```{-|
Copyright  :  (C) 2013-2016, University of Twente
License    :  BSD2 (see the file LICENSE)
Maintainer :  Christiaan Baaij <christiaan.baaij@gmail.com>

Fixed point numbers

* The 'Num' operators for the given types saturate on overflow,
and use truncation as the rounding method.
* 'Fixed' has an instance for 'Fractional' meaning you use fractional
literals @(3.75 :: 'SFixed' 4 18)@.
* Both integer literals and fractional literals are clipped to 'minBound' and
'maxBound'. __NB__ Needs the `-XNegativeLiterals` language extension to work
for signed numbers.
* There is no 'Floating' instance for 'Fixed', but you can use @\$\$('fLit' d)@
to create 'Fixed' point literal from 'Double' constant at compile-time.
* Use <#constraintsynonyms Constraint synonyms> when writing type signatures
for polymorphic functions that use 'Fixed' point numbers.

BEWARE: rounding by truncation can introduce errors larger than naively assumed;
e.g. for /Fixed 16 1/, rounding by truncation turns the real number 4.99 to 4.5,
not 5.0, i.e. an error or 0.49 instead of 0.01

BEWARE: rounding by truncation introduces a sign bias!

* Truncation for positive numbers effectively results in: round towards zero.
* Truncation for negative numbers effectively results in: round towards -infinity.

== Reasoning about precision
Givens the real numbers /A/ and /B/, and the corresponding fixed point numbers
/FA+-da/ and /FB+db/, where /da/ and /db/ denote the (potential) error introduced
by truncation w.r.t. the original /A/ and /B/, the arithmetic operators on fixed
point numbers have the following error propagation properties:

* Addition: /da + db/
* Subtraction: /da - db/
* Multiplication: /FA*db + FB*da + da*db/
* Division: /(FA+da)\/(FB+db) - FA\/FB/

=== Additional error from truncation

Given:

>>> 4.13 :: UFixed 16 3
4.125
>>> 20.9 :: UFixed 16 3
20.875

The expected error that we would get from multiplication is:
/20.875*0.005 + 4.125*0.025 + 0.025*0.005 = 0.207625/

>>> 4.13 * 20.9 :: Double
86.317
>>> (4.13 :: UFixed 16 3) `mul` (20.9 :: UFixed 16 3) :: UFixed 32 6
86.109375
>>> 86.109375 + 0.207625 :: Double
86.317

However the /0.109375/ is smaller than /2^-3/, so the regular multiplication
operator that uses truncation introduces an additional error of /0.109375/:

>>> (4.13 :: UFixed 16 3) * (20.9 :: UFixed 16 3) :: UFixed 16 3
86.0

-}

{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE TemplateHaskell #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE UndecidableInstances #-}

{-# LANGUAGE Trustworthy #-}

{-# OPTIONS_GHC -fplugin=GHC.TypeLits.Normalise #-}
{-# OPTIONS_GHC -fplugin GHC.TypeLits.KnownNat.Solver #-}
{-# OPTIONS_HADDOCK show-extensions #-}

module Clash.Sized.Fixed
( -- * 'SFixed': 'Signed' 'Fixed' point numbers
SFixed, sf, unSF
-- * 'UFixed': 'Unsigned' 'Fixed' point numbers
, UFixed, uf, unUF
-- * Division
, divide
-- * Compile-time 'Double' conversion
, fLit
-- * Run-time 'Double' conversion (not synthesizable)
, fLitR
-- * 'Fixed' point wrapper
, Fixed (..), resizeF, fracShift
-- * Constraint synonyms #constraintsynonyms#
-- \$constraintsynonyms

-- ** Constraint synonyms for 'SFixed'
, NumSFixedC, ENumSFixedC, FracSFixedC, ResizeSFC, DivideSC
-- ** Constraint synonyms for 'UFixed'
, NumUFixedC, ENumUFixedC, FracUFixedC, ResizeUFC, DivideUC
-- ** Constraint synonyms for 'Fixed' wrapper
, NumFixedC, ENumFixedC, FracFixedC, ResizeFC, DivideC
-- * Proxy
, asRepProxy, asIntProxy
)
where

import Control.DeepSeq            (NFData)
import Control.Arrow              ((***), second)
import Data.Bits                  (Bits (..), FiniteBits)
import Data.Data                  (Data)
import Data.Default.Class         (Default (..))
import Data.Either                (isLeft)
import Data.Kind                  (Type)
import Data.List                  (find)
import Data.Proxy                 (Proxy (..))
import Data.Ratio                 ((%), denominator, numerator)
import Data.Typeable              (Typeable, TypeRep, typeRep)
import GHC.TypeLits               (KnownNat, Nat, type (+), natVal)
import GHC.TypeLits.Extra         (Max)
import Language.Haskell.TH        (Q, TExp, TypeQ, appT, conT, litT, mkName,
numTyLit, sigE)
import Test.QuickCheck            (Arbitrary, CoArbitrary)

import Clash.Class.BitPack        (BitPack (..))
import Clash.Class.Num            (ExtendingNum (..), SaturatingNum (..),
SaturationMode (..), boundedAdd, boundedSub,
boundedMul)
import Clash.Class.Resize         (Resize (..))
import Clash.Promoted.Nat         (SNat)
import Clash.Prelude.BitIndex     (msb, split)
import Clash.Prelude.BitReduction (reduceAnd, reduceOr)
import Clash.Sized.BitVector      (BitVector, (++#))
import Clash.Sized.Signed         (Signed)
import Clash.Sized.Unsigned       (Unsigned)
import Clash.XException
(ShowX (..), NFDataX (..), isX, errorX, showsPrecXWith, fromJustX)

{- \$setup
>>> :set -XDataKinds
>>> import Clash.Prelude
>>> let n = \$\$(fLit pi) :: SFixed 4 4
-}

-- | 'Fixed'-point number
--
-- Where:
--
-- * @rep@ is the underlying representation
--
-- * @int@ is the number of bits used to represent the integer part
--
-- * @frac@ is the number of bits used to represent the fractional part
--
-- The 'Num' operators for this type saturate to 'maxBound' on overflow and
-- 'minBound' on underflow, and use truncation as the rounding method.
newtype Fixed (rep :: Nat -> Type) (int :: Nat) (frac :: Nat) =
Fixed { Fixed rep int frac -> rep (int + frac)
unFixed :: rep (int + frac) }

deriving instance NFData (rep (int + frac)) => NFData (Fixed rep int frac)
deriving instance (Typeable rep, Typeable int, Typeable frac
, Data (rep (int + frac))) => Data (Fixed rep int frac)
deriving instance Eq (rep (int + frac))      => Eq (Fixed rep int frac)
deriving instance Ord (rep (int + frac))     => Ord (Fixed rep int frac)
deriving instance Enum (rep (int + frac))    => Enum (Fixed rep int frac)
deriving instance Bounded (rep (int + frac)) => Bounded (Fixed rep int frac)
deriving instance Default (rep (int + frac)) => Default (Fixed rep int frac)
deriving instance Arbitrary (rep (int + frac)) => Arbitrary (Fixed rep int frac)
deriving instance CoArbitrary (rep (int + frac)) => CoArbitrary (Fixed rep int frac)
deriving instance FiniteBits (rep (int + frac)) => FiniteBits (Fixed rep int frac)

-- | Instance functions do not saturate.
-- Meaning that \"@\`shiftL\` 1 == 'satMul' 'SatWrap' 2'@\"
deriving instance Bits (rep (int + frac)) => Bits (Fixed rep int frac)

-- | Signed 'Fixed'-point number, with @int@ integer bits (including sign-bit)
-- and @frac@ fractional bits.
--
-- * The range 'SFixed' @int@ @frac@ numbers is: [-(2^(@int@ -1)) ..
-- 2^(@int@-1) - 2^-@frac@ ]
-- * The resolution of 'SFixed' @int@ @frac@ numbers is: 2^@frac@
-- * The 'Num' operators for this type saturate on overflow,
--   and use truncation as the rounding method.
--
-- >>>  maxBound :: SFixed 3 4
-- 3.9375
-- >>> minBound :: SFixed 3 4
-- -4.0
-- >>> read (show (maxBound :: SFixed 3 4)) :: SFixed 3 4
-- 3.9375
-- >>> 1 + 2 :: SFixed 3 4
-- 3.0
-- >>> 2 + 3 :: SFixed 3 4
-- 3.9375
-- >>> (-2) + (-3) :: SFixed 3 4
-- -4.0
-- >>> 1.375 * (-0.8125) :: SFixed 3 4
-- -1.125
-- >>> (1.375 :: SFixed 3 4) `mul` (-0.8125 :: SFixed 3 4) :: SFixed 6 8
-- -1.1171875
-- >>> (2 :: SFixed 3 4) `add` (3 :: SFixed 3 4) :: SFixed 4 4
-- 5.0
-- >>> (-2 :: SFixed 3 4) `add` (-3 :: SFixed 3 4) :: SFixed 4 4
-- -5.0
type SFixed = Fixed Signed

-- | Unsigned 'Fixed'-point number, with @int@ integer bits and @frac@
-- fractional bits
--
-- * The range 'UFixed' @int@ @frac@ numbers is: [0 .. 2^@int@ - 2^-@frac@ ]
-- * The resolution of 'UFixed' @int@ @frac@ numbers is: 2^@frac@
-- * The 'Num' operators for this type saturate on overflow,
--   and use truncation as the rounding method.
