Data.Type.BitRecords.Arithmetic

Documentation

type family Rem (x :: Nat) (y :: Nat) :: Nat where ...

Get the remainder of the integer division of x and y, such that forall x y. exists k. (Rem x y) == x - y * k The algorithm is: count down x until zero, incrementing the accumulator at each step. Whenever the accumulator is equal to y set it to zero.

If the accumulator has reached y reset it. It is important to do this BEFORE checking if x == y and then returning the accumulator, for the case where x = k * y with k > 0. For example:

 6 Rem 3     = RemImpl 6 3 0
 RemImpl 6 3 0 = RemImpl (6-1) 3 (0+1)   -- RemImpl Clause 4
 RemImpl 5 3 1 = RemImpl (5-1) 3 (1+1)   -- RemImpl Clause 4
 RemImpl 4 3 2 = RemImpl (4-1) 3 (2+1)   -- RemImpl Clause 4
 RemImpl 3 3 3 = RemImpl 3 3 0           -- RemImpl Clause 2 !!!
 RemImpl 3 3 0 = 0                       -- RemImpl Clause 3 !!!

Equations

Rem x 1 = 0

Rem x 0 = TypeError ((Text "divide by zero: " :<>: ShowType x) :<>: Text " `Rem` 0")

Rem x y = RemImpl x y 0

type family RemImpl (x :: Nat) (y :: nat) (acc :: Nat) :: Nat where ...

Equations

RemImpl 0 y acc = acc

RemImpl x y y = RemImpl x y 0

RemImpl y y acc = acc

RemImpl x y acc = RemImpl (x - 1) y (acc + 1)

type RemPow2 x p = FromBits (TakeLastN p (ToBits x RemPow2Bits))

Efficient Rem operation for power of 2 values. Note that x must be representable by RemPow2Bits bits.

type TakeLastN n xs = TakeLastNImplRev n xs '[]

type family TakeLastNImplRev (n :: Nat) (xs :: [t]) (acc :: [t]) :: [t] where ...

Equations

TakeLastNImplRev n '[] acc = TakeLastNImplTakeNRev n acc '[]

TakeLastNImplRev n (x ': xs) acc = TakeLastNImplRev n xs (x ': acc)

type family TakeLastNImplTakeNRev (n :: Nat) (rs :: [t]) (acc :: [t]) :: [t] where ...

Equations

TakeLastNImplTakeNRev n '[] acc = acc

TakeLastNImplTakeNRev 0 rs acc = acc

TakeLastNImplTakeNRev n (r ': rs) acc = TakeLastNImplTakeNRev (n - 1) rs (r ': acc)

type RemPow2Bits = 32

Maximum number of bits an argument x of RemPow2 may occupy.

type Div x y = DivImpl (x - (x `Rem` y)) y 0

Integer division of x and y: Div x y ==> x / y, NOTE This only works for small numbers currently

type family DivImpl (x :: Nat) (y :: nat) (acc :: Nat) :: Nat where ...

Equations

DivImpl 0 y acc = acc

DivImpl x y acc = If ((x + 1) <=? y) acc (DivImpl (x - y) y (acc + 1))

Bit manipulation

type family TestHighBit (x :: Nat) (n :: Nat) :: Bool where ...

Equations

TestHighBit x n = (2 ^ n) <=? x

type ToBits x n = ToBits_ x n False

type family ToBits_ (x :: Nat) (n :: Nat) (started :: Bool) :: [Bool] where ...

Equations

ToBits_ x 0 started = '[]

ToBits_ x n started = ToBitsInner (TestHighBit x (n - 1)) x (n - 1) started

type family ToBitsInner (highBitSet :: Bool) (x :: Nat) (n :: Nat) (started :: Bool) :: [Bool] where ...

Equations

ToBitsInner True x n started = True ': ToBits_ (x - (2 ^ n)) n True

ToBitsInner False x n False = ToBits_ x n False

ToBitsInner False x n True = False ': ToBits_ x n True

type FromBits bits = FromBits_ bits 0

type family FromBits_ (bits :: [Bool]) (acc :: Nat) :: Nat where ...

Equations

FromBits_ '[] acc = acc

FromBits_ (False ': rest) acc = FromBits_ rest (acc + acc)

FromBits_ (True ': rest) acc = FromBits_ rest ((1 + acc) + acc)

type family ShiftBitsR (bits :: [Bool]) (n :: Nat) :: [Bool] where ...

Equations

ShiftBitsR bits 0 = bits

ShiftBitsR '[] n = '[]

ShiftBitsR '[e] 1 = '[]

ShiftBitsR (e ': rest) 1 = e ': ShiftBitsR rest 1

ShiftBitsR (e ': rest) n = ShiftBitsR (ShiftBitsR (e ': rest) 1) (n - 1)

type family GetMostSignificantBitIndex (highestBit :: Nat) (n :: Nat) :: Nat where ...

Equations

GetMostSignificantBitIndex 0 n = 1

GetMostSignificantBitIndex highestBit n = If ((2 ^ (highestBit + 1)) <=? n) (TypeError (((Text "number to big: " :<>: ShowType n) :<>: Text " >= ") :<>: ShowType (2 ^ (highestBit + 1)))) (If ((2 ^ highestBit) <=? n) highestBit (GetMostSignificantBitIndex (highestBit - 1) n))

type family ShiftR (xMaxBits :: Nat) (x :: Nat) (bits :: Nat) :: Nat where ...

Shift a type level natural to the right. This useful for division by powers of two.

Equations

ShiftR xMaxBits x n = FromBits (ShiftBitsR (ToBits x (1 + GetMostSignificantBitIndex xMaxBits x)) n)