mirror of
https://github.com/golang/go
synced 2024-11-22 00:04:41 -07:00
bignum: delete package - functionality subsumed by package big
R=rsc CC=golang-dev https://golang.org/cl/1742045
This commit is contained in:
Robert Griesemer
parent
b2a919fc29
commit
496a935376
@ -63,7 +63,6 @@ DIRS=\
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encoding/hex\
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encoding/pem\
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exec\
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exp/bignum\
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exp/datafmt\
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exp/draw\
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exp/draw/x11\
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@ -1,14 +0,0 @@
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# Copyright 2009 The Go Authors. All rights reserved.
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# Use of this source code is governed by a BSD-style
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# license that can be found in the LICENSE file.
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include ../../../Make.$(GOARCH)
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TARG=exp/bignum
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GOFILES=\
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arith.go\
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bignum.go\
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integer.go\
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rational.go\
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include ../../../Make.pkg
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@ -1,121 +0,0 @@
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// Copyright 2009 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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// Fast versions of the routines in this file are in fast.arith.s.
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// Simply replace this file with arith.s (renamed from fast.arith.s)
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// and the bignum package will build and run on a platform that
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// supports the assembly routines.
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package bignum
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import "unsafe"
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// z1<<64 + z0 = x*y
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func Mul128(x, y uint64) (z1, z0 uint64) {
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// Split x and y into 2 halfwords each, multiply
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// the halfwords separately while avoiding overflow,
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// and return the product as 2 words.
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const (
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W = uint(unsafe.Sizeof(x)) * 8
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W2 = W / 2
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B2 = 1 << W2
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M2 = B2 - 1
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)
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if x < y {
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x, y = y, x
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}
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if x < B2 {
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// y < B2 because y <= x
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// sub-digits of x and y are (0, x) and (0, y)
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// z = z[0] = x*y
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z0 = x * y
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return
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}
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if y < B2 {
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// sub-digits of x and y are (x1, x0) and (0, y)
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// x = (x1*B2 + x0)
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// y = (y1*B2 + y0)
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x1, x0 := x>>W2, x&M2
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// x*y = t2*B2*B2 + t1*B2 + t0
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t0 := x0 * y
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t1 := x1 * y
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// compute result digits but avoid overflow
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// z = z[1]*B + z[0] = x*y
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z0 = t1<<W2 + t0
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z1 = (t1 + t0>>W2) >> W2
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return
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}
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// general case
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// sub-digits of x and y are (x1, x0) and (y1, y0)
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// x = (x1*B2 + x0)
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// y = (y1*B2 + y0)
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x1, x0 := x>>W2, x&M2
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y1, y0 := y>>W2, y&M2
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// x*y = t2*B2*B2 + t1*B2 + t0
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t0 := x0 * y0
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t1 := x1*y0 + x0*y1
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t2 := x1 * y1
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// compute result digits but avoid overflow
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// z = z[1]*B + z[0] = x*y
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z0 = t1<<W2 + t0
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z1 = t2 + (t1+t0>>W2)>>W2
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return
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}
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// z1<<64 + z0 = x*y + c
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func MulAdd128(x, y, c uint64) (z1, z0 uint64) {
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// Split x and y into 2 halfwords each, multiply
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// the halfwords separately while avoiding overflow,
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// and return the product as 2 words.
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const (
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W = uint(unsafe.Sizeof(x)) * 8
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W2 = W / 2
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B2 = 1 << W2
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M2 = B2 - 1
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)
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// TODO(gri) Should implement special cases for faster execution.
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// general case
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// sub-digits of x, y, and c are (x1, x0), (y1, y0), (c1, c0)
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// x = (x1*B2 + x0)
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// y = (y1*B2 + y0)
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x1, x0 := x>>W2, x&M2
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y1, y0 := y>>W2, y&M2
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c1, c0 := c>>W2, c&M2
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// x*y + c = t2*B2*B2 + t1*B2 + t0
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t0 := x0*y0 + c0
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t1 := x1*y0 + x0*y1 + c1
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t2 := x1 * y1
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// compute result digits but avoid overflow
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// z = z[1]*B + z[0] = x*y
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z0 = t1<<W2 + t0
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z1 = t2 + (t1+t0>>W2)>>W2
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return
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}
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// q = (x1<<64 + x0)/y + r
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func Div128(x1, x0, y uint64) (q, r uint64) {
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if x1 == 0 {
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q, r = x0/y, x0%y
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return
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}
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// TODO(gri) Implement general case.
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panic("Div128 not implemented for x > 1<<64-1")
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}
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@ -1,41 +0,0 @@
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// Copyright 2009 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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// This file provides fast assembly versions
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// of the routines in arith.go.
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// func Mul128(x, y uint64) (z1, z0 uint64)
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// z1<<64 + z0 = x*y
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//
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TEXT ·Mul128(SB),7,$0
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MOVQ a+0(FP), AX
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MULQ a+8(FP)
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MOVQ DX, a+16(FP)
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MOVQ AX, a+24(FP)
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RET
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// func MulAdd128(x, y, c uint64) (z1, z0 uint64)
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// z1<<64 + z0 = x*y + c
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//
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TEXT ·MulAdd128(SB),7,$0
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MOVQ a+0(FP), AX
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MULQ a+8(FP)
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ADDQ a+16(FP), AX
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ADCQ $0, DX
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MOVQ DX, a+24(FP)
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MOVQ AX, a+32(FP)
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RET
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// func Div128(x1, x0, y uint64) (q, r uint64)
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// q = (x1<<64 + x0)/y + r
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//
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TEXT ·Div128(SB),7,$0
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MOVQ a+0(FP), DX
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MOVQ a+8(FP), AX
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DIVQ a+16(FP)
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MOVQ AX, a+24(FP)
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MOVQ DX, a+32(FP)
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RET
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File diff suppressed because it is too large
Load Diff
@ -1,681 +0,0 @@
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// Copyright 2009 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package bignum
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import (
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"fmt"
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"testing"
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)
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const (
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sa = "991"
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sb = "2432902008176640000" // 20!
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sc = "933262154439441526816992388562667004907159682643816214685929" +
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"638952175999932299156089414639761565182862536979208272237582" +
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"51185210916864000000000000000000000000" // 100!