--
-- >>> maxBound :: UFixed 3 4
-- 7.9375
-- >>> minBound :: UFixed 3 4
-- 0.0
-- >>> 1 + 2 :: UFixed 3 4
-- 3.0
-- >>> 2 + 6 :: UFixed 3 4
-- 7.9375
-- >>> 1 - 3 :: UFixed 3 4
-- 0.0
-- >>> 1.375 * 0.8125 :: UFixed 3 4
-- 1.0625
-- >>> (1.375 :: UFixed 3 4) `mul` (0.8125 :: UFixed 3 4) :: UFixed 6 8
-- 1.1171875
-- >>> (2 :: UFixed 3 4) `add` (6 :: UFixed 3 4) :: UFixed 4 4
-- 8.0
--
-- However, 'sub' does not saturate to 'minBound' on underflow:
--
-- >>> (1 :: UFixed 3 4) `sub` (3 :: UFixed 3 4) :: UFixed 4 4
-- 14.0
type UFixed = Fixed Unsigned

{-# INLINE sf #-}
-- | Treat a 'Signed' integer as a @Signed@ 'Fixed'-@point@ integer
--
-- >>> sf d4 (-22 :: Signed 7)
-- -1.375
sf
:: SNat frac
-- ^ Position of the virtual @point@
-> Signed (int + frac)
-- ^ The 'Signed' integer
-> SFixed int frac
sf :: SNat frac -> Signed (int + frac) -> SFixed int frac
sf _ fRep :: Signed (int + frac)
fRep = Signed (int + frac) -> SFixed int frac
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
rep (int + frac) -> Fixed rep int frac
Fixed Signed (int + frac)
fRep

{-# INLINE unSF #-}
-- | See the underlying representation of a Signed Fixed-point integer
unSF :: SFixed int frac
-> Signed (int + frac)
unSF :: SFixed int frac -> Signed (int + frac)
unSF (Fixed fRep :: Signed (int + frac)
fRep) = Signed (int + frac)
fRep

{-# INLINE uf #-}
-- | Treat an 'Unsigned' integer as a @Unsigned@ 'Fixed'-@point@ number
--
-- >>> uf d4 (92 :: Unsigned 7)
-- 5.75
uf
:: SNat frac
-- ^ Position of the virtual @point@
-> Unsigned (int + frac)
-- ^ The 'Unsigned' integer
-> UFixed int frac
uf :: SNat frac -> Unsigned (int + frac) -> UFixed int frac
uf _ fRep :: Unsigned (int + frac)
fRep = Unsigned (int + frac) -> UFixed int frac
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
rep (int + frac) -> Fixed rep int frac
Fixed Unsigned (int + frac)
fRep

{-# INLINE unUF #-}
-- | See the underlying representation of an Unsigned Fixed-point integer
unUF :: UFixed int frac
-> Unsigned (int + frac)
unUF :: UFixed int frac -> Unsigned (int + frac)
unUF (Fixed fRep :: Unsigned (int + frac)
fRep) = Unsigned (int + frac)
fRep

{-# INLINE asRepProxy #-}
-- | 'Fixed' as a 'Proxy' for it's representation type @rep@
asRepProxy :: Fixed rep int frac -> Proxy rep
asRepProxy :: Fixed rep int frac -> Proxy rep
asRepProxy _ = Proxy rep
forall k (t :: k). Proxy t
Proxy

{-# INLINE asIntProxy #-}
-- | 'Fixed' as a 'Proxy' for the number of integer bits @int@
asIntProxy :: Fixed rep int frac -> Proxy int
asIntProxy :: Fixed rep int frac -> Proxy int
asIntProxy _ = Proxy int
forall k (t :: k). Proxy t
Proxy

-- | Get the position of the virtual @point@ of a 'Fixed'-@point@ number
fracShift :: KnownNat frac => Fixed rep int frac -> Int
fracShift :: Fixed rep int frac -> Int
fracShift fx :: Fixed rep int frac
fx = Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Fixed rep int frac -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal Fixed rep int frac
fx)

instance ( size ~ (int + frac), KnownNat frac, Integral (rep size)
) => Show (Fixed rep int frac) where
show :: Fixed rep int frac -> String
show f :: Fixed rep int frac
f@(Fixed fRep :: rep (int + frac)
fRep) =
String
i String -> ShowS
forall a. [a] -> [a] -> [a]
++ "." String -> ShowS
forall a. [a] -> [a] -> [a]
++ ((Int -> ShowS) -> (Int, String) -> String
forall a b c. (a -> b -> c) -> (a, b) -> c
uncurry Int -> ShowS
pad ((Int, String) -> String)
-> (Ratio Integer -> (Int, String)) -> Ratio Integer -> String
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (Ratio Integer -> String) -> (Int, Ratio Integer) -> (Int, String)
forall (a :: Type -> Type -> Type) b c d.
Arrow a =>
a b c -> a (d, b) (d, c)
second (Integer -> String
forall a. Show a => a -> String
show (Integer -> String)
-> (Ratio Integer -> Integer) -> Ratio Integer -> String
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Ratio Integer -> Integer
forall a. Ratio a -> a
numerator) ((Int, Ratio Integer) -> (Int, String))
-> (Ratio Integer -> (Int, Ratio Integer))
-> Ratio Integer
-> (Int, String)
forall b c a. (b -> c) -> (a -> b) -> a -> c
.
Maybe (Int, Ratio Integer) -> (Int, Ratio Integer)
forall a. HasCallStack => Maybe a -> a
fromJustX (Maybe (Int, Ratio Integer) -> (Int, Ratio Integer))
-> (Ratio Integer -> Maybe (Int, Ratio Integer))
-> Ratio Integer
-> (Int, Ratio Integer)
forall b c a. (b -> c) -> (a -> b) -> a -> c
. ((Int, Ratio Integer) -> Bool)
-> [(Int, Ratio Integer)] -> Maybe (Int, Ratio Integer)
forall (t :: Type -> Type) a.
Foldable t =>
(a -> Bool) -> t a -> Maybe a
find ((Integer -> Integer -> Bool
forall a. Eq a => a -> a -> Bool
==1) (Integer -> Bool)
-> ((Int, Ratio Integer) -> Integer)
-> (Int, Ratio Integer)
-> Bool
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Ratio Integer -> Integer
forall a. Ratio a -> a
denominator (Ratio Integer -> Integer)
-> ((Int, Ratio Integer) -> Ratio Integer)
-> (Int, Ratio Integer)
-> Integer
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (Int, Ratio Integer) -> Ratio Integer
forall a b. (a, b) -> b
snd) ([(Int, Ratio Integer)] -> Maybe (Int, Ratio Integer))
-> (Ratio Integer -> [(Int, Ratio Integer)])
-> Ratio Integer
-> Maybe (Int, Ratio Integer)
forall b c a. (b -> c) -> (a -> b) -> a -> c
.
((Int, Ratio Integer) -> (Int, Ratio Integer))
-> (Int, Ratio Integer) -> [(Int, Ratio Integer)]
forall a. (a -> a) -> a -> [a]
iterate (Int -> Int
forall a. Enum a => a -> a
succ (Int -> Int)
-> (Ratio Integer -> Ratio Integer)
-> (Int, Ratio Integer)
-> (Int, Ratio Integer)
forall (a :: Type -> Type -> Type) b c b' c'.
Arrow a =>
a b c -> a b' c' -> a (b, b') (c, c')
*** (Ratio Integer -> Ratio Integer -> Ratio Integer
forall a. Num a => a -> a -> a
*10)) ((Int, Ratio Integer) -> [(Int, Ratio Integer)])
-> (Ratio Integer -> (Int, Ratio Integer))
-> Ratio Integer
-> [(Int, Ratio Integer)]
forall b c a. (b -> c) -> (a -> b) -> a -> c
. (,) 0 (Ratio Integer -> String) -> Ratio Integer -> String
forall a b. (a -> b) -> a -> b
\$ (Integer
nom Integer -> Integer -> Ratio Integer
forall a. Integral a => a -> a -> Ratio a
% Integer
denom))
where
pad :: Int -> ShowS
pad n :: Int
n str :: String
str = Int -> Char -> String
forall a. Int -> a -> [a]
replicate (Int
n Int -> Int -> Int
forall a. Num a => a -> a -> a
- String -> Int
forall (t :: Type -> Type) a. Foldable t => t a -> Int
length String
str) '0' String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
str

nF :: Int
nF        = Fixed rep int frac -> Int
forall (frac :: Nat) (rep :: Nat -> Type) (int :: Nat).
KnownNat frac =>
Fixed rep int frac -> Int
fracShift Fixed rep int frac
f
fRepI :: Integer
fRepI     = rep size -> Integer
forall a. Integral a => a -> Integer
toInteger rep size
rep (int + frac)
fRep
fRepI_abs :: Integer
fRepI_abs = Integer -> Integer
forall a. Num a => a -> a
abs Integer
fRepI
i :: String
i         = if Integer
fRepI Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
< 0 then '-' Char -> ShowS
forall a. a -> [a] -> [a]
: Integer -> String
forall a. Show a => a -> String
show (Integer
fRepI_abs Integer -> Int -> Integer
forall a. Bits a => a -> Int -> a
`shiftR` Int
nF)
else Integer -> String
forall a. Show a => a -> String
show (Integer
fRepI Integer -> Int -> Integer
forall a. Bits a => a -> Int -> a
`shiftR` Int
nF)
nom :: Integer
nom       = if Integer
fRepI Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
< 0 then Integer
fRepI_abs Integer -> Integer -> Integer
forall a. Bits a => a -> a -> a
.&. ((2 Integer -> Int -> Integer
forall a b. (Num a, Integral b) => a -> b -> a
^ Int
nF) Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- 1)
else Integer
fRepI Integer -> Integer -> Integer
forall a. Bits a => a -> a -> a
.&. ((2 Integer -> Int -> Integer
forall a b. (Num a, Integral b) => a -> b -> a
^ Int
nF) Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- 1)
denom :: Integer
denom     = 2 Integer -> Int -> Integer
forall a b. (Num a, Integral b) => a -> b -> a
^ Int
nF

instance ( size ~ (int + frac), KnownNat frac, Integral (rep size)
) => ShowX (Fixed rep int frac) where
showsPrecX :: Int -> Fixed rep int frac -> ShowS
showsPrecX = (Int -> Fixed rep int frac -> ShowS)
-> Int -> Fixed rep int frac -> ShowS
forall a. (Int -> a -> ShowS) -> Int -> a -> ShowS
showsPrecXWith Int -> Fixed rep int frac -> ShowS
forall a. Show a => Int -> a -> ShowS
showsPrec

instance NFDataX (rep (int + frac)) => NFDataX (Fixed rep int frac) where
deepErrorX :: String -> Fixed rep int frac
deepErrorX = rep (int + frac) -> Fixed rep int frac
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
rep (int + frac) -> Fixed rep int frac
Fixed (rep (int + frac) -> Fixed rep int frac)
-> (String -> rep (int + frac)) -> String -> Fixed rep int frac
forall b c a. (b -> c) -> (a -> b) -> a -> c
. String -> rep (int + frac)
forall a. HasCallStack => String -> a
errorX
rnfX :: Fixed rep int frac -> ()
rnfX f :: Fixed rep int frac
f@(~(Fixed x :: rep (int + frac)
x)) = if Either String (Fixed rep int frac) -> Bool
forall a b. Either a b -> Bool
isLeft (Fixed rep int frac -> Either String (Fixed rep int frac)
forall a. a -> Either String a
isX Fixed rep int frac
f) then () else rep (int + frac) -> ()
forall a. NFDataX a => a -> ()
rnfX rep (int + frac)
x
hasUndefined :: Fixed rep int frac -> Bool
hasUndefined f :: Fixed rep int frac
f@(~(Fixed x :: rep (int + frac)
x)) = if Either String (Fixed rep int frac) -> Bool
forall a b. Either a b -> Bool
isLeft (Fixed rep int frac -> Either String (Fixed rep int frac)
forall a. a -> Either String a
isX Fixed rep int frac
f) then Bool
True else rep (int + frac) -> Bool
forall a. NFDataX a => a -> Bool
hasUndefined rep (int + frac)
x
ensureSpine :: Fixed rep int frac -> Fixed rep int frac
ensureSpine ~(Fixed x :: rep (int + frac)
x) = rep (int + frac) -> Fixed rep int frac
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
rep (int + frac) -> Fixed rep int frac
Fixed rep (int + frac)
x

-- | None of the 'Read' class' methods are synthesizable.
instance (size ~ (int + frac), KnownNat frac, Bounded (rep size), Integral (rep size))
=> Read (Fixed rep int frac) where
readPrec :: ReadPrec (Fixed rep int frac)
readPrec = Double -> Fixed rep int frac
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat)
(size :: Nat).
