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Commit 496a9353 authored by Robert Griesemer's avatar Robert Griesemer
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bignum: delete package - functionality subsumed by package big 

R=rsc
CC=golang-dev
https://golang.org/cl/1742045
parent b2a919fc
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src/pkg/Makefile
View file @ 496a9353
......@@ -63,7 +63,6 @@ DIRS=\
encoding/hex\
encoding/pem\
exec\
exp/bignum\
exp/datafmt\
exp/draw\
exp/draw/x11\
......
src/pkg/exp/bignum/Makefile deleted 100644 → 0
View file @ b2a919fc
# 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.
include ../../../Make.$(GOARCH)
TARG=exp/bignum
GOFILES=\
arith.go\
bignum.go\
integer.go\
rational.go\
include ../../../Make.pkg
src/pkg/exp/bignum/arith.go deleted 100644 → 0
View file @ b2a919fc
// 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/exp/bignum/arith_amd64.s deleted 100644 → 0
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// 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 ·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 ·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 ·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/exp/bignum/bignum.go deleted 100644 → 0
View file @ b2a919fc
// 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.
// A package for arbitrary precision arithmethic.
// It implements the following numeric types:
//
// - Natural unsigned integers
// - Integer signed integers
// - Rational rational numbers
//
// This package has been designed for ease of use but the functions it provides
// are likely to be quite slow. It may be deprecated eventually. Use package
// big instead, if possible.
//
package bignum
import (
"fmt"
)
// TODO(gri) Complete the set of in-place operations.
// ----------------------------------------------------------------------------
// Internal representation
//
// A natural number of the form
//
// x = x[n-1]*B^(n-1) + x[n-2]*B^(n-2) + ... + x[1]*B + x[0]
//
// with 0 <= x[i] < B and 0 <= i < n is stored in a slice of length n,
// with the digits x[i] as the slice elements.
//
// A natural number is normalized if the slice contains no leading 0 digits.
// During arithmetic operations, denormalized values may occur but are
// always normalized before returning the final result. The normalized
// representation of 0 is the empty slice (length = 0).
//
// The operations for all other numeric types are implemented on top of
// the operations for natural numbers.
//
// The base B is chosen as large as possible on a given platform but there
// are a few constraints besides the size of the largest unsigned integer
// type available:
//
// 1) To improve conversion speed between strings and numbers, the base B
// is chosen such that division and multiplication by 10 (for decimal
// string representation) can be done without using extended-precision
// arithmetic. This makes addition, subtraction, and conversion routines
// twice as fast. It requires a ``buffer'' of 4 bits per operand digit.
// That is, the size of B must be 4 bits smaller then the size of the
// type (digit) in which these operations are performed. Having this
// buffer also allows for trivial (single-bit) carry computation in
// addition and subtraction (optimization suggested by Ken Thompson).
//
// 2) Long division requires extended-precision (2-digit) division per digit.
// Instead of sacrificing the largest base type for all other operations,
// for division the operands are unpacked into ``half-digits'', and the
// results are packed again. For faster unpacking/packing, the base size
// in bits must be even.
type (
digit uint64
digit2 uint32 // half-digits for division
)
const (
logW = 64 // word width
logH = 4 // bits for a hex digit (= small number)
logB = logW - logH // largest bit-width available
// half-digits
_W2 = logB / 2 // width
_B2 = 1 << _W2 // base
_M2 = _B2 - 1 // mask
// full digits
_W = _W2 * 2 // width
_B = 1 << _W // base
_M = _B - 1 // mask
)
// ----------------------------------------------------------------------------
// Support functions
func assert(p bool) {
if !p {
panic("assert failed")
}
}
func isSmall(x digit) bool { return x < 1<<logH }
// For debugging. Keep around.
/*
func dump(x Natural) {
print("[", len(x), "]");
for i := len(x) - 1; i >= 0; i-- {
print(" ", x[i]);
}
println();
}
*/
// ----------------------------------------------------------------------------
// Natural numbers
// Natural represents an unsigned integer value of arbitrary precision.
//
type Natural []digit
// Nat creates a small natural number with value x.
//
func Nat(x uint64) Natural {
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)}
}
// compute number of digits required to represent x
// (this is usually 1 or 2, but the algorithm works
// for any base)
n := 0
for t := x; t > 0; t >>= _W {
n++
}
// split x into digits
z := make(Natural, n)
for i := 0; i < n; i++ {
z[i] = digit(x & _M)
x >>= _W
}
return z
}
// Value returns the lowest 64bits of x.
