src/pkg/big/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. Needed for platforms without
// assembly implementations of these routines.

package big

// TODO(gri) Decide if Word needs to remain exported.

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(x, y)
	if z0 = zz0 + c; z0 < zz0 {
		z1++
	}
	return
}


// Length of x in bits.
func bitLen(x Word) (n int) {
	for ; x >= 0x100; x >>= 8 {
		n += 8
	}
	for ; x > 0; x >>= 1 {
		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
}


// Number of leading zeros in x.
func leadingZeros(x Word) uint {
	return uint(_W - bitLen(x))
}


// 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 := leadingZeros(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

again1:
	if q1 >= _B2 || q1*vn0 > _B2*rhat+un1 {
		q1--
		rhat += vn1
		if rhat < _B2 {
			goto again1
		}
	}

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

again2:
	if q0 >= _B2 || q0*vn0 > _B2*rhat+un0 {
		q0--
		rhat += vn1
		if rhat < _B2 {
			goto again2
		}
	}

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


func addVV_g(z, x, y []Word) (c Word) {
	for i := range z {
		c, z[i] = addWW_g(x[i], y[i], c)
	}
	return
}


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


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
}


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
}


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
}


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
}