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sp = "170141183460469231731687303715884105727" // prime
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)
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func natFromString(s string, base uint, slen *int) Natural {
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x, _, len := NatFromString(s, base)
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if slen != nil {
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*slen = len
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}
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return x
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}
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func intFromString(s string, base uint, slen *int) *Integer {
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x, _, len := IntFromString(s, base)
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if slen != nil {
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*slen = len
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}
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return x
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}
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func ratFromString(s string, base uint, slen *int) *Rational {
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x, _, len := RatFromString(s, base)
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if slen != nil {
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*slen = len
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}
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return x
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}
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var (
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nat_zero = Nat(0)
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nat_one = Nat(1)
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nat_two = Nat(2)
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a = natFromString(sa, 10, nil)
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b = natFromString(sb, 10, nil)
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c = natFromString(sc, 10, nil)
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p = natFromString(sp, 10, nil)
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int_zero = Int(0)
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int_one = Int(1)
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int_two = Int(2)
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ip = intFromString(sp, 10, nil)
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rat_zero = Rat(0, 1)
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rat_half = Rat(1, 2)
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rat_one = Rat(1, 1)
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rat_two = Rat(2, 1)
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)
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var test_msg string
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var tester *testing.T
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func test(n uint, b bool) {
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if !b {
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tester.Fatalf("TEST failed: %s (%d)", test_msg, n)
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}
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}
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func nat_eq(n uint, x, y Natural) {
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if x.Cmp(y) != 0 {
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tester.Fatalf("TEST failed: %s (%d)\nx = %v\ny = %v", test_msg, n, &x, &y)
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}
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}
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func int_eq(n uint, x, y *Integer) {
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if x.Cmp(y) != 0 {
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tester.Fatalf("TEST failed: %s (%d)\nx = %v\ny = %v", test_msg, n, x, y)
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}
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}
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func rat_eq(n uint, x, y *Rational) {
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if x.Cmp(y) != 0 {
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tester.Fatalf("TEST failed: %s (%d)\nx = %v\ny = %v", test_msg, n, x, y)
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}
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}
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func TestNatConv(t *testing.T) {
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tester = t
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test_msg = "NatConvA"
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type entry1 struct {
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x uint64
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s string
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}
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tab := []entry1{
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entry1{0, "0"},
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entry1{255, "255"},
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entry1{65535, "65535"},
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entry1{4294967295, "4294967295"},
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entry1{18446744073709551615, "18446744073709551615"},
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}
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for i, e := range tab {
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test(100+uint(i), Nat(e.x).String() == e.s)
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test(200+uint(i), natFromString(e.s, 0, nil).Value() == e.x)
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}
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test_msg = "NatConvB"
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for i := uint(0); i < 100; i++ {
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test(i, Nat(uint64(i)).String() == fmt.Sprintf("%d", i))
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}
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test_msg = "NatConvC"
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z := uint64(7)
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for i := uint(0); i <= 64; i++ {
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test(i, Nat(z).Value() == z)
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z <<= 1
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}
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test_msg = "NatConvD"
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nat_eq(0, a, Nat(991))
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nat_eq(1, b, Fact(20))
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nat_eq(2, c, Fact(100))
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test(3, a.String() == sa)
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test(4, b.String() == sb)
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test(5, c.String() == sc)
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test_msg = "NatConvE"
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var slen int
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nat_eq(10, natFromString("0", 0, nil), nat_zero)
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nat_eq(11, natFromString("123", 0, nil), Nat(123))
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nat_eq(12, natFromString("077", 0, nil), Nat(7*8+7))
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nat_eq(13, natFromString("0x1f", 0, nil), Nat(1*16+15))
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nat_eq(14, natFromString("0x1fg", 0, &slen), Nat(1*16+15))
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test(4, slen == 4)
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test_msg = "NatConvF"
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tmp := c.Mul(c)
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for base := uint(2); base <= 16; base++ {
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nat_eq(base, natFromString(tmp.ToString(base), base, nil), tmp)
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}
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test_msg = "NatConvG"
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x := Nat(100)
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y, _, _ := NatFromString(fmt.Sprintf("%b", &x), 2)
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nat_eq(100, y, x)
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}
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func abs(x int64) uint64 {
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if x < 0 {
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x = -x
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}
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return uint64(x)
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}
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func TestIntConv(t *testing.T) {
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tester = t