//
func (x Natural) Value() uint64 {
// single-digit values
n := len(x)
switch n {
case 0:
return 0
case 1:
return uint64(x[0])
}
// multi-digit values
// (this is usually 1 or 2, but the algorithm works
// for any base)
z := uint64(0)
s := uint(0)
for i := 0; i < n && s < 64; i++ {
z += uint64(x[i]) << s
s += _W
}
return z
}
// Predicates
// IsEven returns true iff x is divisible by 2.
//
func (x Natural) IsEven() bool { return len(x) == 0 || x[0]&1 == 0 }
// IsOdd returns true iff x is not divisible by 2.
//
func (x Natural) IsOdd() bool { return len(x) > 0 && x[0]&1 != 0 }
// IsZero returns true iff x == 0.
//
func (x Natural) IsZero() bool { return len(x) == 0 }
// Operations
//
// Naming conventions
//
// c carry
// x, y operands
// z result
// n, m len(x), len(y)
func normalize(x Natural) Natural {
n := len(x)
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 make(Natural, n, size)
}
// Nadd sets *zp to the sum x + y.
// *zp may be x or y.
//
func Nadd(zp *Natural, x, y Natural) {
n := len(x)
m := len(y)
if n < m {
Nadd(zp, y, x)
return
}
z := nalloc(*zp, n+1)
c := digit(0)
i := 0
for i < m {
t := c + x[i] + y[i]
c, z[i] = t>>_W, t&_M
i++
}
for i < n {
t := c + x[i]
c, z[i] = t>>_W, t&_M
i++
}
if c != 0 {
z[i] = c
i++
}
*zp = 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
}
// 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 Nsub(zp *Natural, x, y Natural) {
n := len(x)
m := len(y)
if n < m {
panic("underflow")
}
z := nalloc(*zp, n)
c := digit(0)
i := 0
for i < m {
t := c + x[i] - y[i]
c, z[i] = digit(int64(t)>>_W), t&_M // requires arithmetic shift!
i++
}
for i < n {
t := c + x[i]
c, z[i] = digit(int64(t)>>_W), t&_M // requires arithmetic shift!
i++
}
if int64(c) < 0 {
panic("underflow")
}
*zp = normalize(z)
}
// 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 z1 = (x*y + c) div B, z0 = (x*y + c) mod B.
//
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)
}
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
}
// 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
}
c := mul1(*z, *z, digit(d))
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
}
}
// 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)
}
// Mul1 returns the product x * d.
//
func (x Natural) Mul1(d uint64) Natural {
switch {
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))
}
z := make(Natural, len(x)+1)
c := mul1(z, x, digit(d))
z[len(x)] = c
return normalize(z)
}
// Mul returns the product x * y.
//
func (x Natural) Mul(y Natural) Natural {
n := len(x)
m := len(y)
if n < m {
return y.Mul(x)
}
if m == 0 {
return Nat(0)
}
if m == 1 && y[0] < _B {
return x.Mul1(uint64(y[0]))
}
z := make(Natural, n+m)
for j := 0; j < m; j++ {
d := y[j]
if d != 0 {
c := digit(0)
for i := 0; i < n; i++ {
c, z[i+j] = muladd11(x[i], d, z[i+j]+c)
}
z[n+j] = c
}
}
return normalize(z)
}
// DivMod needs multi-precision division, which is not available if digit
// is already using the largest uint size. Instead, unpack each operand
// into operands with twice as many digits of half the size (digit2), do
// DivMod, and then pack the results again.