(size ~ (int + frac), KnownNat frac, Bounded (rep size),
Integral (rep size)) =>
Double -> Fixed rep int frac
fLitR (Double -> Fixed rep int frac)
-> ReadPrec Double -> ReadPrec (Fixed rep int frac)
forall (f :: Type -> Type) a b. Functor f => (a -> b) -> f a -> f b
forall a. Read a => ReadPrec a

{- \$constraintsynonyms
Writing polymorphic functions over fixed point numbers can be a potentially
verbose due to the many class constraints induced by the functions and operators
of this module.

Writing a simple multiply-and-accumulate function can already give rise to many
lines of constraints:

@
mac :: ( 'GHC.TypeLits.KnownNat' frac
, 'GHC.TypeLits.KnownNat' (frac + frac)
, 'GHC.TypeLits.KnownNat' (int + frac)
, 'GHC.TypeLits.KnownNat' (1 + (int + frac))
, 'GHC.TypeLits.KnownNat' ((int + frac) + (int + frac))
, ((int + int) + (frac + frac)) ~ ((int + frac) + (int + frac))
)
=> 'SFixed' int frac
-> 'SFixed' int frac
-> 'SFixed' int frac
-> 'SFixed' int frac
mac s x y = s + (x * y)
@

But with constraint synonyms, you can write the type signature like this:

@
mac1 :: 'NumSFixedC' int frac
=> 'SFixed' int frac
-> 'SFixed' int frac
-> 'SFixed' int frac
-> 'SFixed' int frac
mac1 s x y = s + (x * y)
@

Where 'NumSFixedC' refers to the @Constraints@ needed by the operators of
the 'Num' class for the 'SFixed' datatype.

Although the number of constraints for the @mac@ function defined earlier might
be considered small, here is an \"this way lies madness\" example where you
really want to use constraint kinds:

@
mac2 :: ( 'GHC.TypeLits.KnownNat' frac1
, 'GHC.TypeLits.KnownNat' frac2
, 'GHC.TypeLits.KnownNat' frac3
, 'GHC.TypeLits.KnownNat' (Max frac1 frac2)
, 'GHC.TypeLits.KnownNat' (int1 + frac1)
, 'GHC.TypeLits.KnownNat' (int2 + frac2)
, 'GHC.TypeLits.KnownNat' (int3 + frac3)
, 'GHC.TypeLits.KnownNat' (frac1 + frac2)
, 'GHC.TypeLits.KnownNat' (Max (frac1 + frac2) frac3)
, 'GHC.TypeLits.KnownNat' (((int1 + int2) + (frac1 + frac2)) + (int3 + frac3))
, 'GHC.TypeLits.KnownNat' ((int1 + int2) + (frac1 + frac2))
, 'GHC.TypeLits.KnownNat' (1 + Max (int1 + frac1) (int2 + frac2))
, 'GHC.TypeLits.KnownNat' (1 + Max (int1 + int2) int3 + Max (frac1 + frac2) frac3)
, 'GHC.TypeLits.KnownNat' ((1 + Max int1 int2) + Max frac1 frac2)
, 'GHC.TypeLits.KnownNat' ((1 + Max ((int1 + int2) + (frac1 + frac2)) (int3 + frac3)))
, ((int1 + frac1) + (int2 + frac2)) ~ ((int1 + int2) + (frac1 + frac2))
, (((int1 + int2) + int3) + ((frac1 + frac2) + frac3)) ~ (((int1 + int2) + (frac1 + frac2)) + (int3 + frac3))
)
=> 'SFixed' int1 frac1
-> 'SFixed' int2 frac2
-> 'SFixed' int3 frac3
-> 'SFixed' (1 + Max (int1 + int2) int3) (Max (frac1 + frac2) frac3)
mac2 x y s = (x \`mul\` y) \`add\` s
@

Which, with the proper constraint kinds can be reduced to:

@
mac3 :: ( 'ENumSFixedC' int1 frac1 int2 frac2
, 'ENumSFixedC' (int1 + int2) (frac1 + frac2) int3 frac3
)
=> 'SFixed' int1 frac1
-> 'SFixed' int2 frac2
-> 'SFixed' int3 frac3
-> 'SFixed' (1 + Max (int1 + int2) int3) (Max (frac1 + frac2) frac3)
mac3 x y s = (x \`mul\` y) \`add\` s
@
-}

-- | Constraint for the 'ExtendingNum' instance of 'Fixed'
type ENumFixedC rep int1 frac1 int2 frac2
= ( Bounded  (rep ((1 + Max int1 int2) + Max frac1 frac2))
, Num      (rep ((1 + Max int1 int2) + Max frac1 frac2))
, Bits     (rep ((1 + Max int1 int2) + Max frac1 frac2))
, ExtendingNum (rep (int1 + frac1)) (rep (int2 + frac2))
, MResult (rep (int1 + frac1)) (rep (int2 + frac2)) ~
rep ((int1 + int2) + (frac1 + frac2))
, KnownNat int1
, KnownNat int2
, KnownNat frac1
, KnownNat frac2
, Resize   rep
)

-- | Constraint for the 'ExtendingNum' instance of 'SFixed'
type ENumSFixedC int1 frac1 int2 frac2
= ( KnownNat (int2 + frac2)
, KnownNat (1 + Max int1 int2 + Max frac1 frac2)
, KnownNat (Max frac1 frac2)
, KnownNat (1 + Max int1 int2)
, KnownNat (int1 + frac1)
, KnownNat frac2
, KnownNat int2
, KnownNat frac1
, KnownNat int1
)

-- | Constraint for the 'ExtendingNum' instance of 'UFixed'
type ENumUFixedC int1 frac1 int2 frac2 =
ENumSFixedC int1 frac1 int2 frac2

-- | When used in a polymorphic setting, use the following
-- <Clash-Sized-Fixed.html#constraintsynonyms Constraint synonyms> for less
-- verbose type signatures:
--
-- * @'ENumFixedC'  rep frac1 frac2 size1 size2@ for: 'Fixed'
-- * @'ENumSFixedC' int1 frac1 int2 frac2@       for: 'SFixed'
-- * @'ENumUFixedC' int1 frac1 int2 frac2@       for: 'UFixed'
instance ENumFixedC rep int1 frac1 int2 frac2 =>
ExtendingNum (Fixed rep int1 frac1) (Fixed rep int2 frac2) where
type AResult (Fixed rep int1 frac1) (Fixed rep int2 frac2) =
Fixed rep (1 + Max int1 int2) (Max frac1 frac2)
add :: Fixed rep int1 frac1
-> Fixed rep int2 frac2
-> AResult (Fixed rep int1 frac1) (Fixed rep int2 frac2)
add (Fixed f1 :: rep (int1 + frac1)
f1) (Fixed f2 :: rep (int2 + frac2)
f2) =
let sh1 :: Int
sh1 = Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Proxy (Max frac1 frac2) -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Proxy (Max frac1 frac2)
forall k (t :: k). Proxy t
Proxy @(Max frac1 frac2)) Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Proxy frac1 -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Proxy frac1
forall k (t :: k). Proxy t
Proxy @frac1)) :: Int
f1R :: rep ((1 + Max int1 int2) + Max frac1 frac2)
f1R = rep ((1 + Max int1 int2) + Max frac1 frac2)
-> Int -> rep ((1 + Max int1 int2) + Max frac1 frac2)
forall a. Bits a => a -> Int -> a
shiftL (rep (int1 + frac1) -> rep ((1 + Max int1 int2) + Max frac1 frac2)
forall (f :: Nat -> Type) (a :: Nat) (b :: Nat).
(Resize f, KnownNat a, KnownNat b) =>
f a -> f b
resize rep (int1 + frac1)
f1) Int
sh1 :: rep ((1 + Max int1 int2) + (Max frac1 frac2))
sh2 :: Int
sh2 = Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Proxy (Max frac1 frac2) -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Proxy (Max frac1 frac2)
forall k (t :: k). Proxy t
Proxy @(Max frac1 frac2)) Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Proxy frac2 -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Proxy frac2
forall k (t :: k). Proxy t
Proxy @frac2)) :: Int
f2R :: rep ((1 + Max int1 int2) + Max frac1 frac2)
f2R = rep ((1 + Max int1 int2) + Max frac1 frac2)
-> Int -> rep ((1 + Max int1 int2) + Max frac1 frac2)
forall a. Bits a => a -> Int -> a
shiftL (rep (int2 + frac2) -> rep ((1 + Max int1 int2) + Max frac1 frac2)
forall (f :: Nat -> Type) (a :: Nat) (b :: Nat).
(Resize f, KnownNat a, KnownNat b) =>
f a -> f b
resize rep (int2 + frac2)
f2) Int
sh2 :: rep ((1 + Max int1 int2) + (Max frac1 frac2))
in  rep ((1 + Max int1 int2) + Max frac1 frac2)
-> Fixed rep (1 + Max int1 int2) (Max frac1 frac2)
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
rep (int + frac) -> Fixed rep int frac
Fixed (rep ((1 + Max int1 int2) + Max frac1 frac2)
f1R rep ((1 + Max int1 int2) + Max frac1 frac2)
-> rep ((1 + Max int1 int2) + Max frac1 frac2)
-> rep ((1 + Max int1 int2) + Max frac1 frac2)
forall a. Num a => a -> a -> a
+ rep ((1 + Max int1 int2) + Max frac1 frac2)
f2R)
sub :: Fixed rep int1 frac1
-> Fixed rep int2 frac2
-> AResult (Fixed rep int1 frac1) (Fixed rep int2 frac2)
sub (Fixed f1 :: rep (int1 + frac1)
f1) (Fixed f2 :: rep (int2 + frac2)
f2) =
let sh1 :: Int
sh1 = Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Proxy (Max frac1 frac2) -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Proxy (Max frac1 frac2)
forall k (t :: k). Proxy t
Proxy @(Max frac1 frac2)) Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Proxy frac1 -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Proxy frac1
forall k (t :: k). Proxy t
Proxy @frac1)) :: Int
f1R :: rep ((1 + Max int1 int2) + Max frac1 frac2)
f1R = rep ((1 + Max int1 int2) + Max frac1 frac2)
-> Int -> rep ((1 + Max int1 int2) + Max frac1 frac2)
forall a. Bits a => a -> Int -> a
shiftL (rep (int1 + frac1) -> rep ((1 + Max int1 int2) + Max frac1 frac2)
forall (f :: Nat -> Type) (a :: Nat) (b :: Nat).
(Resize f, KnownNat a, KnownNat b) =>
f a -> f b
resize rep (int1 + frac1)
f1) Int
sh1 :: rep ((1 + Max int1 int2) + (Max frac1 frac2))
sh2 :: Int
sh2 = Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Proxy (Max frac1 frac2) -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Proxy (Max frac1 frac2)
forall k (t :: k). Proxy t
Proxy @(Max frac1 frac2)) Integer -> Integer -> Integer
forall a. Num a => a -> a -> a
- Proxy frac2 -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Proxy frac2
forall k (t :: k). Proxy t
Proxy @frac2)) :: Int
f2R :: rep ((1 + Max int1 int2) + Max frac1 frac2)
f2R = rep ((1 + Max int1 int2) + Max frac1 frac2)
-> Int -> rep ((1 + Max int1 int2) + Max frac1 frac2)
forall a. Bits a => a -> Int -> a
shiftL (rep (int2 + frac2) -> rep ((1 + Max int1 int2) + Max frac1 frac2)
forall (f :: Nat -> Type) (a :: Nat) (b :: Nat).