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test_msg = "IntConvA"
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type entry2 struct {
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x int64
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s string
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}
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tab := []entry2{
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entry2{0, "0"},
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entry2{-128, "-128"},
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entry2{127, "127"},
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entry2{-32768, "-32768"},
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entry2{32767, "32767"},
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entry2{-2147483648, "-2147483648"},
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entry2{2147483647, "2147483647"},
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entry2{-9223372036854775808, "-9223372036854775808"},
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entry2{9223372036854775807, "9223372036854775807"},
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}
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for i, e := range tab {
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test(100+uint(i), Int(e.x).String() == e.s)
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test(200+uint(i), intFromString(e.s, 0, nil).Value() == e.x)
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test(300+uint(i), Int(e.x).Abs().Value() == abs(e.x))
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}
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test_msg = "IntConvB"
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var slen int
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int_eq(0, intFromString("0", 0, nil), int_zero)
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int_eq(1, intFromString("-0", 0, nil), int_zero)
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int_eq(2, intFromString("123", 0, nil), Int(123))
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int_eq(3, intFromString("-123", 0, nil), Int(-123))
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int_eq(4, intFromString("077", 0, nil), Int(7*8+7))
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int_eq(5, intFromString("-077", 0, nil), Int(-(7*8 + 7)))
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int_eq(6, intFromString("0x1f", 0, nil), Int(1*16+15))
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int_eq(7, intFromString("-0x1f", 0, &slen), Int(-(1*16 + 15)))
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test(7, slen == 5)
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int_eq(8, intFromString("+0x1f", 0, &slen), Int(+(1*16 + 15)))
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test(8, slen == 5)
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int_eq(9, intFromString("0x1fg", 0, &slen), Int(1*16+15))
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test(9, slen == 4)
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int_eq(10, intFromString("-0x1fg", 0, &slen), Int(-(1*16 + 15)))
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test(10, slen == 5)
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}
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func TestRatConv(t *testing.T) {
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tester = t
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test_msg = "RatConv"
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var slen int
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rat_eq(0, ratFromString("0", 0, nil), rat_zero)
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rat_eq(1, ratFromString("0/1", 0, nil), rat_zero)
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rat_eq(2, ratFromString("0/01", 0, nil), rat_zero)
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rat_eq(3, ratFromString("0x14/10", 0, &slen), rat_two)
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test(4, slen == 7)
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rat_eq(5, ratFromString("0.", 0, nil), rat_zero)
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rat_eq(6, ratFromString("0.001f", 10, nil), Rat(1, 1000))
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rat_eq(7, ratFromString(".1", 0, nil), Rat(1, 10))
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rat_eq(8, ratFromString("10101.0101", 2, nil), Rat(0x155, 1<<4))
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rat_eq(9, ratFromString("-0003.145926", 10, &slen), Rat(-3145926, 1000000))
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test(10, slen == 12)
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rat_eq(11, ratFromString("1e2", 0, nil), Rat(100, 1))
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rat_eq(12, ratFromString("1e-2", 0, nil), Rat(1, 100))
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rat_eq(13, ratFromString("1.1e2", 0, nil), Rat(110, 1))
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rat_eq(14, ratFromString(".1e2x", 0, &slen), Rat(10, 1))
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test(15, slen == 4)
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}
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func add(x, y Natural) Natural {
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z1 := x.Add(y)
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z2 := y.Add(x)
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if z1.Cmp(z2) != 0 {
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tester.Fatalf("addition not symmetric:\n\tx = %v\n\ty = %t", x, y)
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}
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return z1
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}
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func sum(n uint64, scale Natural) Natural {
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s := nat_zero
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for ; n > 0; n-- {
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s = add(s, Nat(n).Mul(scale))
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}
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return s
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}
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func TestNatAdd(t *testing.T) {
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tester = t
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test_msg = "NatAddA"
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nat_eq(0, add(nat_zero, nat_zero), nat_zero)
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nat_eq(1, add(nat_zero, c), c)
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test_msg = "NatAddB"
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for i := uint64(0); i < 100; i++ {
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t := Nat(i)
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nat_eq(uint(i), sum(i, c), t.Mul(t).Add(t).Shr(1).Mul(c))
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}
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}
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func mul(x, y Natural) Natural {
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z1 := x.Mul(y)
|
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z2 := y.Mul(x)
|
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if z1.Cmp(z2) != 0 {
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tester.Fatalf("multiplication not symmetric:\n\tx = %v\n\ty = %t", x, y)
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}
|
||||
if !x.IsZero() && z1.Div(x).Cmp(y) != 0 {
|
||||
tester.Fatalf("multiplication/division not inverse (A):\n\tx = %v\n\ty = %t", x, y)
|
||||
}
|
||||
if !y.IsZero() && z1.Div(y).Cmp(x) != 0 {
|
||||
tester.Fatalf("multiplication/division not inverse (B):\n\tx = %v\n\ty = %t", x, y)
|
||||
}
|
||||
return z1
|
||||
}
|
||||
|
||||
|
||||
func TestNatSub(t *testing.T) {
|
||||
tester = t
|
||||
test_msg = "NatSubA"
|
||||
nat_eq(0, nat_zero.Sub(nat_zero), nat_zero)
|
||||
nat_eq(1, c.Sub(nat_zero), c)
|
||||
|
||||
test_msg = "NatSubB"
|
||||
for i := uint64(0); i < 100; i++ {
|
||||
t := sum(i, c)
|
||||
for j := uint64(0); j <= i; j++ {
|
||||
t = t.Sub(mul(Nat(j), c))
|
||||
}
|
||||
nat_eq(uint(i), t, nat_zero)
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
func TestNatMul(t *testing.T) {
|
||||
tester = t
|
||||
test_msg = "NatMulA"
|
||||
nat_eq(0, mul(c, nat_zero), nat_zero)
|
||||
nat_eq(1, mul(c, nat_one), c)
|
||||
|
||||
test_msg = "NatMulB"
|
||||
nat_eq(0, b.Mul(MulRange(0, 100)), nat_zero)
|
||||
nat_eq(1, b.Mul(MulRange(21, 100)), c)
|
||||
|
||||
test_msg = "NatMulC"
|
||||
const n = 100
|
||||
p := b.Mul(c).Shl(n)
|
||||
for i := uint(0); i < n; i++ {
|
||||
nat_eq(i, mul(b.Shl(i), c.Shl(n-i)), p)
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
func TestNatDiv(t *testing.T) {
|
||||
tester = t
|
||||
test_msg = "NatDivA"
|
||||
nat_eq(0, c.Div(nat_one), c)
|
||||
nat_eq(1, c.Div(Nat(100)), Fact(99))
|
||||
nat_eq(2, b.Div(c), nat_zero)
|
||||
nat_eq(4, nat_one.Shl(100).Div(nat_one.Shl(90)), nat_one.Shl(10))
|
||||
nat_eq(5, c.Div(b), MulRange(21, 100))
|
||||
|
||||