func unpack(x Natural) []digit2 {
n := len(x)
z := make([]digit2, n*2+1) // add space for extra digit (used by DivMod)
for i := 0; i < n; i++ {
t := x[i]
z[i*2] = digit2(t & _M2)
z[i*2+1] = digit2(t >> _W2 & _M2)
}
// normalize result
k := 2 * n
for k > 0 && z[k-1] == 0 {
k--
}
return z[0:k] // trim leading 0's
}
func pack(x []digit2) Natural {
n := (len(x) + 1) / 2
z := make(Natural, n)
if len(x)&1 == 1 {
// handle odd len(x)
n--
z[n] = digit(x[n*2])
}
for i := 0; i < n; i++ {
z[i] = digit(x[i*2+1])<<_W2 | digit(x[i*2])
}
return normalize(z)
}
func mul21(z, x []digit2, y digit2) digit2 {
c := digit(0)
f := digit(y)
for i := 0; i < len(x); i++ {
t := c + digit(x[i])*f
c, z[i] = t>>_W2, digit2(t&_M2)
}
return digit2(c)
}
func div21(z, x []digit2, y digit2) digit2 {
c := digit(0)
d := digit(y)
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)
}
// divmod returns q and r with x = y*q + r and 0 <= r < y.
// x and y are destroyed in the process.
//
// The algorithm used here is based on 1). 2) describes the same algorithm
// in C. A discussion and summary of the relevant theorems can be found in
// 3). 3) also describes an easier way to obtain the trial digit - however
// it relies on tripple-precision arithmetic which is why Knuth's method is
// used here.
//
// 1) D. Knuth, The Art of Computer Programming. Volume 2. Seminumerical
// Algorithms. Addison-Wesley, Reading, 1969.
// (Algorithm D, Sec. 4.3.1)
//
// 2) Henry S. Warren, Jr., Hacker's Delight. Addison-Wesley, 2003.
// (9-2 Multiword Division, p.140ff)
//
// 3) P. Brinch Hansen, ``Multiple-length division revisited: A tour of the
// minefield''. Software - Practice and Experience 24, (June 1994),
// 579-601. John Wiley & Sons, Ltd.
func divmod(x, y []digit2) ([]digit2, []digit2) {
n := len(x)
m := len(y)
if m == 0 {
panic("division by zero")
}
assert(n+1 <= cap(x)) // space for one extra digit
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] = div21(x[1:n+1], x[0:n], y[0])
} else if m > n {
// y > x => quotient = 0, remainder = x
// TODO in this case we shouldn't even unpack x and y
m = n
} else {
// general case
assert(2 <= m && m <= n)
// normalize x and y
// TODO Instead of multiplying, it would be sufficient to
// shift y such that the normalization condition is
// satisfied (as done in Hacker's Delight).
f := _B2 / (digit(y[m-1]) + 1)
if f != 1 {
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])
for i := n - m; i >= 0; i-- {
k := i + m
// compute trial digit (Knuth)
var q digit
{
x0, x1, x2 := digit(x[k]), digit(x[k-1]), digit(x[k-2])
if x0 != y1 {
q = (x0<<_W2 + x1) / y1
} else {
q = _B2 - 1
}
for y2*q > (x0<<_W2+x1-y1*q)<<_W2+x2 {
q--
}
}
// subtract y*q
c := digit(0)
for j := 0; j < m; j++ {
t := c + digit(x[i+j]) - digit(y[j])*q
c, x[i+j] = digit(int64(t)>>_W2), digit2(t&_M2) // requires arithmetic shift!
}
x[k] = digit2((c + digit(x[k])) & _M2)
// correct if trial digit was too large
if x[k] != 0 {
// add y
c := digit(0)
for j := 0; j < m; j++ {
t := c + digit(x[i+j]) + digit(y[j])
c, x[i+j] = t>>_W2, digit2(t&_M2)
}
x[k] = digit2((c + digit(x[k])) & _M2)
assert(x[k] == 0)
// correct trial digit
q--
}
x[k] = digit2(q)
}
// undo normalization for remainder
if f != 1 {
c := div21(x[0:m], x[0:m], digit2(f))
assert(c == 0)
}
}
return x[m : n+1], x[0:m]
}
// Div returns the quotient q = x / y for y > 0,
// with x = y*q + r and 0 <= r < y.
// If y == 0, a division-by-zero run-time error occurs.
//
func (x Natural) Div(y Natural) Natural {
q, _ := divmod(unpack(x), unpack(y))
return pack(q)
}
// Mod returns the modulus r of the division x / y for y > 0,
// with x = y*q + r and 0 <= r < y.
// If y == 0, a division-by-zero run-time error occurs.
//
func (x Natural) Mod(y Natural) Natural {
_, r := divmod(unpack(x), unpack(y))
return pack(r)
}
// DivMod returns the pair (x.Div(y), x.Mod(y)) for y > 0.
// If y == 0, a division-by-zero run-time error occurs.
