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+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+// This file provides Go implementations of elementary multi-precision
+// arithmetic operations on word vectors. These have the suffix _g.
+// These are needed for platforms without assembly implementations of these routines.
+// This file also contains elementary operations that can be implemented
+// sufficiently efficiently in Go.
+
+package big
+
+import "math/bits"
+
+// A Word represents a single digit of a multi-precision unsigned integer.
+type Word uint
+
+const (
+ _S = _W / 8 // word size in bytes
+
+ _W = bits.UintSize // word size in bits
+ _B = 1 << _W // digit base
+ _M = _B - 1 // digit mask
+)
+
+// Many of the loops in this file are of the form
+// for i := 0; i < len(z) && i < len(x) && i < len(y); i++
+// i < len(z) is the real condition.
+// However, checking i < len(x) && i < len(y) as well is faster than
+// having the compiler do a bounds check in the body of the loop;
+// remarkably it is even faster than hoisting the bounds check
+// out of the loop, by doing something like
+// _, _ = x[len(z)-1], y[len(z)-1]
+// There are other ways to hoist the bounds check out of the loop,
+// but the compiler's BCE isn't powerful enough for them (yet?).
+// See the discussion in CL 164966.
+
+// ----------------------------------------------------------------------------
+// Elementary operations on words
+//
+// These operations are used by the vector operations below.
+
+// z1<<_W + z0 = x*y
+func mulWW_g(x, y Word) (z1, z0 Word) {
+ hi, lo := bits.Mul(uint(x), uint(y))
+ return Word(hi), Word(lo)
+}
+
+// z1<<_W + z0 = x*y + c
+func mulAddWWW_g(x, y, c Word) (z1, z0 Word) {
+ hi, lo := bits.Mul(uint(x), uint(y))
+ var cc uint
+ lo, cc = bits.Add(lo, uint(c), 0)
+ return Word(hi + cc), Word(lo)
+}
+
+// nlz returns the number of leading zeros in x.
+// Wraps bits.LeadingZeros call for convenience.
+func nlz(x Word) uint {
+ return uint(bits.LeadingZeros(uint(x)))
+}
+
+// The resulting carry c is either 0 or 1.
+func addVV_g(z, x, y []Word) (c Word) {
+ // The comment near the top of this file discusses this for loop condition.
+ for i := 0; i < len(z) && i < len(x) && i < len(y); i++ {
+ zi, cc := bits.Add(uint(x[i]), uint(y[i]), uint(c))
+ z[i] = Word(zi)
+ c = Word(cc)
+ }
+ return
+}
+
+// The resulting carry c is either 0 or 1.
+func subVV_g(z, x, y []Word) (c Word) {
+ // The comment near the top of this file discusses this for loop condition.
+ for i := 0; i < len(z) && i < len(x) && i < len(y); i++ {
+ zi, cc := bits.Sub(uint(x[i]), uint(y[i]), uint(c))
+ z[i] = Word(zi)
+ c = Word(cc)
+ }
+ return
+}
+
+// The resulting carry c is either 0 or 1.
+func addVW_g(z, x []Word, y Word) (c Word) {
+ c = y
+ // The comment near the top of this file discusses this for loop condition.
+ for i := 0; i < len(z) && i < len(x); i++ {
+ zi, cc := bits.Add(uint(x[i]), uint(c), 0)
+ z[i] = Word(zi)
+ c = Word(cc)
+ }
+ return
+}
+
+// addVWlarge is addVW, but intended for large z.
+// The only difference is that we check on every iteration
+// whether we are done with carries,
+// and if so, switch to a much faster copy instead.
+// This is only a good idea for large z,
+// because the overhead of the check and the function call
+// outweigh the benefits when z is small.
+func addVWlarge(z, x []Word, y Word) (c Word) {
+ c = y
+ // The comment near the top of this file discusses this for loop condition.