test_msg = "NatDivB"
|
||||
const n = 100
|
||||
p := Fact(n)
|
||||
for i := uint(0); i < n; i++ {
|
||||
nat_eq(100+i, p.Div(MulRange(1, i)), MulRange(i+1, n))
|
||||
}
|
||||
|
||||
// a specific test case that exposed a bug in package big
|
||||
test_msg = "NatDivC"
|
||||
x := natFromString("69720375229712477164533808935312303556800", 10, nil)
|
||||
y := natFromString("3099044504245996706400", 10, nil)
|
||||
q := natFromString("22497377864108980962", 10, nil)
|
||||
r := natFromString("0", 10, nil)
|
||||
qc, rc := x.DivMod(y)
|
||||
nat_eq(0, q, qc)
|
||||
nat_eq(1, r, rc)
|
||||
}
|
||||
|
||||
|
||||
func TestIntQuoRem(t *testing.T) {
|
||||
tester = t
|
||||
test_msg = "IntQuoRem"
|
||||
type T struct {
|
||||
x, y, q, r int64
|
||||
}
|
||||
a := []T{
|
||||
T{+8, +3, +2, +2},
|
||||
T{+8, -3, -2, +2},
|
||||
T{-8, +3, -2, -2},
|
||||
T{-8, -3, +2, -2},
|
||||
T{+1, +2, 0, +1},
|
||||
T{+1, -2, 0, +1},
|
||||
T{-1, +2, 0, -1},
|
||||
T{-1, -2, 0, -1},
|
||||
}
|
||||
for i := uint(0); i < uint(len(a)); i++ {
|
||||
e := &a[i]
|
||||
x, y := Int(e.x).Mul(ip), Int(e.y).Mul(ip)
|
||||
q, r := Int(e.q), Int(e.r).Mul(ip)
|
||||
qq, rr := x.QuoRem(y)
|
||||
int_eq(4*i+0, x.Quo(y), q)
|
||||
int_eq(4*i+1, x.Rem(y), r)
|
||||
int_eq(4*i+2, qq, q)
|
||||
int_eq(4*i+3, rr, r)
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
func TestIntDivMod(t *testing.T) {
|
||||
tester = t
|
||||
test_msg = "IntDivMod"
|
||||
type T struct {
|
||||
x, y, q, r int64
|
||||
}
|
||||
a := []T{
|
||||
T{+8, +3, +2, +2},
|
||||
T{+8, -3, -2, +2},
|
||||
T{-8, +3, -3, +1},
|
||||
T{-8, -3, +3, +1},
|
||||
T{+1, +2, 0, +1},
|
||||
T{+1, -2, 0, +1},
|
||||
T{-1, +2, -1, +1},
|
||||
T{-1, -2, +1, +1},
|
||||
}
|
||||
for i := uint(0); i < uint(len(a)); i++ {
|
||||
e := &a[i]
|
||||
x, y := Int(e.x).Mul(ip), Int(e.y).Mul(ip)
|
||||
q, r := Int(e.q), Int(e.r).Mul(ip)
|
||||
qq, rr := x.DivMod(y)
|
||||
int_eq(4*i+0, x.Div(y), q)
|
||||
int_eq(4*i+1, x.Mod(y), r)
|
||||
int_eq(4*i+2, qq, q)
|
||||
int_eq(4*i+3, rr, r)
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
func TestNatMod(t *testing.T) {
|
||||
tester = t
|
||||
test_msg = "NatModA"
|
||||
for i := uint(0); ; i++ {
|
||||
d := nat_one.Shl(i)
|
||||
if d.Cmp(c) < 0 {
|
||||
nat_eq(i, c.Add(d).Mod(c), d)
|
||||
} else {
|
||||
nat_eq(i, c.Add(d).Div(c), nat_two)
|
||||
nat_eq(i, c.Add(d).Mod(c), d.Sub(c))
|
||||
break
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
func TestNatShift(t *testing.T) {
|
||||
tester = t
|
||||
test_msg = "NatShift1L"
|
||||
test(0, b.Shl(0).Cmp(b) == 0)
|
||||
test(1, c.Shl(1).Cmp(c) > 0)
|
||||
|
||||
test_msg = "NatShift1R"
|
||||
test(3, b.Shr(0).Cmp(b) == 0)
|
||||
test(4, c.Shr(1).Cmp(c) < 0)
|
||||
|
||||
test_msg = "NatShift2"
|
||||
for i := uint(0); i < 100; i++ {
|
||||
test(i, c.Shl(i).Shr(i).Cmp(c) == 0)
|
||||
}
|
||||
|
||||
test_msg = "NatShift3L"
|
||||
{
|
||||
const m = 3
|
||||
p := b
|
||||
f := Nat(1 << m)
|
||||
for i := uint(0); i < 100; i++ {
|
||||
nat_eq(i, b.Shl(i*m), p)
|
||||
p = mul(p, f)
|
||||
}
|
||||
}
|
||||
|
||||
test_msg = "NatShift3R"
|
||||
{
|
||||
p := c
|
||||
for i := uint(0); !p.IsZero(); i++ {
|
||||
nat_eq(i, c.Shr(i), p)
|
||||
p = p.Shr(1)
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
func TestIntShift(t *testing.T) {
|
||||
tester = t
|
||||
test_msg = "IntShift1L"
|
||||
test(0, ip.Shl(0).Cmp(ip) == 0)
|
||||
test(1, ip.Shl(1).Cmp(ip) > 0)
|
||||
|
||||
test_msg = "IntShift1R"
|
||||
test(0, ip.Shr(0).Cmp(ip) == 0)
|
||||
test(1, ip.Shr(1).Cmp(ip) < 0)
|
||||
|
||||
test_msg = "IntShift2"
|
||||
for i := uint(0); i < 100; i++ {
|
||||
test(i, ip.Shl(i).Shr(i).Cmp(ip) == 0)
|
||||
}
|
||||
|
||||
test_msg = "IntShift3L"
|
||||
{
|
||||
const m = 3
|
||||
p := ip
|
||||
f := Int(1 << m)
|
||||
for i := uint(0); i < 100; i++ {
|
||||
int_eq(i, ip.Shl(i*m), p)
|
||||
p = p.Mul(f)
|
||||
}
|
||||
}
|
||||
|
||||
test_msg = "IntShift3R"
|
||||
{
|
||||
p := ip
|
||||
for i := uint(0); p.IsPos(); i++ {
|
||||
int_eq(i, ip.Shr(i), p)
|
||||
p = p.Shr(1)
|
||||
}
|
||||
}
|
||||
|
||||
test_msg = "IntShift4R"
|
||||
int_eq(0, Int(-43).Shr(1), Int(-43>>1))
|
||||
int_eq(0, Int(-1024).Shr(100), Int(-1))
|
||||
int_eq(1, ip.Neg().Shr(10), ip.Neg().Div(Int(1).Shl(10)))
|
||||
}
|
||||
|
||||
|
||||
func TestNatBitOps(t *testing.T) {
|
||||
tester = t
|
||||
|
||||
x := uint64(0xf08e6f56bd8c3941)
|
||||
y := uint64(0x3984ef67834bc)
|
||||
|
||||
bx := Nat(x)
|
||||
by := Nat(y)
|
||||
|
||||
test_msg = "NatAnd"
|
||||
bz := Nat(x & y)
|
||||
for i := uint(0); i < 100; i++ {
|
||||
nat_eq(i, bx.Shl(i).And(by.Shl(i)), bz.Shl(i))
|
||||
}
|
||||
|
||||
test_msg = "NatAndNot"
|
||||
bz = Nat(x &^ y)
|
||||
for i := uint(0); i < 100; i++ {
|
||||
nat_eq(i, bx.Shl(i).AndNot(by.Shl(i)), bz.Shl(i))
|
||||
}
|
||||
|
||||
test_msg = "NatOr"
|
||||
bz = Nat(x | y)
|
||||
for i := uint(0); i < 100; i++ {
|
||||
nat_eq(i, bx.Shl(i).Or(by.Shl(i)), bz.Shl(i))
|
||||
}
|
||||
|
||||
test_msg = "NatXor"
|
||||
bz = Nat(x ^ y)
|
||||
for i := uint(0); i < 100; i++ {
|
||||
nat_eq(i, bx.Shl(i).Xor(by.Shl(i)), bz.Shl(i))
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
func TestIntBitOps1(t *testing.T) {
|
||||
tester = t
|
||||
test_msg = "IntBitOps1"
|
||||
type T struct {
|
||||
x, y int64
|
||||
}
|
||||
a := []T{
|
||||
T{+7, +3},
|
||||
T{+7, -3},
|
||||
T{-7, +3},
|
||||
T{-7, -3},
|
||||
}
|
||||
for i := uint(0); i < uint(len(a)); i++ {
|
||||
e := &a[i]
|
||||
int_eq(4*i+0, Int(e.x).And(Int(e.y)), Int(e.x&e.y))
|
||||
int_eq(4*i+1, Int(e.x).AndNot(Int(e.y)), Int(e.x&^e.y))
|
||||
int_eq(4*i+2, Int(e.x).Or(Int(e.y)), Int(e.x|e.y))
|
||||
int_eq(4*i+3, Int(e.x).Xor(Int(e.y)), Int(e.x^e.y))
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
func TestIntBitOps2(t *testing.T) {
|
||||
tester = t
|
||||
|
||||
test_msg = "IntNot"
|
||||
int_eq(0, Int(-2).Not(), Int(1))
|
||||
int_eq(0, Int(-1).Not(), Int(0))
|
||||
int_eq(0, Int(0).Not(), Int(-1))
|
||||
int_eq(0, Int(1).Not(), Int(-2))
|
||||
int_eq(0, Int(2).Not(), Int(-3))
|
||||
|
||||
test_msg = "IntAnd"
|
||||
for x := int64(-15); x < 5; x++ {
|
||||
bx := Int(x)
|
||||
for y := int64(-5); y < 15; y++ {
|
||||
by := Int(y)
|
||||
for i := uint(50); i < 70; i++ { // shift across 64bit boundary
|
||||
int_eq(i, bx.Shl(i).And(by.Shl(i)), Int(x&y).Shl(i))
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
test_msg = "IntAndNot"
|
||||
for x := int64(-15); x < 5; x++ {
|
||||
bx := Int(x)
|
||||
for y := int64(-5); y < 15; y++ {
|
||||
by := Int(y)
|
||||
for i := uint(50); i < 70; i++ { // shift across 64bit boundary
|
||||
int_eq(2*i+0, bx.Shl(i).AndNot(by.Shl(i)), Int(x&^y).Shl(i))
|
||||
int_eq(2*i+1, bx.Shl(i).And(by.Shl(i).Not()), Int(x&^y).Shl(i))
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
test_msg = "IntOr"
|
||||
for x := int64(-15); x < 5; x++ {
|
||||
bx := Int(x)
|
||||
for y := int64(-5); y < 15; y++ {
|
||||
by := Int(y)
|
||||
for i := uint(50); i < 70; i++ { // shift across 64bit boundary
|
||||
int_eq(i, bx.Shl(i).Or(by.Shl(i)), Int(x|y).Shl(i))
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
test_msg = "IntXor"
|
||||
for x := int64(-15); x < 5; x++ {
|
||||
bx := Int(x)
|
||||
for y := int64(-5); y < 15; y++ {
|
||||
by := Int(y)
|
||||
for i := uint(50); i < 70; i++ { // shift across 64bit boundary
|
||||
int_eq(i, bx.Shl(i).Xor(by.Shl(i)), Int(x^y).Shl(i))
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
func TestNatCmp(t *testing.T) {
|
||||
tester = t
|
||||
test_msg = "NatCmp"
|
||||
test(0, a.Cmp(a) == 0)
|
||||
test(1, a.Cmp(b) < 0)
|
||||
test(2, b.Cmp(a) > 0)
|
||||
test(3, a.Cmp(c) < 0)
|
||||
d := c.Add(b)
|
||||
test(4, c.Cmp(d) < 0)
|
||||
test(5, d.Cmp(c) > 0)
|
||||
}
|
||||
|
||||
|
||||
func TestNatLog2(t *testing.T) {
|
||||
tester = t
|
||||
test_msg = "NatLog2A"
|
||||
test(0, nat_one.Log2() == 0)
|
||||
test(1, nat_two.Log2() == 1)
|
||||
test(2, Nat(3).Log2() == 1)
|
||||
test(3, Nat(4).Log2() == 2)
|
||||
|
||||
test_msg = "NatLog2B"
|
||||
for i := uint(0); i < 100; i++ {
|
||||
test(i, nat_one.Shl(i).Log2() == i)
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
func TestNatGcd(t *testing.T) {
|
||||
tester = t
|
||||
test_msg = "NatGcdA"
|
||||
f := Nat(99991)
|
||||
nat_eq(0, b.Mul(f).Gcd(c.Mul(f)), MulRange(1, 20).Mul(f))
|
||||
}
|
||||
|
||||
|
||||
func TestNatPow(t *testing.T) {
|
||||
tester = t
|
||||
test_msg = "NatPowA"
|
||||
nat_eq(0, nat_two.Pow(0), nat_one)
|
||||
|
||||
test_msg = "NatPowB"
|
||||
for i := uint(0); i < 100; i++ {
|
||||
nat_eq(i, nat_two.Pow(i), nat_one.Shl(i))
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
func TestNatPop(t *testing.T) {
|
||||
tester = t
|
||||
test_msg = "NatPopA"
|
||||
test(0, nat_zero.Pop() == 0)
|
||||
test(1, nat_one.Pop() == 1)
|
||||
test(2, Nat(10).Pop() == 2)
|
||||
test(3, Nat(30).Pop() == 4)
|
||||
test(4, Nat(0x1248f).Shl(33).Pop() == 8)
|
||||
|
||||
test_msg = "NatPopB"
|
||||
for i := uint(0); i < 100; i++ {
|
||||
test(i, nat_one.Shl(i).Sub(nat_one).Pop() == i)
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
func TestIssue571(t *testing.T) {
|
||||
const min_float = "4.940656458412465441765687928682213723651e-324"
|
||||
RatFromString(min_float, 10) // this must not crash
|
||||
}
|
||||
@ -1,520 +0,0 @@
|
||||
// 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.