//
func (x Natural) DivMod(y Natural) (Natural, Natural) {
q, r := divmod(unpack(x), unpack(y))
return pack(q), pack(r)
}
func shl(z, x Natural, s uint) digit {
assert(s <= _W)
n := len(x)
c := digit(0)
for i := 0; i < n; i++ {
c, z[i] = x[i]>>(_W-s), x[i]<<s&_M|c
}
return c
}
// Shl implements ``shift left'' x << s. It returns x * 2^s.
//
func (x Natural) Shl(s uint) Natural {
n := uint(len(x))
m := n + s/_W
z := make(Natural, m+1)
z[m] = shl(z[m-n:m], x, s%_W)
return normalize(z)
}
func shr(z, x Natural, s uint) digit {
assert(s <= _W)
n := len(x)
c := digit(0)
for i := n - 1; i >= 0; i-- {
c, z[i] = x[i]<<(_W-s)&_M, x[i]>>s|c
}
return c
}
// Shr implements ``shift right'' x >> s. It returns x / 2^s.
//
func (x Natural) Shr(s uint) Natural {
n := uint(len(x))
m := n - s/_W
if m > n { // check for underflow
m = 0
}
z := make(Natural, m)
shr(z, x[n-m:n], s%_W)
return normalize(z)
}
// And returns the ``bitwise and'' x & y for the 2's-complement representation of x and y.
//
func (x Natural) And(y Natural) Natural {
n := len(x)
m := len(y)
if n < m {
return y.And(x)
}
z := make(Natural, m)
for i := 0; i < m; i++ {
z[i] = x[i] & y[i]
}
// upper bits are 0
return normalize(z)
}
func copy(z, x Natural) {
for i, e := range x {
z[i] = e
}
}
// AndNot returns the ``bitwise clear'' x &^ y for the 2's-complement representation of x and y.
//
func (x Natural) AndNot(y Natural) Natural {
n := len(x)
m := len(y)
if n < m {
m = n
}
z := make(Natural, n)
for i := 0; i < m; i++ {
z[i] = x[i] &^ y[i]
}
copy(z[m:n], x[m:n])
return normalize(z)
}
// Or returns the ``bitwise or'' x | y for the 2's-complement representation of x and y.
//
func (x Natural) Or(y Natural) Natural {
n := len(x)
m := len(y)
if n < m {
return y.Or(x)
}
z := make(Natural, n)
for i := 0; i < m; i++ {
z[i] = x[i] | y[i]
}
copy(z[m:n], x[m:n])
return z
}
// Xor returns the ``bitwise exclusive or'' x ^ y for the 2's-complement representation of x and y.
//
func (x Natural) Xor(y Natural) Natural {
n := len(x)
m := len(y)
if n < m {
return y.Xor(x)
}
z := make(Natural, n)
for i := 0; i < m; i++ {
z[i] = x[i] ^ y[i]
}
copy(z[m:n], x[m:n])
return normalize(z)
}
// 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 Natural) Cmp(y Natural) int {
n := len(x)
m := len(y)
if n != m || n == 0 {
return n - m
}
i := n - 1
for i > 0 && x[i] == y[i] {
i--
}
d := 0
switch {
case x[i] < y[i]:
d = -1
case x[i] > y[i]:
d = 1
}
return d
}
// log2 computes the binary logarithm of x for x > 0.
// The result is the integer n for which 2^n <= x < 2^(n+1).
// If x == 0 a run-time error occurs.
//
func log2(x uint64) uint {
assert(x > 0)
n := uint(0)
for x > 0 {
x >>= 1
n++
}
return n - 1
}
// Log2 computes the binary logarithm of x for x > 0.
// The result is the integer n for which 2^n <= x < 2^(n+1).
// If x == 0 a run-time error occurs.
//
func (x Natural) Log2() uint {
n := len(x)
if n > 0 {
return (uint(n)-1)*_W + log2(uint64(x[n-1]))
}
panic("Log2(0)")
}
// Computes x = x div d in place (modifies x) for small d's.
// Returns updated x and x mod d.
//
func divmod1(x Natural, d digit) (Natural, digit) {
assert(0 < d && isSmall(d-1))
c := digit(0)
for i := len(x) - 1; i >= 0; i-- {
t := c<<_W + x[i]
c, x[i] = t%d, t/d
}
return normalize(x), c
}
// ToString converts x to a string for a given base, with 2 <= base <= 16.