(Resize f, KnownNat a, KnownNat b) =>
f a -> f b
resize rep (int2 + frac2)
f2) Int
sh2 :: rep ((1 + Max int1 int2) + (Max frac1 frac2))
in  rep ((1 + Max int1 int2) + Max frac1 frac2)
-> Fixed rep (1 + Max int1 int2) (Max frac1 frac2)
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
rep (int + frac) -> Fixed rep int frac
Fixed (rep ((1 + Max int1 int2) + Max frac1 frac2)
f1R rep ((1 + Max int1 int2) + Max frac1 frac2)
-> rep ((1 + Max int1 int2) + Max frac1 frac2)
-> rep ((1 + Max int1 int2) + Max frac1 frac2)
forall a. Num a => a -> a -> a
- rep ((1 + Max int1 int2) + Max frac1 frac2)
f2R)
type MResult (Fixed rep int1 frac1) (Fixed rep int2 frac2) =
Fixed rep (int1 + int2) (frac1 + frac2)
mul :: Fixed rep int1 frac1
-> Fixed rep int2 frac2
-> MResult (Fixed rep int1 frac1) (Fixed rep int2 frac2)
mul (Fixed fRep1 :: rep (int1 + frac1)
fRep1) (Fixed fRep2 :: rep (int2 + frac2)
fRep2) = rep ((int1 + int2) + (frac1 + frac2))
-> Fixed rep (int1 + int2) (frac1 + frac2)
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
rep (int + frac) -> Fixed rep int frac
Fixed (rep (int1 + frac1)
-> rep (int2 + frac2)
-> MResult (rep (int1 + frac1)) (rep (int2 + frac2))
forall a b. ExtendingNum a b => a -> b -> MResult a b
mul rep (int1 + frac1)
fRep1 rep (int2 + frac2)
fRep2)

-- | Constraint for the 'Num' instance of 'Fixed'
type NumFixedC rep int frac
= ( SaturatingNum (rep (int + frac))
, ExtendingNum (rep (int + frac)) (rep (int + frac))
, MResult (rep (int + frac)) (rep (int + frac)) ~
rep ((int + int) + (frac + frac))
, BitSize (rep ((int + int) + (frac + frac))) ~
(int + ((int + frac) + frac))
, BitPack (rep ((int + int) + (frac + frac)))
, Bits    (rep ((int + int) + (frac + frac)))
, BitPack (rep (int + frac))
, Enum    (rep (int + frac))
, Bits    (rep (int + frac))
, Ord     (rep (int + frac))
, Integral (rep (int + frac))
, Resize  rep
, KnownNat int
, KnownNat frac
)

-- | Constraint for the 'Num' instance of 'SFixed'
type NumSFixedC int frac =
( KnownNat ((int + int) + (frac + frac))
, KnownNat (frac + frac)
, KnownNat (int + int)
, KnownNat (int + frac)
, KnownNat frac
, KnownNat int
)

-- | Constraint for the 'Num' instance of 'UFixed'
type NumUFixedC int frac =
NumSFixedC int frac

-- | The operators of this instance saturate on overflow, and use truncation as
-- the rounding method.
--
-- When used in a polymorphic setting, use the following
-- <Clash-Sized-Fixed.html#constraintsynonyms Constraint synonyms> for less
-- verbose type signatures:
--
-- * @'NumFixedC' frac rep size@ for: @'Fixed' frac rep size@
-- * @'NumSFixedC' int frac@     for: @'SFixed' int frac@
-- * @'NumUFixedC' int frac@     for: @'UFixed' int frac@
instance (NumFixedC rep int frac) => Num (Fixed rep int frac) where
+ :: Fixed rep int frac -> Fixed rep int frac -> Fixed rep int frac
(+)              = Fixed rep int frac -> Fixed rep int frac -> Fixed rep int frac
forall a. SaturatingNum a => a -> a -> a
* :: Fixed rep int frac -> Fixed rep int frac -> Fixed rep int frac
(*)              = Fixed rep int frac -> Fixed rep int frac -> Fixed rep int frac
forall a. SaturatingNum a => a -> a -> a
boundedMul
(-)              = Fixed rep int frac -> Fixed rep int frac -> Fixed rep int frac
forall a. SaturatingNum a => a -> a -> a
boundedSub
negate :: Fixed rep int frac -> Fixed rep int frac
negate           = Fixed rep int frac -> Fixed rep int frac -> Fixed rep int frac
forall a. SaturatingNum a => a -> a -> a
boundedSub (rep (int + frac) -> Fixed rep int frac
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
rep (int + frac) -> Fixed rep int frac
Fixed 0)
abs :: Fixed rep int frac -> Fixed rep int frac
abs    (Fixed a :: rep (int + frac)
a) = rep (int + frac) -> Fixed rep int frac
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
rep (int + frac) -> Fixed rep int frac
Fixed (rep (int + frac) -> rep (int + frac)
forall a. Num a => a -> a
abs rep (int + frac)
a)
signum :: Fixed rep int frac -> Fixed rep int frac
signum (Fixed a :: rep (int + frac)
a)
| rep (int + frac)
a rep (int + frac) -> rep (int + frac) -> Bool
forall a. Eq a => a -> a -> Bool
== 0       = 0
| rep (int + frac)
a rep (int + frac) -> rep (int + frac) -> Bool
forall a. Ord a => a -> a -> Bool
<  0       = -1
| Bool
otherwise    = 1
fromInteger :: Integer -> Fixed rep int frac
fromInteger i :: Integer
i    = let fSH :: Int
fSH = Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Proxy frac -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Proxy frac
forall k (t :: k). Proxy t
Proxy @frac))
res :: Integer
res = Integer
i Integer -> Int -> Integer
forall a. Bits a => a -> Int -> a
`shiftL` Int
fSH
rMax :: Integer
rMax = rep (int + frac) -> Integer
forall a. Integral a => a -> Integer
toInteger (rep (int + frac)
forall a. Bounded a => a
maxBound :: rep (int + frac))
rMin :: Integer
rMin = rep (int + frac) -> Integer
forall a. Integral a => a -> Integer
toInteger (rep (int + frac)
forall a. Bounded a => a
minBound :: rep (int + frac))
sat :: Integer
sat | Integer
res Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
> Integer
rMax = Integer
rMax
| Integer
res Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
< Integer
rMin = Integer
rMin
| Bool
otherwise  = Integer
res
in  rep (int + frac) -> Fixed rep int frac
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
rep (int + frac) -> Fixed rep int frac
Fixed (Integer -> rep (int + frac)
forall a. Num a => Integer -> a
fromInteger Integer
sat)

instance (BitPack (rep (int + frac))) => BitPack (Fixed rep int frac) where
type BitSize (Fixed rep int frac) = BitSize (rep (int + frac))
pack :: Fixed rep int frac -> BitVector (BitSize (Fixed rep int frac))
pack   (Fixed fRep :: rep (int + frac)
fRep) = rep (int + frac) -> BitVector (BitSize (rep (int + frac)))
forall a. BitPack a => a -> BitVector (BitSize a)
pack rep (int + frac)
fRep
unpack :: BitVector (BitSize (Fixed rep int frac)) -> Fixed rep int frac
unpack bv :: BitVector (BitSize (Fixed rep int frac))
bv           = rep (int + frac) -> Fixed rep int frac
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
rep (int + frac) -> Fixed rep int frac
Fixed (BitVector (BitSize (rep (int + frac))) -> rep (int + frac)
forall a. BitPack a => BitVector (BitSize a) -> a
unpack BitVector (BitSize (rep (int + frac)))
BitVector (BitSize (Fixed rep int frac))
bv)

instance (Lift (rep (int + frac)), KnownNat frac, KnownNat int, Typeable rep) =>
Lift (Fixed rep int frac) where
lift :: Fixed rep int frac -> Q Exp
lift f :: Fixed rep int frac
f@(Fixed fRep :: rep (int + frac)
fRep) = Q Exp -> TypeQ -> Q Exp
sigE [| Fixed fRep |]
(TypeRep -> Integer -> Integer -> TypeQ
decFixed (Proxy rep -> TypeRep
forall k (proxy :: k -> Type) (a :: k).
Typeable a =>
proxy a -> TypeRep
typeRep (Fixed rep int frac -> Proxy rep
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
Fixed rep int frac -> Proxy rep
asRepProxy Fixed rep int frac
f))
(Proxy int -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Fixed rep int frac -> Proxy int
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
Fixed rep int frac -> Proxy int
asIntProxy Fixed rep int frac
f))
(Fixed rep int frac -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal Fixed rep int frac
f))

decFixed :: TypeRep -> Integer -> Integer -> TypeQ
decFixed :: TypeRep -> Integer -> Integer -> TypeQ
decFixed r :: TypeRep
r i :: Integer
i f :: Integer
f = do
(TypeQ -> TypeQ -> TypeQ) -> TypeQ -> [TypeQ] -> TypeQ
forall (t :: Type -> Type) b a.
