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Commit 8db86824 authored by Robert Griesemer's avatar Robert Griesemer
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- factored out 128-bit muladd and div into arith.go 

- wrote corresponding fast versions in fast.arith.s
- implemented in-place operations for some routines
- updated existing code to be compatible with in-place
  routines

These changes allow the pidigits benchmark to run
approx. 30% faster. Enabling the assembly routines
in fast.arith.s will give another approx. 3%.

R=r
DELTA=486  (252 added, 68 deleted, 166 changed)
OCL=32980
CL=33003
parent ea8197cb
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Showing with 379 additions and 193 deletions
src/pkg/bignum/Makefile
View file @ 8db86824
......@@ -4,7 +4,7 @@
# DO NOT EDIT. Automatically generated by gobuild.
# gobuild -m bignum.go integer.go rational.go >Makefile
# gobuild -m arith.go bignum.go integer.go rational.go >Makefile
D=
......@@ -33,30 +33,37 @@ coverage: packages
$(AS) $*.s
O1=\
bignum.$O\
arith.$O\
O2=\
integer.$O\
bignum.$O\
O3=\
integer.$O\
O4=\
rational.$O\
phases: a1 a2 a3
phases: a1 a2 a3 a4
_obj$D/bignum.a: phases
a1: $(O1)
$(AR) grc _obj$D/bignum.a bignum.$O
$(AR) grc _obj$D/bignum.a arith.$O
rm -f $(O1)
a2: $(O2)
$(AR) grc _obj$D/bignum.a integer.$O
$(AR) grc _obj$D/bignum.a bignum.$O
rm -f $(O2)
a3: $(O3)
$(AR) grc _obj$D/bignum.a rational.$O
$(AR) grc _obj$D/bignum.a integer.$O
rm -f $(O3)
a4: $(O4)
$(AR) grc _obj$D/bignum.a rational.$O
rm -f $(O4)
newpkg: clean
mkdir -p _obj$D
......@@ -66,6 +73,7 @@ $(O1): newpkg
$(O2): a1
$(O3): a2
$(O4): a3
$(O5): a4
nuke: clean
rm -f $(GOROOT)/pkg/$(GOOS)_$(GOARCH)$D/bignum.a
......
src/pkg/bignum/arith.go 0 → 100644
View file @ 8db86824
// 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.
// Fast versions of the routines in this file are in fast.arith.s.
// Simply replace this file with arith.s (renamed from fast.arith.s)
// and the bignum package will build and run on a platform that
// supports the assembly routines.
package bignum
import "unsafe"
// z1<<64 + z0 = x*y
func Mul128(x, y uint64) (z1, z0 uint64) {
// Split x and y into 2 halfwords each, multiply
// the halfwords separately while avoiding overflow,
// and return the product as 2 words.
const (
W = uint(unsafe.Sizeof(x))*8;
W2 = W/2;
B2 = 1<<W2;
M2 = B2-1;
)
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;
t2 := x1*y1;
// compute result digits but avoid overflow
// z = z[1]*B + z[0] = x*y
z0 = t1<<W2 + t0;
z1 = t2 + (t1 + t0>>W2) >> W2;
return;
}
// z1<<64 + z0 = x*y + c
func MulAdd128(x, y, c uint64) (z1, z0 uint64) {
// Split x and y into 2 halfwords each, multiply
// the halfwords separately while avoiding overflow,
// and return the product as 2 words.
const (
W = uint(unsafe.Sizeof(x))*8;
W2 = W/2;
B2 = 1<<W2;
M2 = B2-1;
)
// 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
t0 := x0*y0 + c0;
t1 := x1*y0 + x0*y1 + c1;
t2 := x1*y1;
// compute result digits but avoid overflow
// z = z[1]*B + z[0] = x*y
z0 = t1<<W2 + t0;
z1 = t2 + (t1 + t0>>W2) >> W2;
return;
}
// q = (x1<<64 + x0)/y + r
func Div128(x1, x0, y uint64) (q, r uint64) {
if x1 == 0 {
q, r = x0/y, x0%y;
return;
}
// TODO(gri) Implement general case.
panic("Div128 not implemented for x > 1<<64-1");
}
src/pkg/bignum/bignum.go
View file @ 8db86824
......@@ -11,8 +11,12 @@
//
package bignum
import "fmt"
import (
"bignum";
"fmt";
)
// TODO(gri) Complete the set of in-place operations.