+ for i := 0; i < len(z) && i < len(x); i++ {
+ if c == 0 {
+ copy(z[i:], x[i:])
+ return
+ }
+ zi, cc := bits.Add(uint(x[i]), uint(c), 0)
+ z[i] = Word(zi)
+ c = Word(cc)
+ }
+ return
+}
+
+func subVW_g(z, x []Word, y Word) (c Word) {
+ c = y
+ // The comment near the top of this file discusses this for loop condition.
+ for i := 0; i < len(z) && i < len(x); i++ {
+ zi, cc := bits.Sub(uint(x[i]), uint(c), 0)
+ z[i] = Word(zi)
+ c = Word(cc)
+ }
+ return
+}
+
+// subVWlarge is to subVW as addVWlarge is to addVW.
+func subVWlarge(z, x []Word, y Word) (c Word) {
+ c = y
+ // The comment near the top of this file discusses this for loop condition.
+ for i := 0; i < len(z) && i < len(x); i++ {
+ if c == 0 {
+ copy(z[i:], x[i:])
+ return
+ }
+ zi, cc := bits.Sub(uint(x[i]), uint(c), 0)
+ z[i] = Word(zi)
+ c = Word(cc)
+ }
+ return
+}
+
+func shlVU_g(z, x []Word, s uint) (c Word) {
+ if s == 0 {
+ copy(z, x)
+ return
+ }
+ if len(z) == 0 {
+ return
+ }
+ s &= _W - 1 // hint to the compiler that shifts by s don't need guard code
+ ŝ := _W - s
+ ŝ &= _W - 1 // ditto
+ c = x[len(z)-1] >> ŝ
+ for i := len(z) - 1; i > 0; i-- {
+ z[i] = x[i]<<s | x[i-1]>>ŝ
+ }
+ z[0] = x[0] << s
+ return
+}
+
+func shrVU_g(z, x []Word, s uint) (c Word) {
+ if s == 0 {
+ copy(z, x)
+ return
+ }
+ if len(z) == 0 {
+ return
+ }
+ s &= _W - 1 // hint to the compiler that shifts by s don't need guard code
+ ŝ := _W - s
+ ŝ &= _W - 1 // ditto
+ c = x[0] << ŝ
+ for i := 0; i < len(z)-1; i++ {
+ z[i] = x[i]>>s | x[i+1]<<ŝ
+ }
+ z[len(z)-1] = x[len(z)-1] >> s
+ return
+}
+
+func mulAddVWW_g(z, x []Word, y, r Word) (c Word) {
+ c = r
+ // The comment near the top of this file discusses this for loop condition.
+ for i := 0; i < len(z) && i < len(x); i++ {
+ c, z[i] = mulAddWWW_g(x[i], y, c)
+ }
+ return
+}
+
+func addMulVVW_g(z, x []Word, y Word) (c Word) {
+ // The comment near the top of this file discusses this for loop condition.
+ for i := 0; i < len(z) && i < len(x); i++ {
+ z1, z0 := mulAddWWW_g(x[i], y, z[i])
+ lo, cc := bits.Add(uint(z0), uint(c), 0)
+ c, z[i] = Word(cc), Word(lo)
+ c += z1
+ }
+ return
+}
+
+// q = ( x1 << _W + x0 - r)/y. m = floor(( _B^2 - 1 ) / d - _B). Requiring x1<y.
+// An approximate reciprocal with a reference to "Improved Division by Invariant Integers
+// (IEEE Transactions on Computers, 11 Jun. 2010)"
+func divWW(x1, x0, y, m Word) (q, r Word) {
+ s := nlz(y)
+ if s != 0 {
+ x1 = x1<<s | x0>>(_W-s)
+ x0 <<= s
+ y <<= s
+ }
+ d := uint(y)
+ // We know that
+ // m = ⎣(B^2-1)/d⎦-B
+ // ⎣(B^2-1)/d⎦ = m+B
+ // (B^2-1)/d = m+B+delta1 0 <= delta1 <= (d-1)/d
+ // B^2/d = m+B+delta2 0 <= delta2 <= 1
+ // The quotient we're trying to compute is
+ // quotient = ⎣(x1*B+x0)/d⎦
+ // = ⎣(x1*B*(B^2/d)+x0*(B^2/d))/B^2⎦
+ // = ⎣(x1*B*(m+B+delta2)+x0*(m+B+delta2))/B^2⎦
+ // = ⎣(x1*m+x1*B+x0)/B + x0*m/B^2 + delta2*(x1*B+x0)/B^2⎦
+ // The latter two terms of this three-term sum are between 0 and 1.