Foldable t =>
(b -> a -> b) -> b -> t a -> b
foldl TypeQ -> TypeQ -> TypeQ
appT (Name -> TypeQ
conT ''Fixed) [ Name -> TypeQ
conT (String -> Name
mkName (TypeRep -> String
forall a. Show a => a -> String
show TypeRep
r))
, TyLitQ -> TypeQ
litT (Integer -> TyLitQ
numTyLit Integer
i)
, TyLitQ -> TypeQ
litT (Integer -> TyLitQ
numTyLit Integer
f)
]

-- | Constraint for the 'resizeF' function
type ResizeFC rep int1 frac1 int2 frac2
= ( Resize   rep
, Ord      (rep (int1 + frac1))
, Num      (rep (int1 + frac1))
, Bits     (rep (int1 + frac1))
, Bits     (rep (int2 + frac2))
, Bounded  (rep (int2 + frac2))
, KnownNat int1
, KnownNat frac1
, KnownNat int2
, KnownNat frac2
)

-- | Constraint for the 'resizeF' function, specialized for 'SFixed'
type ResizeSFC int1 frac1 int2 frac2
= ( KnownNat int1
, KnownNat frac1
, KnownNat int2
, KnownNat frac2
, KnownNat (int2 + frac2)
, KnownNat (int1 + frac1)
)

-- | Constraint for the 'resizeF' function, specialized for 'UFixed'
type ResizeUFC int1 frac1 int2 frac2 =
ResizeSFC int1 frac1 int2 frac2

{-# INLINE resizeF #-}
-- | Saturating resize operation, truncates for rounding
--
-- >>> 0.8125 :: SFixed 3 4
-- 0.8125
-- >>> resizeF (0.8125 :: SFixed 3 4) :: SFixed 2 3
-- 0.75
-- >>> 3.4 :: SFixed 3 4
-- 3.375
-- >>> resizeF (3.4 :: SFixed 3 4) :: SFixed 2 3
-- 1.875
-- >>> maxBound :: SFixed 2 3
-- 1.875
--
-- When used in a polymorphic setting, use the following
-- <#constraintsynonyms Constraint synonyms> for less verbose type signatures:
--
-- * @'ResizeFC' rep int1 frac1 int2 frac2@ for:
--   @'Fixed' rep int1 frac1 -> 'Fixed' rep int2 frac2@
--
-- * @'ResizeSFC' int1 frac1 int2 frac2@ for:
--   @'SFixed' int1 frac1 -> 'SFixed' int2 frac2@
--
-- * @'ResizeUFC' rep int1 frac1 int2 frac2@ for:
--   @'UFixed' int1 frac1 -> 'UFixed' int2 frac2@
resizeF
:: forall rep int1 frac1 int2 frac2
. ResizeFC rep int1 frac1 int2 frac2
=> Fixed rep int1 frac1
-> Fixed rep int2 frac2
resizeF :: Fixed rep int1 frac1 -> Fixed rep int2 frac2
resizeF (Fixed fRep :: rep (int1 + frac1)
fRep) = rep (int2 + frac2) -> Fixed rep int2 frac2
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
rep (int + frac) -> Fixed rep int frac
Fixed rep (int2 + frac2)
sat
where
fMin :: rep (int2 + frac2)
fMin  = rep (int2 + frac2)
forall a. Bounded a => a
minBound :: rep (int2 + frac2)
fMax :: rep (int2 + frac2)
fMax  = rep (int2 + frac2)
forall a. Bounded a => a
maxBound :: rep (int2 + frac2)
argSZ :: Integer
argSZ = Proxy (int1 + frac1) -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Proxy (int1 + frac1)
forall k (t :: k). Proxy t
Proxy @(int1 + frac1))
resSZ :: Integer
resSZ = Proxy (int2 + frac2) -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Proxy (int2 + frac2)
forall k (t :: k). Proxy t
Proxy @(int2 + frac2))

argFracSZ :: Int
argFracSZ = Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Proxy frac1 -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Proxy frac1
forall k (t :: k). Proxy t
Proxy @frac1))
resFracSZ :: Int
resFracSZ = Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Proxy frac2 -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Proxy frac2
forall k (t :: k). Proxy t
Proxy @frac2))

-- All size and frac comparisons and related if-then-else statements should
-- be optimized away by the compiler
sat :: rep (int2 + frac2)
sat = if Integer
argSZ Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
<= Integer
resSZ
-- if the argument is smaller than the result, resize before shift
then if Int
argFracSZ Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<= Int
resFracSZ
then rep (int1 + frac1) -> rep (int2 + frac2)
forall (f :: Nat -> Type) (a :: Nat) (b :: Nat).
(Resize f, KnownNat a, KnownNat b) =>
f a -> f b
resize rep (int1 + frac1)
fRep rep (int2 + frac2) -> Int -> rep (int2 + frac2)
forall a. Bits a => a -> Int -> a
`shiftL` (Int
resFracSZ Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
argFracSZ)
else rep (int1 + frac1) -> rep (int2 + frac2)
forall (f :: Nat -> Type) (a :: Nat) (b :: Nat).
(Resize f, KnownNat a, KnownNat b) =>
f a -> f b
resize rep (int1 + frac1)
fRep rep (int2 + frac2) -> Int -> rep (int2 + frac2)
forall a. Bits a => a -> Int -> a
`shiftR` (Int
argFracSZ Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
resFracSZ)
-- if the argument is bigger than the result, shift before resize
else let mask :: rep (int1 + frac1)
mask = rep (int1 + frac1) -> rep (int1 + frac1)
forall a. Bits a => a -> a
complement (rep (int2 + frac2) -> rep (int1 + frac1)
forall (f :: Nat -> Type) (a :: Nat) (b :: Nat).
(Resize f, KnownNat a, KnownNat b) =>
f a -> f b
resize rep (int2 + frac2)
fMax) :: rep (int1 + frac1)
in if Int
argFracSZ Int -> Int -> Bool
forall a. Ord a => a -> a -> Bool
<= Int
resFracSZ
then let shiftedL :: rep (int1 + frac1)
shiftedL         = rep (int1 + frac1)
fRep rep (int1 + frac1) -> Int -> rep (int1 + frac1)
forall a. Bits a => a -> Int -> a
`shiftL`
(Int
resFracSZ Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
argFracSZ)
shiftedL_masked :: rep (int1 + frac1)
shiftedL_masked  = rep (int1 + frac1)
shiftedL rep (int1 + frac1) -> rep (int1 + frac1) -> rep (int1 + frac1)
forall a. Bits a => a -> a -> a
.&. rep (int1 + frac1)
shiftedL_resized :: rep (int2 + frac2)
shiftedL_resized = rep (int1 + frac1) -> rep (int2 + frac2)
forall (f :: Nat -> Type) (a :: Nat) (b :: Nat).
(Resize f, KnownNat a, KnownNat b) =>
f a -> f b
resize rep (int1 + frac1)
shiftedL
in if rep (int1 + frac1)
fRep rep (int1 + frac1) -> rep (int1 + frac1) -> Bool
forall a. Ord a => a -> a -> Bool
>= 0
then if rep (int1 + frac1)
shiftedL_masked rep (int1 + frac1) -> rep (int1 + frac1) -> Bool
forall a. Eq a => a -> a -> Bool
== 0
then rep (int2 + frac2)
shiftedL_resized
else rep (int2 + frac2)
fMax
else if rep (int1 + frac1)
shiftedL_masked rep (int1 + frac1) -> rep (int1 + frac1) -> Bool
forall a. Eq a => a -> a -> Bool
== rep (int1 + frac1)
then rep (int2 + frac2)
shiftedL_resized
else rep (int2 + frac2)
fMin
else let shiftedR :: rep (int1 + frac1)
shiftedR         = rep (int1 + frac1)
fRep rep (int1 + frac1) -> Int -> rep (int1 + frac1)
forall a. Bits a => a -> Int -> a
`shiftR`
(Int
argFracSZ Int -> Int -> Int
forall a. Num a => a -> a -> a
- Int
resFracSZ)
shiftedR_masked :: rep (int1 + frac1)
shiftedR_masked  = rep (int1 + frac1)
shiftedR rep (int1 + frac1) -> rep (int1 + frac1) -> rep (int1 + frac1)
forall a. Bits a => a -> a -> a
.&. rep (int1 + frac1)
shiftedR_resized :: rep (int2 + frac2)
shiftedR_resized = rep (int1 + frac1) -> rep (int2 + frac2)
forall (f :: Nat -> Type) (a :: Nat) (b :: Nat).
(Resize f, KnownNat a, KnownNat b) =>
f a -> f b
resize rep (int1 + frac1)
shiftedR
in if rep (int1 + frac1)
fRep rep (int1 + frac1) -> rep (int1 + frac1) -> Bool
forall a. Ord a => a -> a -> Bool
>= 0
then if rep (int1 + frac1)
shiftedR_masked rep (int1 + frac1) -> rep (int1 + frac1) -> Bool
forall a. Eq a => a -> a -> Bool
== 0
then rep (int2 + frac2)
shiftedR_resized
else rep (int2 + frac2)
fMax
else if rep (int1 + frac1)
shiftedR_masked rep (int1 + frac1) -> rep (int1 + frac1) -> Bool
forall a. Eq a => a -> a -> Bool
== rep (int1 + frac1)
then rep (int2 + frac2)
shiftedR_resized
else rep (int2 + frac2)
fMin

-- | Convert, at compile-time, a 'Double' /constant/ to a 'Fixed'-point /literal/.
-- The conversion saturates on overflow, and uses truncation as its rounding
-- method.
--
-- So when you type:
--
-- @
-- n = \$\$('fLit' pi) :: 'SFixed' 4 4
-- @
--
-- The compiler sees:
--
-- @
-- n = 'Fixed' (fromInteger 50) :: 'SFixed' 4 4
-- @
--
-- Upon evaluation you see that the value is rounded / truncated in accordance
-- to the fixed point representation:
--
-- >>> n
-- 3.125
--
-- Further examples:
--
-- >>> sin 0.5 :: Double
-- 0.479425538604203
-- >>> \$\$(fLit (sin 0.5)) :: SFixed 1 8
-- 0.4765625
-- >>> atan 0.2 :: Double
-- 0.19739555984988078
-- >>> \$\$(fLit (atan 0.2)) :: SFixed 1 8
-- 0.1953125
-- >>> \$\$(fLit (atan 0.2)) :: SFixed 1 20
-- 0.19739532470703125
fLit
:: forall rep int frac size
. ( size ~ (int + frac)
, KnownNat frac
, Bounded (rep size)
, Integral (rep size) )
=> Double
-> Q (TExp (Fixed rep int frac))
fLit :: Double -> Q (TExp (Fixed rep int frac))
fLit a :: Double
a = [|| Fixed (fromInteger sat) ||]
where
rMax :: Integer
rMax      = rep size -> Integer
forall a. Integral a => a -> Integer
toInteger (rep size
forall a. Bounded a => a
maxBound :: rep size)
rMin :: Integer
rMin      = rep size -> Integer
forall a. Integral a => a -> Integer
toInteger (rep size
forall a. Bounded a => a
minBound :: rep size)
sat :: Integer
sat       = if Integer
truncated Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
> Integer
rMax
then Integer
rMax
else if Integer
truncated Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
< Integer
rMin
then Integer
rMin
else Integer
truncated
truncated :: Integer
truncated = Double -> Integer
forall a b. (RealFrac a, Integral b) => a -> b
truncate Double
shifted :: Integer
shifted :: Double
shifted   = Double
a Double -> Double -> Double
forall a. Num a => a -> a -> a
* (2 Double -> Integer -> Double
forall a b. (Num a, Integral b) => a -> b -> a
^ (Proxy frac -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Proxy frac
forall k (t :: k). Proxy t
Proxy @frac)))

-- | Convert, at run-time, a 'Double' to a 'Fixed'-point.
--
-- __NB__: this functions is /not/ synthesizable
--
-- = Creating data-files #creatingdatafiles#
--
-- An example usage of this function is for example to convert a data file
-- containing 'Double's to a data file with ASCI-encoded binary numbers to be
-- used by a synthesizable function like 'Clash.Prelude.ROM.File.asyncRomFile'.