|
||||
|
||||
// Integer numbers
|
||||
//
|
||||
// Integers are normalized if the mantissa is normalized and the sign is
|
||||
// false for mant == 0. Use MakeInt to create normalized Integers.
|
||||
|
||||
package bignum
|
||||
|
||||
import (
|
||||
"fmt"
|
||||
)
|
||||
|
||||
// TODO(gri) Complete the set of in-place operations.
|
||||
|
||||
// Integer represents a signed integer value of arbitrary precision.
|
||||
//
|
||||
type Integer struct {
|
||||
sign bool
|
||||
mant Natural
|
||||
}
|
||||
|
||||
|
||||
// MakeInt makes an integer given a sign and a mantissa.
|
||||
// The number is positive (>= 0) if sign is false or the
|
||||
// mantissa is zero; it is negative otherwise.
|
||||
//
|
||||
func MakeInt(sign bool, mant Natural) *Integer {
|
||||
if mant.IsZero() {
|
||||
sign = false // normalize
|
||||
}
|
||||
return &Integer{sign, mant}
|
||||
}
|
||||
|
||||
|
||||
// Int creates a small integer with value x.
|
||||
//
|
||||
func Int(x int64) *Integer {
|
||||
var ux uint64
|
||||
if x < 0 {
|
||||
// For the most negative x, -x == x, and
|
||||
// the bit pattern has the correct value.
|
||||
ux = uint64(-x)
|
||||
} else {
|
||||
ux = uint64(x)
|
||||
}
|
||||
return MakeInt(x < 0, Nat(ux))
|
||||
}
|
||||
|
||||
|
||||
// Value returns the value of x, if x fits into an int64;
|
||||
// otherwise the result is undefined.
|
||||
//
|
||||
func (x *Integer) Value() int64 {
|
||||
z := int64(x.mant.Value())
|
||||
if x.sign {
|
||||
z = -z
|
||||
}
|
||||
return z
|
||||
}
|
||||
|
||||
|
||||
// Abs returns the absolute value of x.
|
||||
//
|
||||
func (x *Integer) Abs() Natural { return x.mant }
|
||||
|
||||
|
||||
// Predicates
|
||||
|
||||
// IsEven returns true iff x is divisible by 2.
|
||||
//
|
||||
func (x *Integer) IsEven() bool { return x.mant.IsEven() }
|
||||
|
||||
|
||||
// IsOdd returns true iff x is not divisible by 2.
|
||||
//
|
||||
func (x *Integer) IsOdd() bool { return x.mant.IsOdd() }
|
||||
|
||||
|
||||
// IsZero returns true iff x == 0.
|
||||
//
|
||||
func (x *Integer) IsZero() bool { return x.mant.IsZero() }
|
||||
|
||||
|
||||
// IsNeg returns true iff x < 0.
|
||||
//
|
||||
func (x *Integer) IsNeg() bool { return x.sign && !x.mant.IsZero() }
|
||||
|
||||
|
||||
// IsPos returns true iff x >= 0.
|
||||
//
|
||||
func (x *Integer) IsPos() bool { return !x.sign && !x.mant.IsZero() }
|
||||
|
||||
|
||||
// Operations
|
||||
|
||||
// Neg returns the negated value of x.
|
||||
//
|
||||
func (x *Integer) Neg() *Integer { return MakeInt(!x.sign, x.mant) }
|
||||
|
||||
|
||||
// Iadd sets z to the sum x + y.
|
||||
// z must exist and may be x or y.
|
||||
//
|
||||
func Iadd(z, x, y *Integer) {
|
||||
if x.sign == y.sign {
|
||||
// x + y == x + y
|
||||
// (-x) + (-y) == -(x + y)
|
||||
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.sign = x.sign
|
||||
Nsub(&z.mant, x.mant, y.mant)
|
||||
} else {
|
||||
z.sign = !x.sign
|
||||
Nsub(&z.mant, y.mant, x.mant)
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
// Add returns the sum x + y.
|
||||
//
|
||||
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.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.sign = x.sign
|
||||
Nsub(&z.mant, x.mant, y.mant)
|
||||
} else {
|
||||
z.sign = !x.sign
|
||||
Nsub(&z.mant, y.mant, x.mant)
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
// 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)
|
||||
}
|
||||
z.sign = z.sign != (d < 0)
|
||||
Nscale(&z.mant, f)
|
||||
}
|
||||
|
||||
|
||||
// Mul1 returns the product x * d.
|
||||
//
|
||||
func (x *Integer) Mul1(d int64) *Integer {
|
||||
f := uint64(d)
|
||||
if d < 0 {
|
||||
f = uint64(-d)
|
||||
}
|
||||
return MakeInt(x.sign != (d < 0), x.mant.Mul1(f))
|
||||
}
|
||||
|
||||
|
||||
// Mul returns the product x * y.
|
||||
//
|
||||
func (x *Integer) Mul(y *Integer) *Integer {
|
||||
// x * y == x * y
|
||||
// x * (-y) == -(x * y)
|
||||
// (-x) * y == -(x * y)
|
||||
// (-x) * (-y) == x * y
|
||||
return MakeInt(x.sign != y.sign, x.mant.Mul(y.mant))
|
||||
}
|
||||
|
||||
|
||||
// MulNat returns the product x * y, where y is a (unsigned) natural number.
|
||||
//
|
||||
func (x *Integer) MulNat(y Natural) *Integer {
|
||||
// x * y == x * y
|
||||
// (-x) * y == -(x * y)
|
||||
return MakeInt(x.sign, x.mant.Mul(y))
|
||||
}
|
||||
|
||||
|
||||
// Quo returns the quotient q = x / y for y != 0.
|
||||
// If y == 0, a division-by-zero run-time error occurs.
|
||||
//
|
||||
// Quo and Rem implement T-division and modulus (like C99):
|
||||
//
|
||||
// q = x.Quo(y) = trunc(x/y) (truncation towards zero)
|
||||
// r = x.Rem(y) = x - y*q
|
||||
//
|
||||
// (Daan Leijen, ``Division and Modulus for Computer Scientists''.)
|
||||
//
|
||||
func (x *Integer) Quo(y *Integer) *Integer {
|
||||
// x / y == x / y
|
||||
// x / (-y) == -(x / y)
|
||||
// (-x) / y == -(x / y)
|
||||
// (-x) / (-y) == x / y
|
||||
return MakeInt(x.sign != y.sign, x.mant.Div(y.mant))
|
||||
}
|
||||
|
||||
|
||||
// Rem returns the remainder r of the division x / y for y != 0,
|
||||
// with r = x - y*x.Quo(y). Unless r is zero, its sign corresponds
|
||||
// to the sign of x.
|
||||
// If y == 0, a division-by-zero run-time error occurs.
|
||||
//
|
||||
func (x *Integer) Rem(y *Integer) *Integer {
|
||||
// x % y == x % y
|
||||
// x % (-y) == x % y
|
||||
// (-x) % y == -(x % y)
|
||||
// (-x) % (-y) == -(x % y)
|
||||
return MakeInt(x.sign, x.mant.Mod(y.mant))
|
||||
}
|
||||
|
||||
|
||||
// QuoRem returns the pair (x.Quo(y), x.Rem(y)) for y != 0.
|
||||
// If y == 0, a division-by-zero run-time error occurs.
|
||||
//
|
||||
func (x *Integer) QuoRem(y *Integer) (*Integer, *Integer) {
|
||||
q, r := x.mant.DivMod(y.mant)
|
||||
return MakeInt(x.sign != y.sign, q), MakeInt(x.sign, r)
|
||||
}
|
||||
|
||||
|
||||
// Div returns the quotient q = x / y for y != 0.
|
||||
// If y == 0, a division-by-zero run-time error occurs.
|
||||
//
|
||||
// Div and Mod implement Euclidian division and modulus:
|
||||
//
|
||||
// q = x.Div(y)
|
||||
// r = x.Mod(y) with: 0 <= r < |q| and: x = y*q + r
|
||||
//
|
||||
// (Raymond T. Boute, ``The Euclidian definition of the functions
|
||||
// div and mod''. ACM Transactions on Programming Languages and
|
||||
// Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
|
||||
// ACM press.)
