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Implement multiplication
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Mamy André-Ratsimbazafy
jangko
parent
206ffa92cf
commit
a0dec54c12
@ -105,3 +105,33 @@ iterator leastToMostSig*(cLimbs: var Limbs, aLimbs: Limbs, bLimbs: Limbs): (var
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else:
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else:
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for i in countdown(aLimbs.len-1, 0):
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for i in countdown(aLimbs.len-1, 0):
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yield (cLimbs[i], aLimbs[i], bLimbs[i])
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yield (cLimbs[i], aLimbs[i], bLimbs[i])
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import std/macros
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proc replaceNodes(ast: NimNode, what: NimNode, by: NimNode): NimNode =
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# Replace "what" ident node by "by"
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proc inspect(node: NimNode): NimNode =
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case node.kind:
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of {nnkIdent, nnkSym}:
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if node.eqIdent(what):
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return by
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return node
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of nnkEmpty:
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return node
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of nnkLiterals:
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return node
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else:
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var rTree = node.kind.newTree()
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for child in node:
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rTree.add inspect(child)
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return rTree
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result = inspect(ast)
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macro staticFor*(idx: untyped{nkIdent}, start, stopEx: static int, body: untyped): untyped =
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## staticFor [min inclusive, max exclusive)
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result = newStmtList()
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for i in start ..< stopEx:
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result.add nnkBlockStmt.newTree(
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ident("unrolledIter_" & $idx & $i),
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body.replaceNodes(idx, newLit i)
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)
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@ -8,7 +8,7 @@
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# at your option. This file may not be copied, modified, or distributed except according to those terms.
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# at your option. This file may not be copied, modified, or distributed except according to those terms.
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import
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import
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./datatypes, ./uint_comparison, ./uint_bitwise_ops,
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./datatypes,
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./primitives/addcarry_subborrow
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./primitives/addcarry_subborrow
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# ############ Addition & Substraction ############ #
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# ############ Addition & Substraction ############ #
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@ -7,160 +7,78 @@
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#
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#
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# at your option. This file may not be copied, modified, or distributed except according to those terms.
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# at your option. This file may not be copied, modified, or distributed except according to those terms.
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import macros,
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import
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./conversion,
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./datatypes,
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./initialization,
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./primitives/extended_precision
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./datatypes,
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./uint_comparison,
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./uint_addsub
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# ################### Multiplication ################### #
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# ################### Multiplication ################### #
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{.push raises: [], gcsafe.}
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func lo(x: uint64): uint64 {.inline.} =
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func prod*[rLen, aLen, bLen](r: var Limbs[rLen], a: Limbs[aLen], b: Limbs[bLen]) =
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const
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## Multi-precision multiplication
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p: uint64 = 32
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## r <- a*b
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base: uint64 = 1'u64 shl p
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##
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mask: uint64 = base - 1
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## `a`, `b`, `r` can have a different number of limbs
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result = x and mask
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## if `r`.limbs.len < a.limbs.len + b.limbs.len
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## The result will be truncated, i.e. it will be
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## a * b (mod (2^WordBitwidth)^r.limbs.len)
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func hi(x: uint64): uint64 {.inline.} =
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# We use Product Scanning / Comba multiplication
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const
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var t, u, v = Word(0)
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p = 32
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var z: Limbs[rLen] # zero-init, ensure on stack and removes in-place problems
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result = x shr p
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# No generic, somehow Nim is given ambiguous call with the T: UintImpl overload
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staticFor i, 0, min(a.len+b.len, r.len):
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func extPrecMul*(result: var UintImpl[uint8], x, y: uint8) {.inline.}=
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const ib = min(b.len-1, i)
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## Extended precision multiplication
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const ia = i - ib
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result = cast[type result](x.asDoubleUint * y.asDoubleUint)
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staticFor j, 0, min(a.len - ia, ib+1):
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mulAcc(t, u, v, a[ia+j], b[ib-j])
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func extPrecMul*(result: var UintImpl[uint16], x, y: uint16) {.inline.}=
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z[i] = v
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## Extended precision multiplication
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v = u
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result = cast[type result](x.asDoubleUint * y.asDoubleUint)
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u = t
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t = Word(0)
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func extPrecMul*(result: var UintImpl[uint32], x, y: uint32) {.inline.}=
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r = z
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## Extended precision multiplication
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result = cast[type result](x.asDoubleUint * y.asDoubleUint)
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func extPrecAddMul[T: uint8 or uint16 or uint32](result: var UintImpl[T], x, y: T) {.inline.}=
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func prod_high_words*[rLen, aLen, bLen](
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## Extended precision fused in-place addition & multiplication
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r: var Limbs[rLen],
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result += cast[type result](x.asDoubleUint * y.asDoubleUint)
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a: Limbs[aLen], b: Limbs[bLen],
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lowestWordIndex: static int) =
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template extPrecMulImpl(result: var UintImpl[uint64], op: untyped, u, v: uint64) =
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## Multi-precision multiplication keeping only high words
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const
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## r <- a*b >> (2^WordBitWidth)^lowestWordIndex
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p = 64 div 2
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##
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base: uint64 = 1'u64 shl p
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## `a`, `b`, `r` can have a different number of limbs
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## if `r`.limbs.len < a.limbs.len + b.limbs.len - lowestWordIndex
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var
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## The result will be truncated, i.e. it will be
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x0, x1, x2, x3: uint64
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## a * b >> (2^WordBitWidth)^lowestWordIndex (mod (2^WordBitwidth)^r.limbs.len)
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let
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ul = lo(u)
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uh = hi(u)
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vl = lo(v)
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vh = hi(v)
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x0 = ul * vl
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x1 = ul * vh
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x2 = uh * vl
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x3 = uh * vh
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x1 += hi(x0) # This can't carry
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x1 += x2 # but this can
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if x1 < x2: # if carry, add it to x3
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x3 += base
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op(result.hi, x3 + hi(x1))
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op(result.lo, (x1 shl p) or lo(x0))
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func extPrecMul*(result: var UintImpl[uint64], u, v: uint64) =
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## Extended precision multiplication
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extPrecMulImpl(result, `=`, u, v)
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func extPrecAddMul(result: var UintImpl[uint64], u, v: uint64) =
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## Extended precision fused in-place addition & multiplication
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extPrecMulImpl(result, `+=`, u, v)
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macro eqSym(x, y: untyped): untyped =
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let eq = $x == $y # Unfortunately eqIdent compares to string.