-- For example, given a file @Data.txt@ containing:
--
-- @
-- 1.2 2.0 3.0 4.0
-- -1.0 -2.0 -3.5 -4.0
-- @
--
-- which we want to put in a ROM, interpreting them as @8.8@ signed fixed point
-- numbers. What we do is that we first create a conversion utility,
-- @createRomFile@, which uses 'fLitR':
--
-- @createRomFile.hs@:
--
-- @
-- module Main where
--
-- import Clash.Prelude
-- import System.Environment
-- import qualified Data.List as L
--
-- createRomFile
--   :: KnownNat n
--   => (Double -> BitVector n)
--   -> FilePath
--   -> FilePath
--   -> IO ()
-- createRomFile convert fileR fileW = do
--   f <- readFile fileR
--   let ds :: [Double]
--       ds = L.concat . (L.map . L.map) read . L.map words \$ lines f
--       bvs = L.map (filter (/= '_') . show . convert) ds
--   writeFile fileW (unlines bvs)
--
-- toSFixed8_8 :: Double -> SFixed 8 8
-- toSFixed8_8 = 'fLitR'
--
-- main :: IO ()
-- main = do
--   [fileR,fileW] <- getArgs
--   createRomFile ('pack' . toSFixed8_8) fileR fileW
-- @
--
-- We then compile this to an executable:
--
-- @
-- \\$ clash --make createRomFile.hs
-- @
--
-- We can then use this utility to convert our @Data.txt@ file which contains
-- 'Double's to a @Data.bin@ file which will containing the desired ASCI-encoded
-- binary data:
--
-- @
-- \\$ ./createRomFile \"Data.txt\" \"Data.bin\"
-- @
--
-- Which results in a @Data.bin@ file containing:
--
-- @
-- 0000000100110011
-- 0000001000000000
-- 0000001100000000
-- 0000010000000000
-- 1111111100000000
-- 1111111000000000
-- 1111110010000000
-- 1111110000000000
-- @
--
-- We can then use this @Data.bin@ file in for our ROM:
--
-- @
-- romF :: Unsigned 3 -> Unsigned 3 -> SFixed 8 8
--                      \$ 'Clash.Prelude.ROM.File.asyncRomFile' d8 "Data.bin" ((rowAddr * 4) + colAddr)
-- @
--
-- And see that it works as expected:
--
-- @
-- __>>> romF 1 2__
-- -3.5
-- __>>> romF 0 0__
-- 1.19921875
-- @
--
-- == Using Template Haskell
--
-- For those of us who like to live on the edge, another option is to convert
-- our @Data.txt@ at compile-time using
-- For this we first create a module @CreateRomFileTH.hs@:
--
-- @
-- module CreateRomFileTH (romDataFromFile) where
--
-- import Clash.Prelude
-- import qualified Data.List        as L
-- import Language.Haskell.TH        (ExpQ, litE, stringL)
-- import Language.Haskell.TH.Syntax (qRunIO)
--
-- createRomFile :: KnownNat n => (Double -> BitVector n)
--               -> FilePath -> FilePath -> IO ()
-- createRomFile convert fileR fileW = do
--   f <- readFile fileR
--   let ds :: [Double]
--       ds = L.concat . (L.map . L.map) read . L.map words \$ lines f
--       bvs = L.map (filter (/= '_') . show . convert) ds
--   writeFile fileW (unlines bvs)
--
-- romDataFromFile :: KnownNat n => (Double -> BitVector n) -> String -> ExpQ
-- romDataFromFile convert fileR = do
--   let fileW = fileR L.++ ".bin"
--   bvF <- qRunIO (createRomFile convert fileR fileW)
--   litE (stringL fileW)
-- @
--
-- Instead of first converting @Data.txt@ to @Data.bin@, we will now use the
-- @romDataFromFile@ function to convert @Data.txt@ to a new file in the proper
-- format at compile-time of our new @romF'@ function:
--
-- @
-- import Clash.Prelude
-- import CreateRomFileTH
--
-- toSFixed8_8 :: Double -> SFixed 8 8
-- toSFixed8_8 = 'fLitR'
--
-- romF' :: Unsigned 3 -> Unsigned 3 -> SFixed 8 8
-- romF' rowAddr colAddr = unpack \$
--   asyncRomFile d8
--                \$(romDataFromFile (pack . toSFixed8_8) "Data.txt") -- Template Haskell splice
-- @
--
-- And see that it works just like the @romF@ function from earlier:
--
-- @
-- __>>> romF' 1 2__
-- -3.5
-- __>>> romF' 0 0__
-- 1.19921875
-- @
fLitR
:: forall rep int frac size
. ( size ~ (int + frac)
, KnownNat frac
, Bounded (rep size)
, Integral (rep size))
=> Double
-> Fixed rep int frac
fLitR :: Double -> Fixed rep int frac
fLitR a :: Double
a = rep (int + frac) -> Fixed rep int frac
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
rep (int + frac) -> Fixed rep int frac
Fixed (Integer -> rep size
forall a. Num a => Integer -> a
fromInteger Integer
sat)
where
rMax :: Integer
rMax      = rep size -> Integer
forall a. Integral a => a -> Integer
toInteger (rep size
forall a. Bounded a => a
maxBound :: rep size)
rMin :: Integer
rMin      = rep size -> Integer
forall a. Integral a => a -> Integer
toInteger (rep size
forall a. Bounded a => a
minBound :: rep size)
sat :: Integer
sat       = if Integer
truncated Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
> Integer
rMax
then Integer
rMax
else if Integer
truncated Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
< Integer
rMin
then Integer
rMin
else Integer
truncated
truncated :: Integer
truncated = Double -> Integer
forall a b. (RealFrac a, Integral b) => a -> b
truncate Double
shifted :: Integer
shifted :: Double
shifted   = Double
a Double -> Double -> Double
forall a. Num a => a -> a -> a
* (2 Double -> Integer -> Double
forall a b. (Num a, Integral b) => a -> b -> a
^ (Proxy frac -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Proxy frac
forall k (t :: k). Proxy t
Proxy @frac)))

instance NumFixedC rep int frac => SaturatingNum (Fixed rep int frac) where
-> Fixed rep int frac -> Fixed rep int frac -> Fixed rep int frac
satAdd w :: SaturationMode
w (Fixed a :: rep (int + frac)
a) (Fixed b :: rep (int + frac)
b) = rep (int + frac) -> Fixed rep int frac
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
rep (int + frac) -> Fixed rep int frac
Fixed (SaturationMode
-> rep (int + frac) -> rep (int + frac) -> rep (int + frac)
forall a. SaturatingNum a => SaturationMode -> a -> a -> a
w rep (int + frac)
a rep (int + frac)
b)
satSub :: SaturationMode
-> Fixed rep int frac -> Fixed rep int frac -> Fixed rep int frac
satSub  w :: SaturationMode
w (Fixed a :: rep (int + frac)
a) (Fixed b :: rep (int + frac)
b) = rep (int + frac) -> Fixed rep int frac
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
rep (int + frac) -> Fixed rep int frac
Fixed (SaturationMode
-> rep (int + frac) -> rep (int + frac) -> rep (int + frac)
forall a. SaturatingNum a => SaturationMode -> a -> a -> a
satSub SaturationMode
w rep (int + frac)
a rep (int + frac)
b)

satMul :: SaturationMode
-> Fixed rep int frac -> Fixed rep int frac -> Fixed rep int frac
satMul SatWrap (Fixed a :: rep (int + frac)
a) (Fixed b :: rep (int + frac)
b) =
let res :: MResult (rep (int + frac)) (rep (int + frac))
res  = rep (int + frac)
a rep (int + frac)
-> rep (int + frac)
-> MResult (rep (int + frac)) (rep (int + frac))
forall a b. ExtendingNum a b => a -> b -> MResult a b
`mul` rep (int + frac)
b
sh :: Int
sh   = Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Proxy frac -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Proxy frac
forall k (t :: k). Proxy t
Proxy @frac))
res' :: rep ((int + int) + (frac + frac))
res' = rep ((int + int) + (frac + frac))
-> Int -> rep ((int + int) + (frac + frac))
forall a. Bits a => a -> Int -> a
shiftR rep ((int + int) + (frac + frac))
MResult (rep (int + frac)) (rep (int + frac))
res Int
sh
in  rep (int + frac) -> Fixed rep int frac
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
rep (int + frac) -> Fixed rep int frac
Fixed (rep ((int + int) + (frac + frac)) -> rep (int + frac)
forall (f :: Nat -> Type) (a :: Nat) (b :: Nat).
(Resize f, KnownNat a, KnownNat b) =>
f a -> f b
resize rep ((int + int) + (frac + frac))
res')

satMul SatBound (Fixed a :: rep (int + frac)
a) (Fixed b :: rep (int + frac)
b) =
let res :: MResult (rep (int + frac)) (rep (int + frac))
res     = rep (int + frac)
a rep (int + frac)
-> rep (int + frac)
-> MResult (rep (int + frac)) (rep (int + frac))
forall a b. ExtendingNum a b => a -> b -> MResult a b
`mul` rep (int + frac)
b
sh :: Int
sh      = Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Proxy frac -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Proxy frac
forall k (t :: k). Proxy t
Proxy @frac))
(rL :: BitVector int
rL,rR :: BitVector ((int + frac) + frac)
rR) = rep ((int + int) + (frac + frac))
-> (BitVector int, BitVector ((int + frac) + frac))
forall a (m :: Nat) (n :: Nat).
(BitPack a, BitSize a ~ (m + n), KnownNat n) =>
a -> (BitVector m, BitVector n)
split rep ((int + int) + (frac + frac))
MResult (rep (int + frac)) (rep (int + frac))
res :: (BitVector int, BitVector (int + frac + frac))
in  case rep (int + frac) -> Bool
forall a. Bits a => a -> Bool
isSigned rep (int + frac)
a of
True  -> let overflow :: Bit
overflow = Bit -> Bit
forall a. Bits a => a -> a
complement (BitVector (1 + int) -> Bit
forall a. BitPack a => a -> Bit
reduceOr (Bit -> BitVector (BitSize Bit)
forall a. BitPack a => a -> BitVector (BitSize a)
pack (BitVector ((int + frac) + frac) -> Bit
forall a. BitPack a => a -> Bit
msb BitVector ((int + frac) + frac)
rR) BitVector 1 -> BitVector int -> BitVector (1 + int)
forall (m :: Nat) (n :: Nat).
KnownNat m =>
BitVector n -> BitVector m -> BitVector (n + m)
++# BitVector int -> BitVector (BitSize (BitVector int))
forall a. BitPack a => a -> BitVector (BitSize a)
pack BitVector int
rL)) Bit -> Bit -> Bit
forall a. Bits a => a -> a -> a
.|.
BitVector (1 + int) -> Bit
forall a. BitPack a => a -> Bit
reduceAnd (Bit -> BitVector (BitSize Bit)
forall a. BitPack a => a -> BitVector (BitSize a)
pack (BitVector ((int + frac) + frac) -> Bit
forall a. BitPack a => a -> Bit
msb BitVector ((int + frac) + frac)
rR) BitVector 1 -> BitVector int -> BitVector (1 + int)
forall (m :: Nat) (n :: Nat).
KnownNat m =>
BitVector n -> BitVector m -> BitVector (n + m)
++# BitVector int -> BitVector (BitSize (BitVector int))
forall a. BitPack a => a -> BitVector (BitSize a)
pack BitVector int
rL)
in  case Bit
overflow of
1 -> BitVector (BitSize (Fixed rep int frac)) -> Fixed rep int frac
forall a. BitPack a => BitVector (BitSize a) -> a
unpack (BitVector ((int + frac) + frac)
-> BitVector (BitSize (rep (int + frac)))
forall (f :: Nat -> Type) (a :: Nat) (b :: Nat).
(Resize f, KnownNat a, KnownNat b) =>
f a -> f b
resize (BitVector ((int + frac) + frac)
-> Int -> BitVector ((int + frac) + frac)
forall a. Bits a => a -> Int -> a
shiftR BitVector ((int + frac) + frac)
rR Int
sh))
_ -> case BitVector int -> Bit
forall a. BitPack a => a -> Bit
msb BitVector int
rL of
0 -> Fixed rep int frac
forall a. Bounded a => a
maxBound
_ -> Fixed rep int frac
forall a. Bounded a => a
minBound
False -> case BitVector int
rL of
0 -> BitVector (BitSize (Fixed rep int frac)) -> Fixed rep int frac
forall a. BitPack a => BitVector (BitSize a) -> a
unpack (BitVector ((int + frac) + frac)
-> BitVector (BitSize (rep (int + frac)))
forall (f :: Nat -> Type) (a :: Nat) (b :: Nat).