|
||||
//
|
||||
func (x *Integer) Div(y *Integer) *Integer {
|
||||
q, r := x.QuoRem(y)
|
||||
if r.IsNeg() {
|
||||
if y.IsPos() {
|
||||
q = q.Sub(Int(1))
|
||||
} else {
|
||||
q = q.Add(Int(1))
|
||||
}
|
||||
}
|
||||
return q
|
||||
}
|
||||
|
||||
|
||||
// Mod returns the modulus r of the division x / y for y != 0,
|
||||
// with r = x - y*x.Div(y). r is always positive.
|
||||
// If y == 0, a division-by-zero run-time error occurs.
|
||||
//
|
||||
func (x *Integer) Mod(y *Integer) *Integer {
|
||||
r := x.Rem(y)
|
||||
if r.IsNeg() {
|
||||
if y.IsPos() {
|
||||
r = r.Add(y)
|
||||
} else {
|
||||
r = r.Sub(y)
|
||||
}
|
||||
}
|
||||
return r
|
||||
}
|
||||
|
||||
|
||||
// DivMod returns the pair (x.Div(y), x.Mod(y)).
|
||||
//
|
||||
func (x *Integer) DivMod(y *Integer) (*Integer, *Integer) {
|
||||
q, r := x.QuoRem(y)
|
||||
if r.IsNeg() {
|
||||
if y.IsPos() {
|
||||
q = q.Sub(Int(1))
|
||||
r = r.Add(y)
|
||||
} else {
|
||||
q = q.Add(Int(1))
|
||||
r = r.Sub(y)
|
||||
}
|
||||
}
|
||||
return q, r
|
||||
}
|
||||
|
||||
|
||||
// Shl implements ``shift left'' x << s. It returns x * 2^s.
|
||||
//
|
||||
func (x *Integer) Shl(s uint) *Integer { return MakeInt(x.sign, x.mant.Shl(s)) }
|
||||
|
||||
|
||||
// The bitwise operations on integers are defined on the 2's-complement
|
||||
// representation of integers. From
|
||||
//
|
||||
// -x == ^x + 1 (1) 2's complement representation
|
||||
//
|
||||
// follows:
|
||||
//
|
||||
// -(x) == ^(x) + 1
|
||||
// -(-x) == ^(-x) + 1
|
||||
// x-1 == ^(-x)
|
||||
// ^(x-1) == -x (2)
|
||||
//
|
||||
// Using (1) and (2), operations on negative integers of the form -x are
|
||||
// converted to operations on negated positive integers of the form ~(x-1).
|
||||
|
||||
|
||||
// Shr implements ``shift right'' x >> s. It returns x / 2^s.
|
||||
//
|
||||
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(false, x.mant.Shr(s))
|
||||
}
|
||||
|
||||
|
||||
// Not returns the ``bitwise not'' ^x for the 2's-complement representation of x.
|
||||
func (x *Integer) Not() *Integer {
|
||||
if x.sign {
|
||||
// ^(-x) == ^(^(x-1)) == x-1
|
||||
return MakeInt(false, x.mant.Sub(Nat(1)))
|
||||
}
|
||||
|
||||
// ^x == -x-1 == -(x+1)
|
||||
return MakeInt(true, x.mant.Add(Nat(1)))
|
||||
}
|
||||
|
||||
|
||||
// And returns the ``bitwise and'' x & y for the 2's-complement representation of x and y.
|
||||
//
|
||||
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)))
|
||||
}
|
||||
|
||||
// x & y == x & y
|
||||
return MakeInt(false, x.mant.And(y.mant))
|
||||
}
|
||||
|
||||
// x.sign != y.sign
|
||||
if x.sign {
|
||||
x, y = y, x // & is symmetric
|
||||
}
|
||||
|
||||
// x & (-y) == x & ^(y-1) == x &^ (y-1)
|
||||
return MakeInt(false, x.mant.AndNot(y.mant.Sub(Nat(1))))
|
||||
}
|
||||
|
||||
|
||||
// AndNot returns the ``bitwise clear'' x &^ y for the 2's-complement representation of x and y.
|
||||
//
|
||||
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))))
|
||||
}
|
||||
|
||||
// x &^ y == x &^ y
|
||||
return MakeInt(false, x.mant.AndNot(y.mant))
|
||||
}
|
||||
|
||||
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)))
|
||||
}
|
||||
|
||||
// x &^ (-y) == x &^ ^(y-1) == x & (y-1)
|
||||
return MakeInt(false, x.mant.And(y.mant.Sub(Nat(1))))
|
||||
}
|
||||
|
||||
|
||||
// Or returns the ``bitwise or'' x | y for the 2's-complement representation of x and y.
|
||||
//
|
||||
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)))
|
||||
}
|
||||
|
||||
// x | y == x | y
|
||||
return MakeInt(false, x.mant.Or(y.mant))
|
||||
}
|
||||
|
||||
// x.sign != y.sign
|
||||
if x.sign {
|
||||
x, y = y, x // | or symmetric
|
||||
}
|
||||
|
||||
// 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)))
|
||||
}
|
||||
|
||||
|
||||
// Xor returns the ``bitwise xor'' x | y for the 2's-complement representation of x and y.
|
||||
//
|
||||
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))))
|
||||
}
|
||||
|
||||
// x ^ y == x ^ y
|
||||
return MakeInt(false, x.mant.Xor(y.mant))
|
||||
}
|
||||
|
||||
// x.sign != y.sign
|
||||
if x.sign {
|
||||
x, y = y, x // ^ is symmetric
|
||||
}
|
||||
|
||||
// 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)))
|
||||
}
|
||||
|
||||
|
||||
// Cmp compares x and y. The result is an int value that is
|
||||
//
|
||||
// < 0 if x < y
|
||||
// == 0 if x == y
|
||||
// > 0 if x > y
|
||||
//
|
||||
func (x *Integer) Cmp(y *Integer) int {
|
||||
// x cmp y == x cmp y
|
||||
// x cmp (-y) == x
|
||||
// (-x) cmp y == y
|
||||
// (-x) cmp (-y) == -(x cmp y)
|
||||
var r int
|
||||
switch {
|
||||
case x.sign == y.sign:
|
||||
r = x.mant.Cmp(y.mant)
|
||||
if x.sign {
|
||||
r = -r
|
||||
}
|
||||
case x.sign:
|
||||
r = -1
|
||||
case y.sign:
|
||||
r = 1
|
||||
}
|
||||
return r
|
||||
}
|
||||
|
||||
|
||||
// ToString converts x to a string for a given base, with 2 <= base <= 16.
|
||||
//
|
||||
func (x *Integer) ToString(base uint) string {
|
||||
if x.mant.IsZero() {
|
||||
return "0"
|
||||
}
|
||||
var s string
|
||||
if x.sign {
|
||||
s = "-"
|
||||
}
|
||||
return s + x.mant.ToString(base)
|
||||
}
|
||||
|
||||
|
||||
// String converts x to its decimal string representation.
|
||||
// x.String() is the same as x.ToString(10).
|
||||
//
|
||||
func (x *Integer) String() string { return x.ToString(10) }
|
||||
|
||||
|
||||
// Format is a support routine for fmt.Formatter. It accepts
|
||||
// the formats 'b' (binary), 'o' (octal), and 'x' (hexadecimal).
|
||||
//
|
||||
func (x *Integer) Format(h fmt.State, c int) { fmt.Fprintf(h, "%s", x.ToString(fmtbase(c))) }
|
||||
|
||||
|
||||
// IntFromString returns the integer corresponding to the
|
||||
// longest possible prefix of s representing an integer in a
|
||||
// given conversion base, the actual conversion base used, and
|
||||
// the prefix length. The syntax of integers follows the syntax
|
||||
// of signed integer literals in Go.