// ----------------------------------------------------------------------------
// Internal representation
......@@ -59,7 +63,7 @@ type (
const (
logW = 64;
logW = 64; // word width
logH = 4; // bits for a hex digit (= small number)
logB = logW - logH; // largest bit-width available
......@@ -92,7 +96,7 @@ func isSmall(x digit) bool {
// For debugging. Keep around.
/*
func dump(x []digit) {
func dump(x Natural) {
print("[", len(x), "]");
for i := len(x) - 1; i >= 0; i-- {
print(" ", x[i]);
......@@ -110,26 +114,16 @@ func dump(x []digit) {
type Natural []digit;
// Common small values - allocate once.
var nat [16]Natural;
func init() {
nat[0] = Natural{}; // zero has no digits
for i := 1; i < len(nat); i++ {
nat[i] = Natural{digit(i)};
}
}
// Nat creates a small natural number with value x.
//
func Nat(x uint64) Natural {
// avoid allocation for common small values
if x < uint64(len(nat)) {
return nat[x];
if x == 0 {
return nil; // len == 0
}
// single-digit values
// (note: cannot re-use preallocated values because
// the in-place operations may overwrite them)
if x < _B {
return Natural{digit(x)};
}
......@@ -211,25 +205,40 @@ func (x Natural) IsZero() bool {
func normalize(x Natural) Natural {
n := len(x);
for n > 0 && x[n - 1] == 0 { n-- }
if n < len(x) {
x = x[0 : n]; // trim leading 0's
for n > 0 && x[n-1] == 0 { n-- }
return x[0 : n];
}
// nalloc returns a Natural of n digits. If z is large
// enough, z is resized and returned. Otherwise, a new
// Natural is allocated.
//
func nalloc(z Natural, n int) Natural {
size := n;
if size <= 0 {
size = 4;
}
if size <= cap(z) {
return z[0 : n];
}
return x;
return make(Natural, n, size);
}
// Add returns the sum x + y.
// Nadd sets *zp to the sum x + y.
// *zp may be x or y.
//
func (x Natural) Add(y Natural) Natural {
func Nadd(zp *Natural, x, y Natural) {
n := len(x);
m := len(y);
if n < m {
return y.Add(x);
Nadd(zp, y, x);
return;
}
z := nalloc(*zp, n+1);
c := digit(0);
z := make(Natural, n + 1);
i := 0;
for i < m {
t := c + x[i] + y[i];
......@@ -245,23 +254,32 @@ func (x Natural) Add(y Natural) Natural {
z[i] = c;
i++;
}
*zp = z[0 : i]
}
return z[0 : i];
// Add returns the sum z = x + y.
//
func (x Natural) Add(y Natural) Natural {
var z Natural;
Nadd(&z, x, y);
return z;
}
// Sub returns the difference x - y for x >= y.
// Nsub sets *zp to the difference x - y for x >= y.
// If x < y, an underflow run-time error occurs (use Cmp to test if x >= y).
// *zp may be x or y.
//
func (x Natural) Sub(y Natural) Natural {
func Nsub(zp *Natural, x, y Natural) {
n := len(x);
m := len(y);
if n < m {
panic("underflow")
}
z := nalloc(*zp, n);
c := digit(0);
z := make(Natural, n);
i := 0;
for i < m {
t := c + x[i] - y[i];
......@@ -273,110 +291,104 @@ func (x Natural) Sub(y Natural) Natural {
c, z[i] = digit(int64(t)>>_W), t&_M; // requires arithmetic shift!
i++;
}
for i > 0 && z[i - 1] == 0 { // normalize
i--;
if int64(c) < 0 {
panic("underflow");
}
*zp = normalize(z);
}
return z[0 : i];
// Sub returns the difference x - y for x >= y.
// If x < y, an underflow run-time error occurs (use Cmp to test if x >= y).