(Resize f, KnownNat a, KnownNat b) =>
f a -> f b
resize (BitVector ((int + frac) + frac)
-> Int -> BitVector ((int + frac) + frac)
forall a. Bits a => a -> Int -> a
shiftR BitVector ((int + frac) + frac)
rR Int
sh))
_ -> Fixed rep int frac
forall a. Bounded a => a
maxBound

satMul SatZero (Fixed a :: rep (int + frac)
a) (Fixed b :: rep (int + frac)
b) =
let res :: MResult (rep (int + frac)) (rep (int + frac))
res     = rep (int + frac)
a rep (int + frac)
-> rep (int + frac)
-> MResult (rep (int + frac)) (rep (int + frac))
forall a b. ExtendingNum a b => a -> b -> MResult a b
`mul` rep (int + frac)
b
sh :: Int
sh      = Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Proxy frac -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Proxy frac
forall k (t :: k). Proxy t
Proxy @frac))
(rL :: BitVector int
rL,rR :: BitVector ((int + frac) + frac)
rR) = rep ((int + int) + (frac + frac))
-> (BitVector int, BitVector ((int + frac) + frac))
forall a (m :: Nat) (n :: Nat).
(BitPack a, BitSize a ~ (m + n), KnownNat n) =>
a -> (BitVector m, BitVector n)
split rep ((int + int) + (frac + frac))
MResult (rep (int + frac)) (rep (int + frac))
res :: (BitVector int, BitVector (int + frac + frac))
in  case rep (int + frac) -> Bool
forall a. Bits a => a -> Bool
isSigned rep (int + frac)
a of
True  -> let overflow :: Bit
overflow = Bit -> Bit
forall a. Bits a => a -> a
complement (BitVector (1 + int) -> Bit
forall a. BitPack a => a -> Bit
reduceOr (Bit -> BitVector (BitSize Bit)
forall a. BitPack a => a -> BitVector (BitSize a)
pack (BitVector ((int + frac) + frac) -> Bit
forall a. BitPack a => a -> Bit
msb BitVector ((int + frac) + frac)
rR) BitVector 1 -> BitVector int -> BitVector (1 + int)
forall (m :: Nat) (n :: Nat).
KnownNat m =>
BitVector n -> BitVector m -> BitVector (n + m)
++# BitVector int -> BitVector (BitSize (BitVector int))
forall a. BitPack a => a -> BitVector (BitSize a)
pack BitVector int
rL)) Bit -> Bit -> Bit
forall a. Bits a => a -> a -> a
.|.
BitVector (1 + int) -> Bit
forall a. BitPack a => a -> Bit
reduceAnd (Bit -> BitVector (BitSize Bit)
forall a. BitPack a => a -> BitVector (BitSize a)
pack (BitVector ((int + frac) + frac) -> Bit
forall a. BitPack a => a -> Bit
msb BitVector ((int + frac) + frac)
rR) BitVector 1 -> BitVector int -> BitVector (1 + int)
forall (m :: Nat) (n :: Nat).
KnownNat m =>
BitVector n -> BitVector m -> BitVector (n + m)
++# BitVector int -> BitVector (BitSize (BitVector int))
forall a. BitPack a => a -> BitVector (BitSize a)
pack BitVector int
rL)
in  case Bit
overflow of
1 -> BitVector (BitSize (Fixed rep int frac)) -> Fixed rep int frac
forall a. BitPack a => BitVector (BitSize a) -> a
unpack (BitVector ((int + frac) + frac)
-> BitVector (BitSize (rep (int + frac)))
forall (f :: Nat -> Type) (a :: Nat) (b :: Nat).
(Resize f, KnownNat a, KnownNat b) =>
f a -> f b
resize (BitVector ((int + frac) + frac)
-> Int -> BitVector ((int + frac) + frac)
forall a. Bits a => a -> Int -> a
shiftR BitVector ((int + frac) + frac)
rR Int
sh))
_ -> 0
False -> case BitVector int
rL of
0 -> BitVector (BitSize (Fixed rep int frac)) -> Fixed rep int frac
forall a. BitPack a => BitVector (BitSize a) -> a
unpack (BitVector ((int + frac) + frac)
-> BitVector (BitSize (rep (int + frac)))
forall (f :: Nat -> Type) (a :: Nat) (b :: Nat).
(Resize f, KnownNat a, KnownNat b) =>
f a -> f b
resize (BitVector ((int + frac) + frac)
-> Int -> BitVector ((int + frac) + frac)
forall a. Bits a => a -> Int -> a
shiftR BitVector ((int + frac) + frac)
rR Int
sh))
_ -> 0

satMul SatSymmetric (Fixed a :: rep (int + frac)
a) (Fixed b :: rep (int + frac)
b) =
let res :: MResult (rep (int + frac)) (rep (int + frac))
res     = rep (int + frac)
a rep (int + frac)
-> rep (int + frac)
-> MResult (rep (int + frac)) (rep (int + frac))
forall a b. ExtendingNum a b => a -> b -> MResult a b
`mul` rep (int + frac)
b
sh :: Int
sh      = Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Proxy frac -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Proxy frac
forall k (t :: k). Proxy t
Proxy @frac))
(rL :: BitVector int
rL,rR :: BitVector ((int + frac) + frac)
rR) = rep ((int + int) + (frac + frac))
-> (BitVector int, BitVector ((int + frac) + frac))
forall a (m :: Nat) (n :: Nat).
(BitPack a, BitSize a ~ (m + n), KnownNat n) =>
a -> (BitVector m, BitVector n)
split rep ((int + int) + (frac + frac))
MResult (rep (int + frac)) (rep (int + frac))
res :: (BitVector int, BitVector (int + frac + frac))
in  case rep (int + frac) -> Bool
forall a. Bits a => a -> Bool
isSigned rep (int + frac)
a of
True  -> let overflow :: Bit
overflow = Bit -> Bit
forall a. Bits a => a -> a
complement (BitVector (1 + int) -> Bit
forall a. BitPack a => a -> Bit
reduceOr (Bit -> BitVector (BitSize Bit)
forall a. BitPack a => a -> BitVector (BitSize a)
pack (BitVector ((int + frac) + frac) -> Bit
forall a. BitPack a => a -> Bit
msb BitVector ((int + frac) + frac)
rR) BitVector 1 -> BitVector int -> BitVector (1 + int)
forall (m :: Nat) (n :: Nat).
KnownNat m =>
BitVector n -> BitVector m -> BitVector (n + m)
++# BitVector int -> BitVector (BitSize (BitVector int))
forall a. BitPack a => a -> BitVector (BitSize a)
pack BitVector int
rL)) Bit -> Bit -> Bit
forall a. Bits a => a -> a -> a
.|.
BitVector (1 + int) -> Bit
forall a. BitPack a => a -> Bit
reduceAnd (Bit -> BitVector (BitSize Bit)
forall a. BitPack a => a -> BitVector (BitSize a)
pack (BitVector ((int + frac) + frac) -> Bit
forall a. BitPack a => a -> Bit
msb BitVector ((int + frac) + frac)
rR) BitVector 1 -> BitVector int -> BitVector (1 + int)
forall (m :: Nat) (n :: Nat).
KnownNat m =>
BitVector n -> BitVector m -> BitVector (n + m)
++# BitVector int -> BitVector (BitSize (BitVector int))
forall a. BitPack a => a -> BitVector (BitSize a)
pack BitVector int
rL)
in  case Bit
overflow of
1 -> BitVector (BitSize (Fixed rep int frac)) -> Fixed rep int frac
forall a. BitPack a => BitVector (BitSize a) -> a
unpack (BitVector ((int + frac) + frac)
-> BitVector (BitSize (rep (int + frac)))
forall (f :: Nat -> Type) (a :: Nat) (b :: Nat).
(Resize f, KnownNat a, KnownNat b) =>
f a -> f b
resize (BitVector ((int + frac) + frac)
-> Int -> BitVector ((int + frac) + frac)
forall a. Bits a => a -> Int -> a
shiftR BitVector ((int + frac) + frac)
rR Int
sh))
_ -> case BitVector int -> Bit
forall a. BitPack a => a -> Bit
msb BitVector int
rL of
0 -> Fixed rep int frac
forall a. Bounded a => a
maxBound
_ -> Fixed rep int frac -> Fixed rep int frac
forall a. Enum a => a -> a
succ Fixed rep int frac
forall a. Bounded a => a
minBound
False -> case BitVector int
rL of
0 -> BitVector (BitSize (Fixed rep int frac)) -> Fixed rep int frac
forall a. BitPack a => BitVector (BitSize a) -> a
unpack (BitVector ((int + frac) + frac)
-> BitVector (BitSize (rep (int + frac)))
forall (f :: Nat -> Type) (a :: Nat) (b :: Nat).
(Resize f, KnownNat a, KnownNat b) =>
f a -> f b
resize (BitVector ((int + frac) + frac)
-> Int -> BitVector ((int + frac) + frac)
forall a. Bits a => a -> Int -> a
shiftR BitVector ((int + frac) + frac)
rR Int
sh))
_ -> Fixed rep int frac
forall a. Bounded a => a
maxBound

-- | Constraint for the 'divide' function
type DivideC rep int1 frac1 int2 frac2
= ( Resize   rep
, Integral (rep (((int1 + frac2) + 1) + (int2 + frac1)))
, Bits     (rep (((int1 + frac2) + 1) + (int2 + frac1)))
, KnownNat int1
, KnownNat frac1
, KnownNat int2
, KnownNat frac2
)

-- | Constraint for the 'divide' function, specialized for 'SFixed'
type DivideSC int1 frac1 int2 frac2
= ( KnownNat (((int1 + frac2) + 1) + (int2 + frac1))
, KnownNat frac2
, KnownNat int2
, KnownNat frac1
, KnownNat int1
)

-- | Constraint for the 'divide' function, specialized for 'UFixed'
type DivideUC int1 frac1 int2 frac2 =
DivideSC int1 frac1 int2 frac2

-- | Fixed point division
--
-- When used in a polymorphic setting, use the following
-- <#constraintsynonyms Constraint synonyms> for less verbose type signatures:
--
-- * @'DivideC' rep int1 frac1 int2 frac2@ for:
--   @'Fixed' rep int1 frac1 -> 'Fixed' rep int2 frac2 -> 'Fixed' rep (int1 + frac2 + 1) (int2 + frac1)@
--
-- * @'DivideSC' rep int1 frac1 int2 frac2@ for:
--   @'SFixed' int1 frac1 -> 'SFixed' int2 frac2 -> 'SFixed' (int1 + frac2 + 1) (int2 + frac1)@
--
-- * @'DivideUC' rep int1 frac1 int2 frac2@ for:
--   @'UFixed' int1 frac1 -> 'UFixed' int2 frac2 -> 'UFixed' (int1 + frac2 + 1) (int2 + frac1)@
divide
:: DivideC rep int1 frac1 int2 frac2
=> Fixed rep int1 frac1
-> Fixed rep int2 frac2
-> Fixed rep (int1 + frac2 + 1) (int2 + frac1)
divide :: Fixed rep int1 frac1
-> Fixed rep int2 frac2
-> Fixed rep ((int1 + frac2) + 1) (int2 + frac1)
divide (Fixed fr1 :: rep (int1 + frac1)
fr1) fx2 :: Fixed rep int2 frac2
fx2@(Fixed fr2 :: rep (int2 + frac2)
fr2) =
let int2 :: Int
int2  = Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Proxy int2 -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal (Fixed rep int2 frac2 -> Proxy int2
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
Fixed rep int frac -> Proxy int
asIntProxy Fixed rep int2 frac2
fx2))
frac2 :: Int
frac2 = Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Fixed rep int2 frac2 -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal Fixed rep int2 frac2
fx2)
fr1' :: rep (((int1 + frac2) + 1) + (int2 + frac1))
fr1'  = rep (int1 + frac1) -> rep (((int1 + frac2) + 1) + (int2 + frac1))
forall (f :: Nat -> Type) (a :: Nat) (b :: Nat).