//
func (x Natural) ToString(base uint) string {
if len(x) == 0 {
return "0"
}
// allocate buffer for conversion
assert(2 <= base && base <= 16)
n := (x.Log2()+1)/log2(uint64(base)) + 1 // +1: round up
s := make([]byte, n)
// don't destroy x
t := make(Natural, len(x))
copy(t, x)
// convert
i := n
for !t.IsZero() {
i--
var d digit
t, d = divmod1(t, digit(base))
s[i] = "0123456789abcdef"[d]
}
return string(s[i:n])
}
// String converts x to its decimal string representation.
// x.String() is the same as x.ToString(10).
//
func (x Natural) String() string { return x.ToString(10) }
func fmtbase(c int) uint {
switch c {
case 'b':
return 2
case 'o':
return 8
case 'x':
return 16
}
return 10
}
// Format is a support routine for fmt.Formatter. It accepts
// the formats 'b' (binary), 'o' (octal), and 'x' (hexadecimal).
//
func (x Natural) Format(h fmt.State, c int) { fmt.Fprintf(h, "%s", x.ToString(fmtbase(c))) }
func hexvalue(ch byte) uint {
d := uint(1 << logH)
switch {
case '0' <= ch && ch <= '9':
d = uint(ch - '0')
case 'a' <= ch && ch <= 'f':
d = uint(ch-'a') + 10
case 'A' <= ch && ch <= 'F':
d = uint(ch-'A') + 10
}
return d
}
// 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
// prefix length. The syntax of natural numbers follows the syntax
// of unsigned 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 NatFromString(s string, base uint) (Natural, uint, int) {
// determine base if necessary
i, n := 0, len(s)
if base == 0 {
base = 10
if n > 0 && s[0] == '0' {
if n > 1 && (s[1] == 'x' || s[1] == 'X') {
base, i = 16, 2
} else {
base, i = 8, 1
}
}
}
// convert string
assert(2 <= base && base <= 16)
x := Nat(0)
for ; i < n; i++ {
d := hexvalue(s[i])
if d < base {
x = muladd1(x, digit(base), digit(d))
} else {
break
}
}
return x, base, i
}
// Natural number functions
func pop1(x digit) uint {
n := uint(0)
for x != 0 {
x &= x - 1
n++
}
return n
}
// Pop computes the ``population count'' of (the number of 1 bits in) x.
//
func (x Natural) Pop() uint {
n := uint(0)
for i := len(x) - 1; i >= 0; i-- {
n += pop1(x[i])
}
return n
}
// Pow computes x to the power of n.
//
func (xp Natural) Pow(n uint) Natural {
z := Nat(1)
x := xp
for n > 0 {
// z * x^n == x^n0
if n&1 == 1 {
z = z.Mul(x)
}
x, n = x.Mul(x), n/2
}
return z
}
// MulRange computes the product of all the unsigned integers
// in the range [a, b] inclusively.
//
func MulRange(a, b uint) Natural {
switch {
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)))
}
m := (a + b) >> 1
assert(a <= m && m < b)
return MulRange(a, m).Mul(MulRange(m+1, b))
}
// Fact computes the factorial of n (Fact(n) == MulRange(2, n)).
//
func Fact(n uint) Natural {
// Using MulRange() instead of the basic for-loop
// lead to faster factorial computation.
return MulRange(2, n)
}
// Binomial computes the binomial coefficient of (n, k).
//
func Binomial(n, k uint) Natural { return MulRange(n-k+1, n).Div(MulRange(1, k)) }
// Gcd computes the gcd of x and y.