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result = newLit eq
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func extPrecAddMul[T](result: var UintImpl[UintImpl[T]], u, v: UintImpl[T])
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func extPrecMul*[T](result: var UintImpl[UintImpl[T]], u, v: UintImpl[T])
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# Forward declaration
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template extPrecMulImpl*[T](result: var UintImpl[UintImpl[T]], op: untyped, x, y: UintImpl[T]) =
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# See details at
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# https://en.wikipedia.org/wiki/Karatsuba_algorithm
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# https://locklessinc.com/articles/256bit_arithmetic/
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# https://www.miracl.com/press/missing-a-trick-karatsuba-variations-michael-scott
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#
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#
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# We use the naive school grade multiplication instead of Karatsuba I.e.
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# This is useful for
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# z1 = x.hi * y.lo + x.lo * y.hi (Naive) = (x.lo - x.hi)(y.hi - y.lo) + z0 + z2 (Karatsuba)
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# - Barret reduction
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#
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# - Approximating multiplication by a fractional constant in the form f(a) = K/C * a
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# On modern architecture:
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# with K and C known at compile-time.
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# - addition and multiplication have the same cost
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# 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)
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# - Karatsuba would require to deal with potentially negative intermediate result
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# Precompute P = K*M/C at compile-time
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# and introduce branching
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# and at runtime do P*a/M <=> P*a >> (WordBitWidth*w)
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# - More total operations means more register moves
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# i.e. prod_high_words(result, P, a, w)
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var z1: type x
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# We use Product Scanning / Comba multiplication
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var t, u, v = Word(0) # Will raise warning on empty iterations
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var z: Limbs[rLen] # zero-init, ensure on stack and removes in-place problems
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# Low part and hi part - z0 & z2
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# The previous 2 columns can affect the lowest word due to carries
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when eqSym(op, `+=`):
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# but not the ones before (we accumulate in 3 words (t, u, v))
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extPrecAddMul(result.lo, x.lo, y.lo)
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const w = lowestWordIndex - 2
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extPrecAddMul(result.hi, x.hi, y.hi)
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else:
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extPrecMul(result.lo, x.lo, y.lo)
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extPrecMul(result.hi, x.hi, y.hi)
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## TODO - fuse those parts and reduce the number of carry checks
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staticFor i, max(0, w), min(a.len+b.len, r.len+lowestWordIndex):
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# Middle part - z1 - 1st mul
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const ib = min(b.len-1, i)
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extPrecMul(z1, x.hi, y.lo)
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const ia = i - ib
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result.lo.hi += z1.lo
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staticFor j, 0, min(a.len - ia, ib+1):
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if result.lo.hi < z1.lo:
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mulAcc(t, u, v, a[ia+j], b[ib-j])
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inc result.hi
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result.hi.lo += z1.hi
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when i >= lowestWordIndex:
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if result.hi.lo < z1.hi:
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z[i-lowestWordIndex] = v
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inc result.hi.hi
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v = u
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u = t
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t = Word(0)
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# Middle part - z1 - 2nd mul
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r = z
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extPrecMul(z1, x.lo, y.hi)
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result.lo.hi += z1.lo
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if result.lo.hi < z1.lo:
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inc result.hi
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result.hi.lo += z1.hi
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if result.hi.lo < z1.hi:
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inc result.hi.hi
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func extPrecAddMul[T](result: var UintImpl[UintImpl[T]], u, v: UintImpl[T]) =
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## Extended precision fused in-place addition & multiplication
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extPrecMulImpl(result, `+=`, u, v)
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func extPrecMul*[T](result: var UintImpl[UintImpl[T]], u, v: UintImpl[T]) =
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## Extended precision multiplication
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extPrecMulImpl(result, `=`, u, v)
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func `*`*[T](x, y: UintImpl[T]): UintImpl[T] {.inline.}=
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## Multiplication for multi-precision unsigned uint
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#
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# For our representation, it is similar to school grade multiplication
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# Consider hi and lo as if they were digits
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#
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# 12
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# X 15
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# ------
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# 10 lo*lo -> z0
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# 5 hi*lo -> z1
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# 2 lo*hi -> z1
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# 10 hi*hi -- z2
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# ------
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# 180
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#
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# If T is a type
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# For T * T --> T we don't need to compute z2 as it always overflow
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# For T * T --> 2T (uint64 * uint64 --> uint128) we use extra precision multiplication
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extPrecMul(result, x.lo, y.lo)
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result.hi += x.lo * y.hi + x.hi * y.lo
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