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# Source file src/math/big/arith.go

```  // Copyright 2009 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style

// This file provides Go implementations of elementary multi-precision
// arithmetic operations on word vectors. Needed for platforms without
// assembly implementations of these routines.

package big

// A Word represents a single digit of a multi-precision unsigned integer.
type Word uintptr

const (
// Compute the size _S of a Word in bytes.
_m    = ^Word(0)
_logS = _m>>8&1 + _m>>16&1 + _m>>32&1
_S    = 1 << _logS

_W = _S << 3 // word size in bits
_B = 1 << _W // digit base
_M = _B - 1  // digit mask

_W2 = _W / 2   // half word size in bits
_B2 = 1 << _W2 // half digit base
_M2 = _B2 - 1  // half digit mask
)

// ----------------------------------------------------------------------------
// Elementary operations on words
//
// These operations are used by the vector operations below.

// z1<<_W + z0 = x+y+c, with c == 0 or 1
func addWW_g(x, y, c Word) (z1, z0 Word) {
yc := y + c
z0 = x + yc
if z0 < x || yc < y {
z1 = 1
}
return
}

// z1<<_W + z0 = x-y-c, with c == 0 or 1
func subWW_g(x, y, c Word) (z1, z0 Word) {
yc := y + c
z0 = x - yc
if z0 > x || yc < y {
z1 = 1
}
return
}

// z1<<_W + z0 = x*y
// Adapted from Warren, Hacker's Delight, p. 132.
func mulWW_g(x, y Word) (z1, z0 Word) {
x0 := x & _M2
x1 := x >> _W2
y0 := y & _M2
y1 := y >> _W2
w0 := x0 * y0
t := x1*y0 + w0>>_W2
w1 := t & _M2
w2 := t >> _W2
w1 += x0 * y1
z1 = x1*y1 + w2 + w1>>_W2
z0 = x * y
return
}

// z1<<_W + z0 = x*y + c
func mulAddWWW_g(x, y, c Word) (z1, z0 Word) {
z1, zz0 := mulWW_g(x, y)
if z0 = zz0 + c; z0 < zz0 {
z1++
}
return
}

// Length of x in bits.
func bitLen_g(x Word) (n int) {
for ; x >= 0x8000; x >>= 16 {
n += 16
}
if x >= 0x80 {
x >>= 8
n += 8
}
if x >= 0x8 {
x >>= 4
n += 4
}
if x >= 0x2 {
x >>= 2
n += 2
}
if x >= 0x1 {
n++
}
return
}

// log2 computes the integer binary logarithm of x.
// The result is the integer n for which 2^n <= x < 2^(n+1).
// If x == 0, the result is -1.
func log2(x Word) int {
return bitLen(x) - 1
}

// nlz returns the number of leading zeros in x.
func nlz(x Word) uint {
return uint(_W - bitLen(x))
}

// nlz64 returns the number of leading zeros in x.
func nlz64(x uint64) uint {
switch _W {
case 32:
w := x >> 32
if w == 0 {
return 32 + nlz(Word(x))
}
return nlz(Word(w))
case 64:
return nlz(Word(x))
}
panic("unreachable")
}

// q = (u1<<_W + u0 - r)/y
// Adapted from Warren, Hacker's Delight, p. 152.
func divWW_g(u1, u0, v Word) (q, r Word) {
if u1 >= v {
return 1<<_W - 1, 1<<_W - 1
}

s := nlz(v)
v <<= s

vn1 := v >> _W2
vn0 := v & _M2
un32 := u1<<s | u0>>(_W-s)
un10 := u0 << s
un1 := un10 >> _W2
un0 := un10 & _M2
q1 := un32 / vn1
rhat := un32 - q1*vn1

for q1 >= _B2 || q1*vn0 > _B2*rhat+un1 {
q1--
rhat += vn1
if rhat >= _B2 {
break
}
}

un21 := un32*_B2 + un1 - q1*v
q0 := un21 / vn1
rhat = un21 - q0*vn1

for q0 >= _B2 || q0*vn0 > _B2*rhat+un0 {
q0--
rhat += vn1
if rhat >= _B2 {
break
}
}

return q1*_B2 + q0, (un21*_B2 + un0 - q0*v) >> s
}

// Keep for performance debugging.
// Using addWW_g is likely slower.

// The resulting carry c is either 0 or 1.
func addVV_g(z, x, y []Word) (c Word) {
for i := range z {
c, z[i] = addWW_g(x[i], y[i], c)
}
return
}

for i, xi := range x[:len(z)] {
yi := y[i]
zi := xi + yi + c
z[i] = zi
// see "Hacker's Delight", section 2-12 (overflow detection)
c = (xi&yi | (xi|yi)&^zi) >> (_W - 1)
}
return
}

// The resulting carry c is either 0 or 1.
func subVV_g(z, x, y []Word) (c Word) {
for i := range z {
c, z[i] = subWW_g(x[i], y[i], c)
}
return
}

for i, xi := range x[:len(z)] {
yi := y[i]
zi := xi - yi - c
z[i] = zi
// see "Hacker's Delight", section 2-12 (overflow detection)
c = (yi&^xi | (yi|^xi)&zi) >> (_W - 1)
}
return
}

// The resulting carry c is either 0 or 1.
func addVW_g(z, x []Word, y Word) (c Word) {
c = y
for i := range z {
c, z[i] = addWW_g(x[i], c, 0)
}
return
}

c = y
for i, xi := range x[:len(z)] {
zi := xi + c
z[i] = zi
c = xi &^ zi >> (_W - 1)
}
return
}

func subVW_g(z, x []Word, y Word) (c Word) {
c = y
for i := range z {
c, z[i] = subWW_g(x[i], c, 0)
}
return
}

c = y
for i, xi := range x[:len(z)] {
zi := xi - c
z[i] = zi
c = (zi &^ xi) >> (_W - 1)
}
return
}

func shlVU_g(z, x []Word, s uint) (c Word) {
if n := len(z); n > 0 {
ŝ := _W - s
w1 := x[n-1]
c = w1 >> ŝ
for i := n - 1; i > 0; i-- {
w := w1
w1 = x[i-1]
z[i] = w<<s | w1>>ŝ
}
z[0] = w1 << s
}
return
}

func shrVU_g(z, x []Word, s uint) (c Word) {
if n := len(z); n > 0 {
ŝ := _W - s
w1 := x[0]
c = w1 << ŝ
for i := 0; i < n-1; i++ {
w := w1
w1 = x[i+1]
z[i] = w>>s | w1<<ŝ
}
z[n-1] = w1 >> s
}
return
}

func mulAddVWW_g(z, x []Word, y, r Word) (c Word) {
c = r
for i := range z {
c, z[i] = mulAddWWW_g(x[i], y, c)
}
return
}

// TODO(gri) Remove use of addWW_g here and then we can remove addWW_g and subWW_g.
func addMulVVW_g(z, x []Word, y Word) (c Word) {
for i := range z {
z1, z0 := mulAddWWW_g(x[i], y, z[i])
c, z[i] = addWW_g(z0, c, 0)
c += z1
}
return
}

func divWVW_g(z []Word, xn Word, x []Word, y Word) (r Word) {
r = xn
for i := len(z) - 1; i >= 0; i-- {
z[i], r = divWW_g(r, x[i], y)
}
return
}

```

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