//
func (x Natural) Sub(y Natural) Natural {
var z Natural;
Nsub(&z, x, y);
return z;
}
// Returns c = x*y div B, z = x*y mod B.
// MulAdd128 is defined in arith.go and arith.s .
func MulAdd128(x, y, c uint64) (z1, z0 uint64)
// Returns z1 = (x*y + c) div B, z0 = (x*y + c) mod B.
//
func mul11(x, y digit) (z1, z0 digit) {
// Split x and y into 2 sub-digits each,
// multiply the digits separately while avoiding overflow,
// and return the product as two separate digits.
func muladd11(x, y, c digit) (digit, digit) {
z1, z0 := MulAdd128(uint64(x), uint64(y), uint64(c));
return digit(z1<<(64 - logB) | z0>>logB), digit(z0&_M);
}
// This code also works for non-even bit widths W
// which is why there are separate constants below
// for half-digits.
const W2 = (_W + 1)/2;
const DW = W2*2 - _W; // 0 or 1
const B2 = 1<<W2;
const M2 = _B2 - 1;
if x < y {
x, y = y, x;
func mul1(z, x Natural, y digit) (c digit) {
for i := 0; i < len(x); i++ {
c, z[i] = muladd11(x[i], y, c);
}
return;
}
if x < _B2 {
// y < _B2 because y <= x
// sub-digits of x and y are (0, x) and (0, y)
// x = x
// y = y
t0 := x*y;
// compute result digits but avoid overflow
// z = z1*B + z0 = x*y
z0 = t0 & _M;
z1 = (t0>>W2) >> (_W-W2);
// Nscale sets *z to the scaled value (*z) * d.
//
func Nscale(z *Natural, d uint64) {
switch {
case d == 0:
*z = Nat(0);
return
case d == 1:
return;
case d >= _B:
*z = z.Mul1(d);
return;
}
if y < _B2 {
// split x and y into sub-digits
// sub-digits of y are (x1, x0) and (0, y)
// x = (x1*B2 + x0)
// y = y
x1, x0 := x>>W2, x&M2;
c := mul1(*z, *z, digit(d));
// x*y = t1*B2 + t0
t0 := x0*y;
t1 := x1*y;
if c != 0 {
n := len(*z);
if n >= cap(*z) {
zz := make(Natural, n+1);
for i, d := range *z {
zz[i] = d;
}
*z = zz
} else {
*z = (*z)[0 : n+1];
}
(*z)[n] = c;
}
}
// compute result digits but avoid overflow
// z = z1*B + z0 = x*y
z0 = (t1<<W2 + t0)&_M;
z1 = (t1 + t0>>W2) >> (_W-W2);
return;
// Computes x = x*d + c for small d's.
//
func muladd1(x Natural, d, c digit) Natural {
assert(isSmall(d-1) && isSmall(c));
n := len(x);
z := make(Natural, n + 1);
for i := 0; i < n; i++ {
t := c + x[i]*d;
c, z[i] = t>>_W, t&_M;
}
z[n] = c;
// 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*B2^2 + t1*B2 + t0
t0 := x0*y0;
t1 := x1*y0 + x0*y1;
t2 := x1*y1;
// compute result digits but avoid overflow
// z = z1*B + z0 = x*y
z0 = (t1<<W2 + t0)&_M;
z1 = t2<<DW + (t1 + t0>>W2) >> (_W-W2);
return;
return normalize(z);
}
func (x Natural) Mul(y Natural) Natural
// Mul1 returns the product x * d.
