
x Bool -> Bool -> Bool
&& a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< a
0 = -b
1 b -> b -> b
forall a. Fractional a => a -> a -> a
/ b
0
 | a -> Bool
forall a. RealFloat a => a -> Bool
isInfinite a
x          =  b
1 b -> b -> b
forall a. Fractional a => a -> a -> a
/ b
0
 | a -> Bool
forall a. RealFloat a => a -> Bool
isNegativeZero a
x      = b -> b
forall a. Num a => a -> a
negate b
0
 | Bool
True                  = Rational -> b
forall a. Fractional a => Rational -> a
fromRational (a -> Rational
forall a. Real a => a -> Rational
toRational a
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 <http://smt-lib.org/papers/BTRW14.pdf>, towards the
-- end of section 4.c.
fpRemH :: RealFloat a => a -> a -> a
fpRemH :: a -> a -> a
fpRemH a
x a
y
  | a -> Bool
forall a. RealFloat a => a -> Bool
isInfinite a
x Bool -> Bool -> Bool
|| a -> Bool
forall a. RealFloat a => a -> Bool
isNaN a
x = a
0 a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
0
  | a
y a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0       Bool -> Bool -> Bool
|| a -> Bool
forall a. RealFloat a => a -> Bool
isNaN a
y = a
0 a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
0
  | a -> Bool
forall a. RealFloat a => a -> Bool
isInfinite a
y            = a
x
  | Bool
True                    = a -> a
forall p. (Eq p, Fractional p) => p -> p
pSign (a
x a -> a -> a
forall a. Num a => a -> a -> a
- Rational -> a
forall a. Fractional a => Rational -> a
fromRational (Integer -> Rational
forall a. Num a => Integer -> a
fromInteger Integer
d Rational -> Rational -> Rational
forall a. Num a => a -> a -> a
* Rational
ry))
  where rx, ry, rd :: Rational
        rx :: Rational
rx = a -> Rational
forall a. Real a => a -> Rational
toRational a
x
        ry :: Rational
ry = a -> Rational
forall a. Real a => a -> Rational
toRational a
y
        rd :: Rational
rd = Rational
rx Rational -> Rational -> Rational
forall a. Fractional a => a -> a -> a
/ Rational
ry
        d :: Integer
        d :: Integer
d = Rational -> Integer
forall a b. (RealFrac a, Integral b) => a -> b
round Rational
rd
        -- If the result is 0, make sure we preserve the sign of x
        pSign :: p -> p
pSign p
r
          | p
r p -> p -> Bool
forall a. Eq a => a -> a -> Bool
== p
0 = if a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< a
0 Bool -> Bool -> Bool
|| a -> Bool
forall a. RealFloat a => a -> Bool
isNegativeZero a
x then -p
0.0 else p
0.0
          | Bool
True   = p
r

-- | Convert a float to the nearest integral representable in that type
fpRoundToIntegralH :: RealFloat a => a -> a
fpRoundToIntegralH :: a -> a
fpRoundToIntegralH a
x
  | a -> Bool
forall a. RealFloat a => a -> Bool
isNaN a
x      = a
x
  | a
x a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0       = a
x
  | a -> Bool
forall a. RealFloat a => a -> Bool
isInfinite a
x = a
x
  | Integer
i Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
== Integer
0       = if a
x a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< a
0 Bool -> Bool -> Bool
|| a -> Bool
forall a. RealFloat a => a -> Bool
isNegativeZero a
x then -a
0.0 else a
0.0
  | Bool
True         = Integer -> a
forall a. Num a => Integer -> a
fromInteger Integer
i
  where i :: Integer
        i :: Integer
i = a -> Integer
forall a b. (RealFrac a, Integral b) => a -> b
round a
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 -> a -> Bool
fpIsEqualObjectH a
a a
b
  | a -> Bool
forall a. RealFloat a => a -> Bool
isNaN a
a          = a -> Bool
forall a. RealFloat a => a -> Bool
isNaN a
b
  | a -> Bool
forall a. RealFloat a => a -> Bool
isNegativeZero a
a = a -> Bool
forall a. RealFloat a => a -> Bool
isNegativeZero a
b
  | a -> Bool
forall a. RealFloat a => a -> Bool
isNegativeZero a
b = a -> Bool
forall a. RealFloat a => a -> Bool
isNegativeZero a
a
  | Bool
True             = a
a a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== 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 -> a -> Ordering
fpCompareObjectH a
a a
b
  | a
a a -> a -> Bool
forall a. RealFloat a => a -> a -> Bool
`fpIsEqualObjectH` a
b   = Ordering
EQ
  | a -> Bool
forall a. RealFloat a => a -> Bool
isNaN a
a                  = Ordering
LT
  | a -> Bool
forall a. RealFloat a => a -> Bool
isNaN a
b                  = Ordering
GT
  | a -> Bool
forall a. RealFloat a => a -> Bool
isNegativeZero a
a, a
b a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0 = Ordering
LT
  | a -> Bool
forall a. RealFloat a => a -> Bool
isNegativeZero a
b, a
a a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0 = Ordering
GT
  | Bool
True                     = a
a a -> a -> Ordering
forall a. Ord a => a -> a -> Ordering
`compare` a
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 :: a -> Bool
fpIsNormalizedH a
x = Bool -> Bool
not (a -> Bool
forall a. RealFloat a => a -> Bool
isDenormalized a
x Bool -> Bool -> Bool
|| a -> Bool
forall a. RealFloat a => a -> Bool
isInfinite a
x Bool -> Bool -> Bool
|| a -> Bool
forall a. RealFloat a => a -> Bool
isNaN a
x Bool -> Bool -> Bool
|| a
x a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0)