+ // So we can compute just the first term, and we will be low by at most 2.
+ t1, t0 := bits.Mul(uint(m), uint(x1))
+ _, c := bits.Add(t0, uint(x0), 0)
+ t1, _ = bits.Add(t1, uint(x1), c)
+ // The quotient is either t1, t1+1, or t1+2.
+ // We'll try t1 and adjust if needed.
+ qq := t1
+ // compute remainder r=x-d*q.
+ dq1, dq0 := bits.Mul(d, qq)
+ r0, b := bits.Sub(uint(x0), dq0, 0)
+ r1, _ := bits.Sub(uint(x1), dq1, b)
+ // The remainder we just computed is bounded above by B+d:
+ // r = x1*B + x0 - d*q.
+ // = x1*B + x0 - d*⎣(x1*m+x1*B+x0)/B⎦
+ // = x1*B + x0 - d*((x1*m+x1*B+x0)/B-alpha) 0 <= alpha < 1
+ // = x1*B + x0 - x1*d/B*m - x1*d - x0*d/B + d*alpha
+ // = x1*B + x0 - x1*d/B*⎣(B^2-1)/d-B⎦ - x1*d - x0*d/B + d*alpha
+ // = x1*B + x0 - x1*d/B*⎣(B^2-1)/d-B⎦ - x1*d - x0*d/B + d*alpha
+ // = x1*B + x0 - x1*d/B*((B^2-1)/d-B-beta) - x1*d - x0*d/B + d*alpha 0 <= beta < 1
+ // = x1*B + x0 - x1*B + x1/B + x1*d + x1*d/B*beta - x1*d - x0*d/B + d*alpha
+ // = x0 + x1/B + x1*d/B*beta - x0*d/B + d*alpha
+ // = x0*(1-d/B) + x1*(1+d*beta)/B + d*alpha
+ // < B*(1-d/B) + d*B/B + d because x0<B (and 1-d/B>0), x1<d, 1+d*beta<=B, alpha<1
+ // = B - d + d + d
+ // = B+d
+ // So r1 can only be 0 or 1. If r1 is 1, then we know q was too small.
+ // Add 1 to q and subtract d from r. That guarantees that r is <B, so
+ // we no longer need to keep track of r1.
+ if r1 != 0 {
+ qq++
+ r0 -= d
+ }
+ // If the remainder is still too large, increment q one more time.
+ if r0 >= d {
+ qq++
+ r0 -= d
+ }
+ return Word(qq), Word(r0 >> s)
+}
+
+func divWVW(z []Word, xn Word, x []Word, y Word) (r Word) {
+ r = xn
+ if len(x) == 1 {
+ qq, rr := bits.Div(uint(r), uint(x[0]), uint(y))
+ z[0] = Word(qq)
+ return Word(rr)
+ }
+ rec := reciprocalWord(y)
+ for i := len(z) - 1; i >= 0; i-- {
+ z[i], r = divWW(r, x[i], y, rec)
+ }
+ return r
+}
+
+// reciprocalWord return the reciprocal of the divisor. rec = floor(( _B^2 - 1 ) / u - _B). u = d1 << nlz(d1).
+func reciprocalWord(d1 Word) Word {
+ u := uint(d1 << nlz(d1))
+ x1 := ^u
+ x0 := uint(_M)
+ rec, _ := bits.Div(x1, x0, u) // (_B^2-1)/U-_B = (_B*(_M-C)+_M)/U
+ return Word(rec)
+}
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