//
func (x Natural) Mul1(d uint64) Natural {
switch {
case d == 0: return nat[0];
case d == 0: return Nat(0);
case d == 1: return x;
case isSmall(digit(d)): muladd1(x, digit(d), 0);
case d >= _B: return x.Mul(Nat(d));
}
n := len(x);
z := make(Natural, n + 1);
if d != 0 {
c := digit(0);
for i := 0; i < n; i++ {
// z[i] += c + x[i]*d;
z1, z0 := mul11(x[i], digit(d));
t := c + z[i] + z0;
c, z[i] = t>>_W, t&_M;
c += z1;
}
z[n] = c;
}
z := make(Natural, len(x) + 1);
c := mul1(z, x, digit(d));
z[len(x)] = c;
return normalize(z);
}
......@@ -390,6 +402,10 @@ func (x Natural) Mul(y Natural) Natural {
return y.Mul(x);
}
if m == 0 {
return Nat(0);
}
if m == 1 && y[0] < _B {
return x.Mul1(uint64(y[0]));
}
......@@ -400,11 +416,7 @@ func (x Natural) Mul(y Natural) Natural {
if d != 0 {
c := digit(0);
for i := 0; i < n; i++ {
// z[i+j] += c + x[i]*d;
z1, z0 := mul11(x[i], d);
t := c + z[i+j] + z0;
c, z[i+j] = t>>_W, t&_M;
c += z1;
c, z[i+j] = muladd11(x[i], d, z[i+j] + c);
}
z[n+j] = c;
}
......@@ -450,11 +462,10 @@ func pack(x []digit2) Natural {
}
func mul1(z, x []digit2, y digit2) digit2 {
n := len(x);
func mul21(z, x []digit2, y digit2) digit2 {
c := digit(0);
f := digit(y);
for i := 0; i < n; i++ {
for i := 0; i < len(x); i++ {
t := c + digit(x[i])*f;
c, z[i] = t>>_W2, digit2(t&_M2);
}
......@@ -462,12 +473,11 @@ func mul1(z, x []digit2, y digit2) digit2 {
}
func div1(z, x []digit2, y digit2) digit2 {
n := len(x);
func div21(z, x []digit2, y digit2) digit2 {
c := digit(0);
d := digit(y);
for i := n-1; i >= 0; i-- {
t := c*_B2 + digit(x[i]);
for i := len(x)-1; i >= 0; i-- {
t := c<<_W2 + digit(x[i]);
c, z[i] = t%d, digit2(t/d);
}
return digit2(c);
......@@ -501,13 +511,13 @@ func divmod(x, y []digit2) ([]digit2, []digit2) {
panic("division by zero");
}
assert(n+1 <= cap(x)); // space for one extra digit
x = x[0 : n + 1];
x = x[0 : n+1];
assert(x[n] == 0);
if m == 1 {
// division by single digit
// result is shifted left by 1 in place!
x[0] = div1(x[1 : n+1], x[0 : n], y[0]);
x[0] = div21(x[1 : n+1], x[0 : n], y[0]);
} else if m > n {
// y > x => quotient = 0, remainder = x
......@@ -524,13 +534,12 @@ func divmod(x, y []digit2) ([]digit2, []digit2) {
// satisfied (as done in Hacker's Delight).
f := _B2 / (digit(y[m-1]) + 1);
if f != 1 {
mul1(x, x, digit2(f));
mul1(y, y, digit2(f));
mul21(x, x, digit2(f));
mul21(y, y, digit2(f));
}
assert(_B2/2 <= y[m-1] && y[m-1] < _B2); // incorrect scaling
y1, y2 := digit(y[m-1]), digit(y[m-2]);
d2 := digit(y1)<<_W2 + digit(y2);
for i := n-m; i >= 0; i-- {
k := i+m;
......@@ -540,7 +549,7 @@ func divmod(x, y []digit2) ([]digit2, []digit2) {
if x0 != y1 {
q = (x0<<_W2 + x1)/y1;
} else {
q = _B2 - 1;
q = _B2-1;
}
for y2*q > (x0<<_W2 + x1 - y1*q)<<_W2 + x2 {
q--
......@@ -572,7 +581,7 @@ func divmod(x, y []digit2) ([]digit2, []digit2) {
// undo normalization for remainder
if f != 1 {
c := div1(x[0 : m], x[0 : m], digit2(f));
c := div21(x[0 : m], x[0 : m], digit2(f));
assert(c == 0);
}
}
......@@ -610,7 +619,7 @@ func (x Natural) DivMod(y Natural) (Natural, Natural) {
}
func shl(z, x []digit, s uint) digit {
func shl(z, x Natural, s uint) digit {
assert(s <= _W);
n := len(x);
c := digit(0);
......@@ -634,7 +643,7 @@ func (x Natural) Shl(s uint) Natural {
}
func shr(z, x []digit, s uint) digit {
func shr(z, x Natural, s uint) digit {
assert(s <= _W);
n := len(x);
c := digit(0);
......@@ -680,7 +689,7 @@ func (x Natural) And(y Natural) Natural {
}
func copy(z, x []digit) {
func copy(z, x Natural) {
for i, e := range x {
z[i] = e
}
......@@ -881,23 +890,6 @@ func hexvalue(ch byte) uint {
}
// Computes x = x*d + c for small d's.
