// 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. Needed for platforms without
// assembly implementations of these routines.
package big
import "unsafe"
type Word uintptr
const (
_S = uintptr(unsafe.Sizeof(Word(0))); // TODO(gri) should Sizeof return a uintptr?
_logW = (0x650 >> _S) & 7;
_W = 1 << _logW;
_B = 1 << _W;
_M = _B - 1;
_W2 = _W / 2;
_B2 = 1 << _W2;
_M2 = _B2 - 1;
)
// ----------------------------------------------------------------------------
// 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
func mulWW_g(x, y Word) (z1, z0 Word) {
// Split x and y into 2 halfWords each, multiply
// the halfWords separately while avoiding overflow,
// and return the product as 2 Words.
if x < y {
x, y = y, x
}
if x < _B2 {
// y < _B2 because y <= x
// sub-digits of x and y are (0, x) and (0, y)
// z = z[0] = x*y
z0 = x * y;
return;
}
if y < _B2 {
// sub-digits of x and y are (x1, x0) and (0, y)
// x = (x1*_B2 + x0)
// y = (y1*_B2 + y0)
x1, x0 := x>>_W2, x&_M2;
// x*y = t2*_B2*_B2 + t1*_B2 + t0
t0 := x0 * y;
t1 := x1 * y;
// compute result digits but avoid overflow
// z = z[1]*_B + z[0] = x*y
z0 = t1<<_W2 + t0;
z1 = (t1 + t0>>_W2) >> _W2;
return;
}
// general case
// sub-digits of x and y are (x1, x0) and (y1, y0)
// x = (x1*_B2 + x0)
// y = (y1*_B2 + y0)
x1, x0 := x>>_W2, x&_M2;
y1, y0 := y>>_W2, y&_M2;
// x*y = t2*_B2*_B2 + t1*_B2 + t0
t0 := x0 * y0;
// t1 := x1*y0 + x0*y1;
var c Word;
t1 := x1 * y0;
t1a := t1;
t1 += x0 * y1;
if t1 < t1a {
c++
}
t2 := x1*y1 + c*_B2;
// compute result digits but avoid overflow
// z = z[1]*_B + z[0] = x*y
// This may overflow, but that's ok because we also sum t1 and t0 above
// and we take care of the overflow there.
z0 = t1<<_W2 + t0;
// z1 = t2 + (t1 + t0>>_W2)>>_W2;
var c3 Word;
z1 = t1 + t0>>_W2;
if z1 < t1 {
c3++
}
z1 >>= _W2;
z1 += c3 * _B2;
z1 += t2;
return;
}
// z1<<_W + z0 = x*y + c
func mulAddWWW_g(x, y, c Word) (z1, z0 Word) {
// Split x and y into 2 halfWords each, multiply
// the halfWords separately while avoiding overflow,
// and return the product as 2 Words.
// TODO(gri) Should implement special cases for faster execution.
// general case
// sub-digits of x, y, and c are (x1, x0), (y1, y0), (c1, c0)
// x = (x1*_B2 + x0)
// y = (y1*_B2 + y0)
x1, x0 := x>>_W2, x&_M2;
y1, y0 := y>>_W2, y&_M2;
c1, c0 := c>>_W2, c&_M2;
// x*y + c = t2*_B2*_B2 + t1*_B2 + t0
// (1<<32-1)^2 == 1<<64 - 1<<33 + 1, so there's space to add c0 in here.
t0 := x0*y0 + c0;
// t1 := x1*y0 + x0*y1 + c1;
var c2 Word; // extra carry
t1 := x1*y0 + c1;
t1a := t1;
t1 += x0 * y1;
if t1 < t1a { // If the number got smaller then we overflowed.
c2++
}
t2 := x1*y1 + c2*_B2;
// compute result digits but avoid overflow
// z = z[1]*_B + z[0] = x*y
// z0 = t1<<_W2 + t0;
// This may overflow, but that's ok because we also sum t1 and t0 below
// and we take care of the overflow there.
z0 = t1<<_W2 + t0;
var c3 Word;
z1 = t1 + t0>>_W2;
if z1 < t1 {
c3++
}
z1 >>= _W2;
z1 += t2 + c3*_B2;
return;
}
// q = (x1<<_W + x0 - r)/y
// The most significant bit of y must be 1.
func divStep(x1, x0, y Word) (q, r Word) {
d1, d0 := y>>_W2, y&_M2;
q1, r1 := x1/d1, x1%d1;
m := q1 * d0;
r1 = r1*_B2 | x0>>_W2;
if r1 < m {
q1--;
r1 += y;
if r1 >= y && r1 < m {
q1--;
r1 += y;
}
}
r1 -= m;
r0 := r1 % d1;
q0 := r1 / d1;
m = q0 * d0;
r0 = r0*_B2 | x0&_M2;
if r0 < m {
q0--;
r0 += y;
if r0 >= y && r0 < m {
q0--;
r0 += y;
}
}
r0 -= m;
q = q1*_B2 | q0;
r = r0;
return;
}
// Number of leading zeros in x.