//
func (x Natural) Gcd(y Natural) Natural {
// Euclidean algorithm.
a, b := x, y
for !b.IsZero() {
a, b = b, a.Mod(b)
}
return a
}
src/pkg/exp/bignum/bignum_test.go deleted 100644 → 0
View file @ b2a919fc
// 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.
package bignum
import (
"fmt"
"testing"
)
const (
sa = "991"
sb = "2432902008176640000" // 20!
sc = "933262154439441526816992388562667004907159682643816214685929" +
"638952175999932299156089414639761565182862536979208272237582" +
"51185210916864000000000000000000000000" // 100!
sp = "170141183460469231731687303715884105727" // prime
)
func natFromString(s string, base uint, slen *int) Natural {
x, _, len := NatFromString(s, base)
if slen != nil {
*slen = len
}
return x
}
func intFromString(s string, base uint, slen *int) *Integer {
x, _, len := IntFromString(s, base)
if slen != nil {
*slen = len
}
return x
}
func ratFromString(s string, base uint, slen *int) *Rational {
x, _, len := RatFromString(s, base)
if slen != nil {
*slen = len
}
return x
}
var (
nat_zero = Nat(0)
nat_one = Nat(1)
nat_two = Nat(2)
a = natFromString(sa, 10, nil)
b = natFromString(sb, 10, nil)
c = natFromString(sc, 10, nil)
p = natFromString(sp, 10, nil)
int_zero = Int(0)
int_one = Int(1)
int_two = Int(2)
ip = intFromString(sp, 10, nil)
rat_zero = Rat(0, 1)
rat_half = Rat(1, 2)
rat_one = Rat(1, 1)
rat_two = Rat(2, 1)
)
var test_msg string
var tester *testing.T
func test(n uint, b bool) {
if !b {
tester.Fatalf("TEST failed: %s (%d)", test_msg, n)
}
}
func nat_eq(n uint, x, y Natural) {
if x.Cmp(y) != 0 {
tester.Fatalf("TEST failed: %s (%d)\nx = %v\ny = %v", test_msg, n, &x, &y)
}
}
func int_eq(n uint, x, y *Integer) {
if x.Cmp(y) != 0 {
tester.Fatalf("TEST failed: %s (%d)\nx = %v\ny = %v", test_msg, n, x, y)
}
}
func rat_eq(n uint, x, y *Rational) {
if x.Cmp(y) != 0 {
tester.Fatalf("TEST failed: %s (%d)\nx = %v\ny = %v", test_msg, n, x, y)
}
}
func TestNatConv(t *testing.T) {
tester = t
test_msg = "NatConvA"
type entry1 struct {
x uint64
s string
}
tab := []entry1{
entry1{0, "0"},
entry1{255, "255"},
entry1{65535, "65535"},
entry1{4294967295, "4294967295"},
entry1{18446744073709551615, "18446744073709551615"},
}
for i, e := range tab {
test(100+uint(i), Nat(e.x).String() == e.s)
test(200+uint(i), natFromString(e.s, 0, nil).Value() == e.x)
}
test_msg = "NatConvB"
for i := uint(0); i < 100; i++ {
test(i, Nat(uint64(i)).String() == fmt.Sprintf("%d", i))
}
test_msg = "NatConvC"
z := uint64(7)
for i := uint(0); i <= 64; i++ {
test(i, Nat(z).Value() == z)
z <<= 1
}
test_msg = "NatConvD"
nat_eq(0, a, Nat(991))
nat_eq(1, b, Fact(20))
nat_eq(2, c, Fact(100))
test(3, a.String() == sa)
test(4, b.String() == sb)
test(5, c.String() == sc)
test_msg = "NatConvE"
var slen int
nat_eq(10, natFromString("0", 0, nil), nat_zero)
nat_eq(11, natFromString("123", 0, nil), Nat(123))
nat_eq(12, natFromString("077", 0, nil), Nat(7*8+7))
nat_eq(13, natFromString("0x1f", 0, nil), Nat(1*16+15))
nat_eq(14, natFromString("0x1fg", 0, &slen), Nat(1*16+15))
test(4, slen == 4)
test_msg = "NatConvF"
tmp := c.Mul(c)
for base := uint(2); base <= 16; base++ {
nat_eq(base, natFromString(tmp.ToString(base), base, nil), tmp)
}
test_msg = "NatConvG"
x := Nat(100)
y, _, _ := NatFromString(fmt.Sprintf("%b", &x), 2)
nat_eq(100, y, x)
}
func abs(x int64) uint64 {
if x < 0 {
x = -x
}
return uint64(x)
}
func TestIntConv(t *testing.T) {