//
func muladd1(x Natural, d, c digit) Natural {
assert(isSmall(d-1) && isSmall(c));
n := len(x);
z := make(Natural, n + 1);
for i := 0; i < n; i++ {
t := c + x[i]*d;
c, z[i] = t>>_W, t&_M;
}
z[n] = c;
return normalize(z);
}
// NatFromString returns the natural number corresponding to the
// longest possible prefix of s representing a natural number in a
// given conversion base, the actual conversion base used, and the
......@@ -924,7 +916,7 @@ func NatFromString(s string, base uint) (Natural, uint, int) {
// convert string
assert(2 <= base && base <= 16);
x := nat[0];
x := Nat(0);
for ; i < n; i++ {
d := hexvalue(s[i]);
if d < base {
......@@ -964,7 +956,7 @@ func (x Natural) Pop() uint {
// Pow computes x to the power of n.
//
func (xp Natural) Pow(n uint) Natural {
z := nat[1];
z := Nat(1);
x := xp;
for n > 0 {
// z * x^n == x^n0
......@@ -982,7 +974,7 @@ func (xp Natural) Pow(n uint) Natural {
//
func MulRange(a, b uint) Natural {
switch {
case a > b: return nat[1];
case a > b: return Nat(1);
case a == b: return Nat(uint64(a));
case a + 1 == b: return Nat(uint64(a)).Mul(Nat(uint64(b)));
}
......
src/pkg/bignum/fast.arith.s 0 → 100644
View file @ 8db86824
// 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 fast assembly versions
// of the routines in arith.go.
// func Mul128(x, y uint64) (z1, z0 uint64)
// z1<<64 + z0 = x*y
//
TEXT bignum·Mul128(SB),7,$0
MOVQ a+0(FP), AX
MULQ a+8(FP)
MOVQ DX, a+16(FP)
MOVQ AX, a+24(FP)
RET
// func MulAdd128(x, y, c uint64) (z1, z0 uint64)
// z1<<64 + z0 = x*y + c
//
TEXT bignum·MulAdd128(SB),7,$0
MOVQ a+0(FP), AX
MULQ a+8(FP)
ADDQ a+16(FP), AX
ADCQ $0, DX
MOVQ DX, a+24(FP)
MOVQ AX, a+32(FP)
RET
// func Div128(x1, x0, y uint64) (q, r uint64)
// q = (x1<<64 + x0)/y + r
//
TEXT bignum·Div128(SB),7,$0
MOVQ a+0(FP), DX
MOVQ a+8(FP), AX
DIVQ a+16(FP)
MOVQ AX, a+24(FP)
MOVQ DX, a+32(FP)
RET
src/pkg/bignum/integer.go
View file @ 8db86824
......@@ -9,9 +9,12 @@
package bignum
import "bignum"
import "fmt"
import (
"bignum";
"fmt";
)
// TODO(gri) Complete the set of in-place operations.
// Integer represents a signed integer value of arbitrary precision.
//
......@@ -113,61 +116,82 @@ func (x *Integer) Neg() *Integer {
}
// Add returns the sum x + y.
// Iadd sets z to the sum x + y.
// z must exist and may be x or y.
