nim-stint/stint/private/uint_mul.nim

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# Stint
# Copyright 2018-Present Status Research & Development GmbH
# Licensed under either of
#
# * Apache License, version 2.0, ([LICENSE-APACHE](LICENSE-APACHE) or http://www.apache.org/licenses/LICENSE-2.0)
# * MIT license ([LICENSE-MIT](LICENSE-MIT) or http://opensource.org/licenses/MIT)
#
# at your option. This file may not be copied, modified, or distributed except according to those terms.
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import
./datatypes,
./primitives/extended_precision
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# Multiplication
# --------------------------------------------------------
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{.push raises: [], gcsafe.}
func prod*[rLen, aLen, bLen: static int](r: var Limbs[rLen], a: Limbs[aLen], b: Limbs[bLen]) =
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## Multi-precision multiplication
## r <- a*b
##
## `a`, `b`, `r` can have a different number of limbs
## if `r`.limbs.len < a.limbs.len + b.limbs.len
## The result will be truncated, i.e. it will be
## a * b (mod (2^WordBitwidth)^r.limbs.len)
# We use Product Scanning / Comba multiplication
var t, u, v = Word(0)
var z: typeof(r) # zero-init, ensure on stack and removes in-place problems
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staticFor i, 0, min(a.len+b.len, r.len):
const ib = min(b.len-1, i)
const ia = i - ib
staticFor j, 0, min(a.len - ia, ib+1):
mulAcc(t, u, v, a[ia+j], b[ib-j])
z[i] = v
v = u
u = t
t = 0
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r = z
func prod_high_words*[rLen, aLen, bLen: static int](
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r: var Limbs[rLen],
a: Limbs[aLen], b: Limbs[bLen],
lowestWordIndex: static int) =
## Multi-precision multiplication keeping only high words
## r <- a*b >> (2^WordBitWidth)^lowestWordIndex
##
## `a`, `b`, `r` can have a different number of limbs
## if `r`.limbs.len < a.limbs.len + b.limbs.len - lowestWordIndex
## The result will be truncated, i.e. it will be
## a * b >> (2^WordBitWidth)^lowestWordIndex (mod (2^WordBitwidth)^r.limbs.len)
#
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# This is useful for
# - Barret reduction
# - Approximating multiplication by a fractional constant in the form f(a) = K/C * a
# with K and C known at compile-time.
# We can instead find a well chosen M = (2^WordBitWidth)^w, with M > C (i.e. M is a power of 2 bigger than C)
# Precompute P = K*M/C at compile-time
# and at runtime do P*a/M <=> P*a >> (WordBitWidth*w)
# i.e. prod_high_words(result, P, a, w)
# We use Product Scanning / Comba multiplication
var t, u, v = Word(0) # Will raise warning on empty iterations
var z: Limbs[rLen] # zero-init, ensure on stack and removes in-place problems
# The previous 2 columns can affect the lowest word due to carries
# but not the ones before (we accumulate in 3 words (t, u, v))
const w = lowestWordIndex - 2
staticFor i, max(0, w), min(a.len+b.len, r.len+lowestWordIndex):
const ib = min(b.len-1, i)
const ia = i - ib
staticFor j, 0, min(a.len - ia, ib+1):
mulAcc(t, u, v, a[ia+j], b[ib-j])
when i >= lowestWordIndex:
z[i-lowestWordIndex] = v
v = u
u = t
t = Word(0)
r = z