tester = t
test_msg = "IntConvA"
type entry2 struct {
x int64
s string
}
tab := []entry2{
entry2{0, "0"},
entry2{-128, "-128"},
entry2{127, "127"},
entry2{-32768, "-32768"},
entry2{32767, "32767"},
entry2{-2147483648, "-2147483648"},
entry2{2147483647, "2147483647"},
entry2{-9223372036854775808, "-9223372036854775808"},
entry2{9223372036854775807, "9223372036854775807"},
}
for i, e := range tab {
test(100+uint(i), Int(e.x).String() == e.s)
test(200+uint(i), intFromString(e.s, 0, nil).Value() == e.x)
test(300+uint(i), Int(e.x).Abs().Value() == abs(e.x))
}
test_msg = "IntConvB"
var slen int
int_eq(0, intFromString("0", 0, nil), int_zero)
int_eq(1, intFromString("-0", 0, nil), int_zero)
int_eq(2, intFromString("123", 0, nil), Int(123))
int_eq(3, intFromString("-123", 0, nil), Int(-123))
int_eq(4, intFromString("077", 0, nil), Int(7*8+7))
int_eq(5, intFromString("-077", 0, nil), Int(-(7*8 + 7)))
int_eq(6, intFromString("0x1f", 0, nil), Int(1*16+15))
int_eq(7, intFromString("-0x1f", 0, &slen), Int(-(1*16 + 15)))
test(7, slen == 5)
int_eq(8, intFromString("+0x1f", 0, &slen), Int(+(1*16 + 15)))
test(8, slen == 5)
int_eq(9, intFromString("0x1fg", 0, &slen), Int(1*16+15))
test(9, slen == 4)
int_eq(10, intFromString("-0x1fg", 0, &slen), Int(-(1*16 + 15)))
test(10, slen == 5)
}
func TestRatConv(t *testing.T) {
tester = t
test_msg = "RatConv"
var slen int
rat_eq(0, ratFromString("0", 0, nil), rat_zero)
rat_eq(1, ratFromString("0/1", 0, nil), rat_zero)
rat_eq(2, ratFromString("0/01", 0, nil), rat_zero)
rat_eq(3, ratFromString("0x14/10", 0, &slen), rat_two)
test(4, slen == 7)
rat_eq(5, ratFromString("0.", 0, nil), rat_zero)
rat_eq(6, ratFromString("0.001f", 10, nil), Rat(1, 1000))
rat_eq(7, ratFromString(".1", 0, nil), Rat(1, 10))
rat_eq(8, ratFromString("10101.0101", 2, nil), Rat(0x155, 1<<4))
rat_eq(9, ratFromString("-0003.145926", 10, &slen), Rat(-3145926, 1000000))
test(10, slen == 12)
rat_eq(11, ratFromString("1e2", 0, nil), Rat(100, 1))
rat_eq(12, ratFromString("1e-2", 0, nil), Rat(1, 100))
rat_eq(13, ratFromString("1.1e2", 0, nil), Rat(110, 1))
rat_eq(14, ratFromString(".1e2x", 0, &slen), Rat(10, 1))
test(15, slen == 4)
}
func add(x, y Natural) Natural {
z1 := x.Add(y)
z2 := y.Add(x)
if z1.Cmp(z2) != 0 {
tester.Fatalf("addition not symmetric:\n\tx = %v\n\ty = %t", x, y)
}
return z1
}
func sum(n uint64, scale Natural) Natural {
s := nat_zero
for ; n > 0; n-- {
s = add(s, Nat(n).Mul(scale))
}
return s
}
func TestNatAdd(t *testing.T) {
tester = t
test_msg = "NatAddA"
nat_eq(0, add(nat_zero, nat_zero), nat_zero)
nat_eq(1, add(nat_zero, c), c)
test_msg = "NatAddB"
for i := uint64(0); i < 100; i++ {
t := Nat(i)
nat_eq(uint(i), sum(i, c), t.Mul(t).Add(t).Shr(1).Mul(c))
}
}
func mul(x, y Natural) Natural {
z1 := x.Mul(y)
z2 := y.Mul(x)
if z1.Cmp(z2) != 0 {
tester.Fatalf("multiplication not symmetric:\n\tx = %v\n\ty = %t", x, y)
}
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
}
src/pkg/exp/bignum/integer.go deleted 100644 → 0
View file @ b2a919fc
// 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
}
src/pkg/exp/bignum/nrdiv_test.go deleted 100644 → 0
View file @ b2a919fc
// 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
}
src/pkg/exp/bignum/rational.go deleted 100644 → 0
View file @ b2a919fc
// 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
}
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