//
func (x *Integer) Add(y *Integer) *Integer {
var z *Integer;
func Iadd(z, x, y *Integer) {
if x.sign == y.sign {
// x + y == x + y
// (-x) + (-y) == -(x + y)
z = MakeInt(x.sign, x.mant.Add(y.mant));
z.sign = x.sign;
Nadd(&z.mant, x.mant, y.mant);
} else {
// x + (-y) == x - y == -(y - x)
// (-x) + y == y - x == -(x - y)
if x.mant.Cmp(y.mant) >= 0 {
z = MakeInt(false, x.mant.Sub(y.mant));
z.sign = x.sign;
Nsub(&z.mant, x.mant, y.mant);
} else {
z = MakeInt(true, y.mant.Sub(x.mant));
z.sign = !x.sign;
Nsub(&z.mant, y.mant, x.mant);
}
}
if x.sign {
z.sign = !z.sign;
}
return z;
}
// Sub returns the difference x - y.
// Add returns the sum x + y.
//
func (x *Integer) Sub(y *Integer) *Integer {
var z *Integer;
func (x *Integer) Add(y *Integer) *Integer {
var z Integer;
Iadd(&z, x, y);
return &z;
}
func Isub(z, x, y *Integer) {
if x.sign != y.sign {
// x - (-y) == x + y
// (-x) - y == -(x + y)
z = MakeInt(false, x.mant.Add(y.mant));
z.sign = x.sign;
Nadd(&z.mant, x.mant, y.mant);
} else {
// x - y == x - y == -(y - x)
// (-x) - (-y) == y - x == -(x - y)
if x.mant.Cmp(y.mant) >= 0 {
z = MakeInt(false, x.mant.Sub(y.mant));
z.sign = x.sign;
Nsub(&z.mant, x.mant, y.mant);
} else {
z = MakeInt(true, y.mant.Sub(x.mant));
z.sign = !x.sign;
Nsub(&z.mant, y.mant, x.mant);
}
}
if x.sign {
z.sign = !z.sign;
}
// Sub returns the difference x - y.
//
func (x *Integer) Sub(y *Integer) *Integer {
var z Integer;
Isub(&z, x, y);
return &z;
}
// Nscale sets *z to the scaled value (*z) * d.
//
func Iscale(z *Integer, d int64) {
f := uint64(d);
if d < 0 {
f = uint64(-d);
}
return z;
z.sign = z.sign != (d < 0);
Nscale(&z.mant, f);
}
// Mul1 returns the product x * d.
//
func (x *Integer) Mul1(d int64) *Integer {
// x * y == x * y
// x * (-y) == -(x * y)
// (-x) * y == -(x * y)
// (-x) * (-y) == x * y
f := uint64(d);
if d < 0 {
f = uint64(-d);
......@@ -326,7 +350,7 @@ func (x *Integer) Shl(s uint) *Integer {
func (x *Integer) Shr(s uint) *Integer {
if x.sign {
// (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
return MakeInt(true, x.mant.Sub(nat[1]).Shr(s).Add(nat[1]));
return MakeInt(true, x.mant.Sub(Nat(1)).Shr(s).Add(Nat(1)));
}
return MakeInt(false, x.mant.Shr(s));
......@@ -337,11 +361,11 @@ func (x *Integer) Shr(s uint) *Integer {
func (x *Integer) Not() *Integer {
if x.sign {
// ^(-x) == ^(^(x-1)) == x-1
return MakeInt(false, x.mant.Sub(nat[1]));
return MakeInt(false, x.mant.Sub(Nat(1)));
}
// ^x == -x-1 == -(x+1)
return MakeInt(true, x.mant.Add(nat[1]));
return MakeInt(true, x.mant.Add(Nat(1)));
}
......@@ -351,7 +375,7 @@ func (x *Integer) And(y *Integer) *Integer {
if x.sign == y.sign {
if x.sign {
// (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
return MakeInt(true, x.mant.Sub(nat[1]).Or(y.mant.Sub(nat[1])).Add(nat[1]));
return MakeInt(true, x.mant.Sub(Nat(1)).Or(y.mant.Sub(Nat(1))).Add(Nat(1)));
}
// x & y == x & y
......@@ -364,7 +388,7 @@ func (x *Integer) And(y *Integer) *Integer {
}
// x & (-y) == x & ^(y-1) == x &^ (y-1)
return MakeInt(false, x.mant.AndNot(y.mant.Sub(nat[1])));
return MakeInt(false, x.mant.AndNot(y.mant.Sub(Nat(1))));
