Editando: arith.go
// 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(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 } if len(x) != len(z) { // This is an invariant guaranteed by the caller. panic("len(x) != len(z)") } 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 := 1; i < len(z); i++ { z[i-1] = x[i-1]>>s | x[i]<<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) } // 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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Kerym Chaeceran