|
||||
//
|
||||
// If the base argument is 0, the string prefix determines the actual
|
||||
// conversion base. A prefix of ``0x'' or ``0X'' selects base 16; the
|
||||
// ``0'' prefix selects base 8. Otherwise the selected base is 10.
|
||||
//
|
||||
func IntFromString(s string, base uint) (*Integer, uint, int) {
|
||||
// skip sign, if any
|
||||
i0 := 0
|
||||
if len(s) > 0 && (s[0] == '-' || s[0] == '+') {
|
||||
i0 = 1
|
||||
}
|
||||
|
||||
mant, base, slen := NatFromString(s[i0:], base)
|
||||
|
||||
return MakeInt(i0 > 0 && s[0] == '-', mant), base, i0 + slen
|
||||
}
|
||||
@ -1,188 +0,0 @@
|
||||
// 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 implements Newton-Raphson division and uses
|
||||
// it as an additional test case for bignum.
|
||||
//
|
||||
// Division of x/y is achieved by computing r = 1/y to
|
||||
// obtain the quotient q = x*r = x*(1/y) = x/y. The
|
||||
// reciprocal r is the solution for f(x) = 1/x - y and
|
||||
// the solution is approximated through iteration. The
|
||||
// iteration does not require division.
|
||||
|
||||
package bignum
|
||||
|
||||
import "testing"
|
||||
|
||||
|
||||
// An fpNat is a Natural scaled by a power of two
|
||||
// (an unsigned floating point representation). The
|
||||
// value of an fpNat x is x.m * 2^x.e .
|
||||
//
|
||||
type fpNat struct {
|
||||
m Natural
|
||||
e int
|
||||
}
|
||||
|
||||
|
||||
// sub computes x - y.
|
||||
func (x fpNat) sub(y fpNat) fpNat {
|
||||
switch d := x.e - y.e; {
|
||||
case d < 0:
|
||||
return fpNat{x.m.Sub(y.m.Shl(uint(-d))), x.e}
|
||||
case d > 0:
|
||||
return fpNat{x.m.Shl(uint(d)).Sub(y.m), y.e}
|
||||
}
|
||||
return fpNat{x.m.Sub(y.m), x.e}
|
||||
}
|
||||
|
||||
|
||||
// mul2 computes x*2.
|
||||
func (x fpNat) mul2() fpNat { return fpNat{x.m, x.e + 1} }
|
||||
|
||||
|
||||
// mul computes x*y.
|
||||
func (x fpNat) mul(y fpNat) fpNat { return fpNat{x.m.Mul(y.m), x.e + y.e} }
|
||||
|
||||
|
||||
// mant computes the (possibly truncated) Natural representation
|
||||
// of an fpNat x.
|
||||
//
|
||||
func (x fpNat) mant() Natural {
|
||||
switch {
|
||||
case x.e > 0:
|
||||
return x.m.Shl(uint(x.e))
|
||||
case x.e < 0:
|
||||
return x.m.Shr(uint(-x.e))
|
||||
}
|
||||
return x.m
|
||||
}
|
||||
|
||||
|
||||
// nrDivEst computes an estimate of the quotient q = x0/y0 and returns q.
|
||||
// q may be too small (usually by 1).
|
||||
//
|
||||
func nrDivEst(x0, y0 Natural) Natural {
|
||||
if y0.IsZero() {
|
||||
panic("division by zero")
|
||||
return nil
|
||||
}
|
||||
// y0 > 0
|
||||
|
||||
if y0.Cmp(Nat(1)) == 0 {
|
||||
return x0
|
||||
}
|
||||
// y0 > 1
|
||||
|
||||
switch d := x0.Cmp(y0); {
|
||||
case d < 0:
|
||||
return Nat(0)
|
||||
case d == 0:
|
||||
return Nat(1)
|
||||
}
|
||||
// x0 > y0 > 1
|
||||
|
||||
// Determine maximum result length.
|
||||
maxLen := int(x0.Log2() - y0.Log2() + 1)
|
||||
|
||||
// In the following, each number x is represented
|
||||
// as a mantissa x.m and an exponent x.e such that
|
||||
// x = xm * 2^x.e.
|
||||
x := fpNat{x0, 0}
|
||||
y := fpNat{y0, 0}
|
||||
|
||||
// Determine a scale factor f = 2^e such that
|
||||
// 0.5 <= y/f == y*(2^-e) < 1.0
|
||||
// and scale y accordingly.
|
||||
e := int(y.m.Log2()) + 1
|
||||
y.e -= e
|
||||
|
||||
// t1
|
||||
var c = 2.9142
|
||||
const n = 14
|
||||
t1 := fpNat{Nat(uint64(c * (1 << n))), -n}
|
||||
|
||||
// Compute initial value r0 for the reciprocal of y/f.
|
||||
// r0 = t1 - 2*y
|
||||
r := t1.sub(y.mul2())
|
||||
two := fpNat{Nat(2), 0}
|
||||
|
||||
// Newton-Raphson iteration
|
||||
p := Nat(0)
|
||||
for i := 0; ; i++ {
|
||||
// check if we are done
|
||||
// TODO: Need to come up with a better test here
|
||||
// as it will reduce computation time significantly.
|
||||
// q = x*r/f
|
||||
q := x.mul(r)
|
||||
q.e -= e
|
||||
res := q.mant()
|
||||
if res.Cmp(p) == 0 {
|
||||
return res
|
||||
}
|
||||
p = res
|
||||
|
||||
// r' = r*(2 - y*r)
|
||||
r = r.mul(two.sub(y.mul(r)))
|
||||
|
||||
// reduce mantissa size
|
||||
// TODO: Find smaller bound as it will reduce
|
||||
// computation time massively.
|
||||
d := int(r.m.Log2()+1) - maxLen
|
||||
if d > 0 {
|
||||
r = fpNat{r.m.Shr(uint(d)), r.e + d}
|
||||
}
|
||||
}
|
||||
|
||||
panic("unreachable")
|
||||
return nil
|
||||
}
|
||||
|
||||
|
||||
func nrdiv(x, y Natural) (q, r Natural) {
|
||||
q = nrDivEst(x, y)
|
||||
r = x.Sub(y.Mul(q))
|
||||
// if r is too large, correct q and r
|
||||
// (usually one iteration)
|
||||
for r.Cmp(y) >= 0 {
|
||||
q = q.Add(Nat(1))
|
||||
r = r.Sub(y)
|
||||
}
|
||||
return
|
||||
}
|
||||
|
||||
|
||||
func div(t *testing.T, x, y Natural) {
|
||||
q, r := nrdiv(x, y)
|
||||
qx, rx := x.DivMod(y)
|
||||
if q.Cmp(qx) != 0 {
|
||||
t.Errorf("x = %s, y = %s, got q = %s, want q = %s", x, y, q, qx)
|
||||
}
|
||||
if r.Cmp(rx) != 0 {
|
||||
t.Errorf("x = %s, y = %s, got r = %s, want r = %s", x, y, r, rx)
|
||||
}
|
||||
}
|
||||
|
||||
|
||||
func idiv(t *testing.T, x0, y0 uint64) { div(t, Nat(x0), Nat(y0)) }
|
||||
|
||||
|
||||
func TestNRDiv(t *testing.T) {
|
||||
idiv(t, 17, 18)
|
||||
idiv(t, 17, 17)
|
||||
idiv(t, 17, 1)
|
||||
idiv(t, 17, 16)
|
||||
idiv(t, 17, 10)
|
||||
idiv(t, 17, 9)
|
||||
idiv(t, 17, 8)
|
||||
idiv(t, 17, 5)
|
||||
idiv(t, 17, 3)
|
||||
idiv(t, 1025, 512)
|
||||
idiv(t, 7489595, 2)
|
||||
idiv(t, 5404679459, 78495)
|
||||
idiv(t, 7484890589595, 7484890589594)
|
||||
div(t, Fact(100), Fact(91))
|
||||
div(t, Fact(1000), Fact(991))
|
||||
//div(t, Fact(10000), Fact(9991)); // takes too long - disabled for now
|
||||
}
|
||||
@ -1,205 +0,0 @@
|
||||
// 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.