}
......@@ -374,7 +398,7 @@ func (x *Integer) AndNot(y *Integer) *Integer {
if x.sign == y.sign {
if x.sign {
// (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
return MakeInt(false, y.mant.Sub(nat[1]).AndNot(x.mant.Sub(nat[1])));
return MakeInt(false, y.mant.Sub(Nat(1)).AndNot(x.mant.Sub(Nat(1))));
}
// x &^ y == x &^ y
......@@ -383,11 +407,11 @@ func (x *Integer) AndNot(y *Integer) *Integer {
if x.sign {
// (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
return MakeInt(true, x.mant.Sub(nat[1]).Or(y.mant).Add(nat[1]));
return MakeInt(true, x.mant.Sub(Nat(1)).Or(y.mant).Add(Nat(1)));
}
// x &^ (-y) == x &^ ^(y-1) == x & (y-1)
return MakeInt(false, x.mant.And(y.mant.Sub(nat[1])));
return MakeInt(false, x.mant.And(y.mant.Sub(Nat(1))));
}
......@@ -397,7 +421,7 @@ func (x *Integer) Or(y *Integer) *Integer {
if x.sign == y.sign {
if x.sign {
// (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
return MakeInt(true, x.mant.Sub(nat[1]).And(y.mant.Sub(nat[1])).Add(nat[1]));
return MakeInt(true, x.mant.Sub(Nat(1)).And(y.mant.Sub(Nat(1))).Add(Nat(1)));
}
// x | y == x | y
......@@ -410,7 +434,7 @@ func (x *Integer) Or(y *Integer) *Integer {
}
// x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
return MakeInt(true, y.mant.Sub(nat[1]).AndNot(x.mant).Add(nat[1]));
return MakeInt(true, y.mant.Sub(Nat(1)).AndNot(x.mant).Add(Nat(1)));
}
......@@ -420,7 +444,7 @@ func (x *Integer) Xor(y *Integer) *Integer {
if x.sign == y.sign {
if x.sign {
// (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
return MakeInt(false, x.mant.Sub(nat[1]).Xor(y.mant.Sub(nat[1])));
return MakeInt(false, x.mant.Sub(Nat(1)).Xor(y.mant.Sub(Nat(1))));
}
// x ^ y == x ^ y
......@@ -433,7 +457,7 @@ func (x *Integer) Xor(y *Integer) *Integer {
}
// x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
return MakeInt(true, x.mant.Xor(y.mant.Sub(nat[1])).Add(nat[1]));
return MakeInt(true, x.mant.Xor(y.mant.Sub(Nat(1))).Add(Nat(1)));
}
......
src/pkg/bignum/rational.go
View file @ 8db86824
......@@ -22,7 +22,7 @@ type Rational struct {
//
func MakeRat(a *Integer, b Natural) *Rational {
f := a.mant.Gcd(b); // f > 0
if f.Cmp(nat[1]) != 0 {
if f.Cmp(Nat(1)) != 0 {
a = MakeInt(a.sign, a.mant.Div(f));
b = b.Div(f);
}
......@@ -75,7 +75,7 @@ func (x *Rational) IsPos() bool {
// in the form x == x'/1; i.e., if x is an integer value.
//
func (x *Rational) IsInt() bool {
return x.b.Cmp(nat[1]) == 0;
return x.b.Cmp(Nat(1)) == 0;
}
......@@ -184,7 +184,7 @@ func (x *Rational) Format(h fmt.State, c int) {
func RatFromString(s string, base uint) (*Rational, uint, int) {
// read numerator
a, abase, alen := IntFromString(s, base);
b := nat[1];
b := Nat(1);
// read denominator or fraction, if any
var blen int;
......@@ -211,7 +211,7 @@ func RatFromString(s string, base uint) (*Rational, uint, int) {
rlen++;
e, _, elen := IntFromString(s[rlen : len(s)], 10);
rlen += elen;
m := nat[10].Pow(uint(e.mant.Value()));
m := Nat(10).Pow(uint(e.mant.Value()));
if e.sign {
b = b.Mul(m);
} else {
......
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