(Resize f, KnownNat a, KnownNat b) =>
f a -> f b
resize rep (int1 + frac1)
fr1
fr2' :: rep (((int1 + frac2) + 1) + (int2 + frac1))
fr2'  = rep (int2 + frac2) -> rep (((int1 + frac2) + 1) + (int2 + frac1))
forall (f :: Nat -> Type) (a :: Nat) (b :: Nat).
(Resize f, KnownNat a, KnownNat b) =>
f a -> f b
resize rep (int2 + frac2)
fr2
fr1SH :: rep (((int1 + frac2) + 1) + (int2 + frac1))
fr1SH = rep (((int1 + frac2) + 1) + (int2 + frac1))
-> Int -> rep (((int1 + frac2) + 1) + (int2 + frac1))
forall a. Bits a => a -> Int -> a
shiftL rep (((int1 + frac2) + 1) + (int2 + frac1))
fr1' ((Int
int2 Int -> Int -> Int
forall a. Num a => a -> a -> a
+ Int
frac2))
res :: rep (((int1 + frac2) + 1) + (int2 + frac1))
res   = rep (((int1 + frac2) + 1) + (int2 + frac1))
fr1SH rep (((int1 + frac2) + 1) + (int2 + frac1))
-> rep (((int1 + frac2) + 1) + (int2 + frac1))
-> rep (((int1 + frac2) + 1) + (int2 + frac1))
forall a. Integral a => a -> a -> a
`quot` rep (((int1 + frac2) + 1) + (int2 + frac1))
fr2'
in  rep (((int1 + frac2) + 1) + (int2 + frac1))
-> Fixed rep ((int1 + frac2) + 1) (int2 + frac1)
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
rep (int + frac) -> Fixed rep int frac
Fixed rep (((int1 + frac2) + 1) + (int2 + frac1))
res

-- | Constraint for the 'Fractional' instance of 'Fixed'
type FracFixedC rep int frac
= ( NumFixedC rep int frac
, DivideC   rep int frac int frac
, Integral  (rep (int + frac))
, KnownNat  int
, KnownNat  frac
)

-- | Constraint for the 'Fractional' instance of 'SFixed'
type FracSFixedC int frac
= ( NumSFixedC int frac
, KnownNat ((int + frac + 1) + (int + frac))
)

-- | Constraint for the 'Fractional' instance of 'UFixed'
type FracUFixedC int frac
= FracSFixedC int frac

-- | The operators of this instance saturate on overflow, and use truncation as
-- the rounding method.
--
-- When used in a polymorphic setting, use the following
-- <Clash-Sized-Fixed.html#constraintsynonyms Constraint synonyms> for less
-- verbose type signatures:
--
-- * @'FracFixedC' frac rep size@ for: @'Fixed' frac rep size@
-- * @'FracSFixedC' int frac@     for: @'SFixed' int frac@
-- * @'FracUFixedC' int frac@     for: @'UFixed' int frac@
instance FracFixedC rep int frac => Fractional (Fixed rep int frac) where
f1 :: Fixed rep int frac
f1 / :: Fixed rep int frac -> Fixed rep int frac -> Fixed rep int frac
/ f2 :: Fixed rep int frac
f2        = Fixed rep ((int + frac) + 1) (int + frac) -> Fixed rep int frac
forall (rep :: Nat -> Type) (int1 :: Nat) (frac1 :: Nat)
(int2 :: Nat) (frac2 :: Nat).
ResizeFC rep int1 frac1 int2 frac2 =>
Fixed rep int1 frac1 -> Fixed rep int2 frac2
resizeF (Fixed rep int frac
-> Fixed rep int frac -> Fixed rep ((int + frac) + 1) (int + frac)
forall (rep :: Nat -> Type) (int1 :: Nat) (frac1 :: Nat)
(int2 :: Nat) (frac2 :: Nat).
DivideC rep int1 frac1 int2 frac2 =>
Fixed rep int1 frac1
-> Fixed rep int2 frac2
-> Fixed rep ((int1 + frac2) + 1) (int2 + frac1)
divide Fixed rep int frac
f1 Fixed rep int frac
f2)
recip :: Fixed rep int frac -> Fixed rep int frac
recip fx :: Fixed rep int frac
fx       = Fixed rep ((int + frac) + 1) (int + frac) -> Fixed rep int frac
forall (rep :: Nat -> Type) (int1 :: Nat) (frac1 :: Nat)
(int2 :: Nat) (frac2 :: Nat).
ResizeFC rep int1 frac1 int2 frac2 =>
Fixed rep int1 frac1 -> Fixed rep int2 frac2
resizeF (Fixed rep int frac
-> Fixed rep int frac -> Fixed rep ((int + frac) + 1) (int + frac)
forall (rep :: Nat -> Type) (int1 :: Nat) (frac1 :: Nat)
(int2 :: Nat) (frac2 :: Nat).
DivideC rep int1 frac1 int2 frac2 =>
Fixed rep int1 frac1
-> Fixed rep int2 frac2
-> Fixed rep ((int1 + frac2) + 1) (int2 + frac1)
divide (1 :: Fixed rep int frac) Fixed rep int frac
fx)
fromRational :: Ratio Integer -> Fixed rep int frac
fromRational r :: Ratio Integer
r = Fixed rep int frac
res
where
res :: Fixed rep int frac
res  = rep (int + frac) -> Fixed rep int frac
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
rep (int + frac) -> Fixed rep int frac
Fixed (Integer -> rep (int + frac)
forall a. Num a => Integer -> a
fromInteger Integer
sat)
sat :: Integer
sat  = if Integer
res' Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
> Integer
rMax
then Integer
rMax
else if Integer
res' Integer -> Integer -> Bool
forall a. Ord a => a -> a -> Bool
< Integer
rMin then Integer
rMin else Integer
res'

rMax :: Integer
rMax = rep (int + frac) -> Integer
forall a. Integral a => a -> Integer
toInteger (rep (int + frac)
forall a. Bounded a => a
maxBound :: rep (int + frac))
rMin :: Integer
rMin = rep (int + frac) -> Integer
forall a. Integral a => a -> Integer
toInteger (rep (int + frac)
forall a. Bounded a => a
minBound :: rep (int + frac))
res' :: Integer
res' = Integer
n Integer -> Integer -> Integer
forall a. Integral a => a -> a -> a
`div` Integer
d

frac :: Int
frac = Integer -> Int
forall a. Num a => Integer -> a
fromInteger (Fixed rep int frac -> Integer
forall (n :: Nat) (proxy :: Nat -> Type).
KnownNat n =>
proxy n -> Integer
natVal Fixed rep int frac
res)
n :: Integer
n    = Ratio Integer -> Integer
forall a. Ratio a -> a
numerator   Ratio Integer
r Integer -> Int -> Integer
forall a. Bits a => a -> Int -> a
`shiftL` (2 Int -> Int -> Int
forall a. Num a => a -> a -> a
* Int
frac)
d :: Integer
d    = Ratio Integer -> Integer
forall a. Ratio a -> a
denominator Ratio Integer
r Integer -> Int -> Integer
forall a. Bits a => a -> Int -> a
`shiftL` Int
frac

instance (NumFixedC rep int frac, Integral (rep (int + frac))) =>
Real (Fixed rep int frac) where
toRational :: Fixed rep int frac -> Ratio Integer
toRational f :: Fixed rep int frac
f@(Fixed fRep :: rep (int + frac)
fRep) = Integer
nom Integer -> Integer -> Ratio Integer
forall a. Integral a => a -> a -> Ratio a
% Integer
denom
where
nF :: Int
nF        = Fixed rep int frac -> Int
forall (frac :: Nat) (rep :: Nat -> Type) (int :: Nat).
KnownNat frac =>
Fixed rep int frac -> Int
fracShift Fixed rep int frac
f
denom :: Integer
denom     = 1 Integer -> Int -> Integer
forall a. Bits a => a -> Int -> a
`shiftL` Int
nF
nom :: Integer
nom       = rep (int + frac) -> Integer
forall a. Integral a => a -> Integer
toInteger rep (int + frac)
fRep

instance (FracFixedC rep int frac, NumFixedC rep int frac, Integral (rep (int + frac))) =>
RealFrac (Fixed rep int frac) where
properFraction :: Fixed rep int frac -> (b, Fixed rep int frac)
properFraction f :: Fixed rep int frac
f@(Fixed fRep :: rep (int + frac)
fRep) = (rep (int + frac) -> b
forall a b. (Integral a, Num b) => a -> b
fromIntegral rep (int + frac)
whole, Fixed rep int frac
fract)
where
whole :: rep (int + frac)
whole = (rep (int + frac)
fRep rep (int + frac) -> Int -> rep (int + frac)
forall a. Bits a => a -> Int -> a
`shiftR` Fixed rep int frac -> Int
forall (frac :: Nat) (rep :: Nat -> Type) (int :: Nat).
KnownNat frac =>
Fixed rep int frac -> Int
fracShift Fixed rep int frac
f) rep (int + frac) -> rep (int + frac) -> rep (int + frac)
forall a. Num a => a -> a -> a
+ rep (int + frac)
offset
fract :: Fixed rep int frac
fract = rep (int + frac) -> Fixed rep int frac
forall (rep :: Nat -> Type) (int :: Nat) (frac :: Nat).
rep (int + frac) -> Fixed rep int frac
Fixed (rep (int + frac) -> Fixed rep int frac)
-> rep (int + frac) -> Fixed rep int frac
forall a b. (a -> b) -> a -> b
\$ rep (int + frac)
fRep rep (int + frac) -> rep (int + frac) -> rep (int + frac)
forall a. Num a => a -> a -> a
- (rep (int + frac)
whole rep (int + frac) -> Int -> rep (int + frac)
forall a. Bits a => a -> Int -> a
`shiftL` Fixed rep int frac -> Int
forall (frac :: Nat) (rep :: Nat -> Type) (int :: Nat).
KnownNat frac =>
Fixed rep int frac -> Int
fracShift Fixed rep int frac
f)
offset :: rep (int + frac)
offset = if Fixed rep int frac
f Fixed rep int frac -> Fixed rep int frac -> Bool
forall a. Ord a => a -> a -> Bool
< 0 then 1 else 0
```