func leadingZeros(x Word) (n uint) {
if x == 0 {
return _W
}
for x&(1<<(_W-1)) == 0 {
n++;
x <<= 1;
}
return;
}
// q = (x1<<_W + x0 - r)/y
func divWW_g(x1, x0, y Word) (q, r Word) {
if x1 == 0 {
q, r = x0/y, x0%y;
return;
}
var q0, q1 Word;
z := leadingZeros(y);
if y > x1 {
if z != 0 {
y <<= z;
x1 = (x1 << z) | (x0 >> (_W - z));
x0 <<= z;
}
q0, x0 = divStep(x1, x0, y);
q1 = 0;
} else {
if z == 0 {
x1 -= y;
q1 = 1;
} else {
z1 := _W - z;
y <<= z;
x2 := x1 >> z1;
x1 = (x1 << z) | (x0 >> z1);
x0 <<= z;
q1, x1 = divStep(x2, x1, y);
}
q0, x0 = divStep(x1, x0, y);
}
r = x0 >> z;
if q1 != 0 {
panic("div out of range")
}
return q0, r;
}
// ----------------------------------------------------------------------------
// Elementary operations on vectors
// All higher-level functions use these elementary vector operations.
// The function pointers f are initialized with default implementations
// f_g, written in Go for portability. The corresponding assembly routines
// f_s should be installed if they exist.
var (
// addVV sets z and returns c such that z+c = x+y.
addVV func(z, x, y *Word, n int) (c Word) = addVV_g;
// subVV sets z and returns c such that z-c = x-y.
subVV func(z, x, y *Word, n int) (c Word) = subVV_g;
// addVW sets z and returns c such that z+c = x-y.
addVW func(z, x *Word, y Word, n int) (c Word) = addVW_g;
// subVW sets z and returns c such that z-c = x-y.
subVW func(z, x *Word, y Word, n int) (c Word) = subVW_g;
// mulAddVWW sets z and returns c such that z+c = x*y + r.
mulAddVWW func(z, x *Word, y, r Word, n int) (c Word) = mulAddVWW_g;
// addMulVVW sets z and returns c such that z+c = z + x*y.
addMulVVW func(z, x *Word, y Word, n int) (c Word) = addMulVVW_g;
// divWVW sets z and returns r such that z-r = (xn<<(n*_W) + x) / y.
divWVW func(z *Word, xn Word, x *Word, y Word, n int) (r Word) = divWVW_g;
)
func init() {
// Uncomment to use generic routines.
//return;
// Install assembly routines.
addVV = addVV_s;
subVV = subVV_s;
addVW = addVW_s;
subVW = subVW_s;
mulAddVWW = mulAddVWW_s;
addMulVVW = addMulVVW_s;
divWVW = divWVW_s;
}
func (p *Word) at(i int) *Word {
return (*Word)(unsafe.Pointer(uintptr(unsafe.Pointer(p)) + uintptr(i)*_S))
}
func addVV_s(z, x, y *Word, n int) (c Word)
func addVV_g(z, x, y *Word, n int) (c Word) {
for i := 0; i < n; i++ {
c, *z.at(i) = addWW_g(*x.at(i), *y.at(i), c)
}
return;
}
func subVV_s(z, x, y *Word, n int) (c Word)
func subVV_g(z, x, y *Word, n int) (c Word) {
for i := 0; i < n; i++ {
c, *z.at(i) = subWW_g(*x.at(i), *y.at(i), c)
}
return;
}
func addVW_s(z, x *Word, y Word, n int) (c Word)
func addVW_g(z, x *Word, y Word, n int) (c Word) {
c = y;
for i := 0; i < n; i++ {
c, *z.at(i) = addWW_g(*x.at(i), c, 0)
}
return;
}
func subVW_s(z, x *Word, y Word, n int) (c Word)
func subVW_g(z, x *Word, y Word, n int) (c Word) {
c = y;
for i := 0; i < n; i++ {
c, *z.at(i) = subWW_g(*x.at(i), c, 0)
}
return;
}
func mulAddVWW_s(z, x *Word, y, r Word, n int) (c Word)
func mulAddVWW_g(z, x *Word, y, r Word, n int) (c Word) {
c = r;
for i := 0; i < n; i++ {
c, *z.at(i) = mulAddWWW_g(*x.at(i), y, c)
}
return;
}
func addMulVVW_s(z, x *Word, y Word, n int) (c Word)
func addMulVVW_g(z, x *Word, y Word, n int) (c Word) {
for i := 0; i < n; i++ {
z1, z0 := mulAddWWW_g(*x.at(i), y, *z.at(i));
c, *z.at(i) = addWW_g(z0, c, 0);
c += z1;
}
return;
}
func divWVW_s(z *Word, xn Word, x *Word, y Word, n int) (r Word)
func divWVW_g(z *Word, xn Word, x *Word, y Word, n int) (r Word) {
r = xn;
for i := n - 1; i >= 0; i-- {
*z.at(i), r = divWW_g(r, *x.at(i), y)
}
return;
}
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