|
||||
|
||||
// Rational numbers
|
||||
|
||||
package bignum
|
||||
|
||||
import "fmt"
|
||||
|
||||
|
||||
// Rational represents a quotient a/b of arbitrary precision.
|
||||
//
|
||||
type Rational struct {
|
||||
a *Integer // numerator
|
||||
b Natural // denominator
|
||||
}
|
||||
|
||||
|
||||
// MakeRat makes a rational number given a numerator a and a denominator b.
|
||||
//
|
||||
func MakeRat(a *Integer, b Natural) *Rational {
|
||||
f := a.mant.Gcd(b) // f > 0
|
||||
if f.Cmp(Nat(1)) != 0 {
|
||||
a = MakeInt(a.sign, a.mant.Div(f))
|
||||
b = b.Div(f)
|
||||
}
|
||||
return &Rational{a, b}
|
||||
}
|
||||
|
||||
|
||||
// Rat creates a small rational number with value a0/b0.
|
||||
//
|
||||
func Rat(a0 int64, b0 int64) *Rational {
|
||||
a, b := Int(a0), Int(b0)
|
||||
if b.sign {
|
||||
a = a.Neg()
|
||||
}
|
||||
return MakeRat(a, b.mant)
|
||||
}
|
||||
|
||||
|
||||
// Value returns the numerator and denominator of x.
|
||||
//
|
||||
func (x *Rational) Value() (numerator *Integer, denominator Natural) {
|
||||
return x.a, x.b
|
||||
}
|
||||
|
||||
|
||||
// Predicates
|
||||
|
||||
// IsZero returns true iff x == 0.
|
||||
//
|
||||
func (x *Rational) IsZero() bool { return x.a.IsZero() }
|
||||
|
||||
|
||||
// IsNeg returns true iff x < 0.
|
||||
//
|
||||
func (x *Rational) IsNeg() bool { return x.a.IsNeg() }
|
||||
|
||||
|
||||
// IsPos returns true iff x > 0.
|
||||
//
|
||||
func (x *Rational) IsPos() bool { return x.a.IsPos() }
|
||||
|
||||
|
||||
// IsInt returns true iff x can be written with a denominator 1
|
||||
// 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 }
|
||||
|
||||
|
||||
// Operations
|
||||
|
||||
// Neg returns the negated value of x.
|
||||
//
|
||||
func (x *Rational) Neg() *Rational { return MakeRat(x.a.Neg(), x.b) }
|
||||
|
||||
|
||||
// Add returns the sum x + y.
|
||||
//
|
||||
func (x *Rational) Add(y *Rational) *Rational {
|
||||
return MakeRat((x.a.MulNat(y.b)).Add(y.a.MulNat(x.b)), x.b.Mul(y.b))
|
||||
}
|
||||
|
||||
|
||||
// Sub returns the difference x - y.
|
||||
//
|
||||
func (x *Rational) Sub(y *Rational) *Rational {
|
||||
return MakeRat((x.a.MulNat(y.b)).Sub(y.a.MulNat(x.b)), x.b.Mul(y.b))
|
||||
}
|
||||
|
||||
|
||||
// Mul returns the product x * y.
|
||||
//
|
||||
func (x *Rational) Mul(y *Rational) *Rational { return MakeRat(x.a.Mul(y.a), x.b.Mul(y.b)) }
|
||||
|
||||
|
||||
// Quo returns the quotient x / y for y != 0.
|
||||
// If y == 0, a division-by-zero run-time error occurs.
|
||||
//
|
||||
func (x *Rational) Quo(y *Rational) *Rational {
|
||||
a := x.a.MulNat(y.b)
|
||||
b := y.a.MulNat(x.b)
|
||||
if b.IsNeg() {
|
||||
a = a.Neg()
|
||||
}
|
||||
return MakeRat(a, b.mant)
|
||||
}
|
||||
|
||||
|
||||
// Cmp compares x and y. The result is an int value
|
||||
//
|
||||
// < 0 if x < y
|
||||
// == 0 if x == y
|
||||
// > 0 if x > y
|
||||
//
|
||||
func (x *Rational) Cmp(y *Rational) int { return (x.a.MulNat(y.b)).Cmp(y.a.MulNat(x.b)) }
|
||||
|
||||
|
||||
// ToString converts x to a string for a given base, with 2 <= base <= 16.
|
||||
// The string representation is of the form "n" if x is an integer; otherwise
|
||||
// it is of form "n/d".
|
||||
//
|
||||
func (x *Rational) ToString(base uint) string {
|
||||
s := x.a.ToString(base)
|
||||
if !x.IsInt() {
|
||||
s += "/" + x.b.ToString(base)
|
||||
}
|
||||
return s
|
||||
}
|
||||
|
||||
|
||||
// String converts x to its decimal string representation.
|
||||
// x.String() is the same as x.ToString(10).
|
||||
//
|
||||
func (x *Rational) String() string { return x.ToString(10) }
|
||||
|
||||
|
||||
// Format is a support routine for fmt.Formatter. It accepts
|
||||
// the formats 'b' (binary), 'o' (octal), and 'x' (hexadecimal).
|
||||
//
|
||||
func (x *Rational) Format(h fmt.State, c int) { fmt.Fprintf(h, "%s", x.ToString(fmtbase(c))) }
|
||||
|
||||
|
||||
// RatFromString returns the rational number corresponding to the
|
||||
// longest possible prefix of s representing a rational number in a
|
||||
// given conversion base, the actual conversion base used, and the
|
||||
// prefix length. The syntax of a rational number is:
|
||||
//
|
||||
// rational = mantissa [exponent] .
|
||||
// mantissa = integer ('/' natural | '.' natural) .
|
||||
// exponent = ('e'|'E') integer .
|
||||
//
|
||||
// If the base argument is 0, the string prefix determines the actual
|
||||
// conversion base for the mantissa. A prefix of ``0x'' or ``0X'' selects
|
||||
// base 16; the ``0'' prefix selects base 8. Otherwise the selected base is 10.
|
||||
// If the mantissa is represented via a division, both the numerator and
|
||||
// denominator may have different base prefixes; in that case the base of
|
||||
// of the numerator is returned. If the mantissa contains a decimal point,
|
||||
// the base for the fractional part is the same as for the part before the
|
||||
// decimal point and the fractional part does not accept a base prefix.
|
||||
// The base for the exponent is always 10.
|
||||
//
|
||||
func RatFromString(s string, base uint) (*Rational, uint, int) {
|
||||
// read numerator
|
||||
a, abase, alen := IntFromString(s, base)
|
||||
b := Nat(1)
|
||||
|
||||
// read denominator or fraction, if any
|
||||
var blen int
|
||||
if alen < len(s) {
|
||||
ch := s[alen]
|
||||
if ch == '/' {
|
||||
alen++
|
||||
b, base, blen = NatFromString(s[alen:], base)
|
||||
} else if ch == '.' {
|
||||
alen++
|
||||
b, base, blen = NatFromString(s[alen:], abase)
|
||||
assert(base == abase)
|
||||
f := Nat(uint64(base)).Pow(uint(blen))
|
||||
a = MakeInt(a.sign, a.mant.Mul(f).Add(b))
|
||||
b = f
|
||||
}
|
||||
}
|
||||
|
||||
// read exponent, if any
|
||||
rlen := alen + blen
|
||||
if rlen < len(s) {
|
||||
ch := s[rlen]
|
||||
if ch == 'e' || ch == 'E' {
|
||||
rlen++
|
||||
e, _, elen := IntFromString(s[rlen:], 10)
|
||||
rlen += elen
|
||||
m := Nat(10).Pow(uint(e.mant.Value()))
|
||||
if e.sign {
|
||||
b = b.Mul(m)
|
||||
} else {
|
||||
a = a.MulNat(m)
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
return MakeRat(a, b), base, rlen
|
||||
}
|
||||
Loading…
Reference in New Issue
Block a user