mirror of https://github.com/golang/go.git
1265 lines
24 KiB
Go
Executable File
1265 lines
24 KiB
Go
Executable File
// 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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// A package for arbitrary precision arithmethic.
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// It implements the following numeric types:
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//
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// - Natural unsigned integer numbers
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// - Integer signed integer numbers
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// - Rational rational numbers
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import Fmt "fmt"
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// ----------------------------------------------------------------------------
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// Internal representation
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//
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// A natural number of the form
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//
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// x = x[n-1]*B^(n-1) + x[n-2]*B^(n-2) + ... + x[1]*B + x[0]
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//
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// with 0 <= x[i] < B and 0 <= i < n is stored in an array of length n,
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// with the digits x[i] as the array elements.
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//
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// A natural number is normalized if the array contains no leading 0 digits.
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// During arithmetic operations, denormalized values may occur which are
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// always normalized before returning the final result. The normalized
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// representation of 0 is the empty array (length = 0).
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//
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// The operations for all other numeric types are implemented on top of
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// the operations for natural numbers.
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//
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// The base B is chosen as large as possible on a given platform but there
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// are a few constraints besides the size of the largest unsigned integer
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// type available:
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//
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// 1) To improve conversion speed between strings and numbers, the base B
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// is chosen such that division and multiplication by 10 (for decimal
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// string representation) can be done without using extended-precision
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// arithmetic. This makes addition, subtraction, and conversion routines
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// twice as fast. It requires a "buffer" of 4 bits per operand digit.
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// That is, the size of B must be 4 bits smaller then the size of the
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// type (Digit) in which these operations are performed. Having this
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// buffer also allows for trivial (single-bit) carry computation in
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// addition and subtraction (optimization suggested by Ken Thompson).
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//
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// 2) Long division requires extended-precision (2-digit) division per digit.
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// Instead of sacrificing the largest base type for all other operations,
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// for division the operands are unpacked into "half-digits", and the
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// results are packed again. For faster unpacking/packing, the base size
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// in bits must be even.
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type (
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Digit uint64;
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Digit2 uint32; // half-digits for division
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)
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const LogW = 64;
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const LogH = 4; // bits for a hex digit (= "small" number)
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const LogB = LogW - LogH; // largest bit-width available
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const (
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// half-digits
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W2 = LogB / 2; // width
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B2 = 1 << W2; // base
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M2 = B2 - 1; // mask
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// full digits
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W = W2 * 2; // width
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B = 1 << W; // base
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M = B - 1; // mask
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)
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// ----------------------------------------------------------------------------
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// Support functions
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func assert(p bool) {
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if !p {
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panic("assert failed");
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}
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}
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func IsSmall(x Digit) bool {
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return x < 1<<LogH;
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}
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export func Dump(x []Digit) {
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print("[", len(x), "]");
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for i := len(x) - 1; i >= 0; i-- {
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print(" ", x[i]);
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}
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println();
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}
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// ----------------------------------------------------------------------------
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// Natural numbers
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//
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// Naming conventions
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//
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// c carry
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// x, y operands
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// z result
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// n, m len(x), len(y)
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export type Natural []Digit;
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var (
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NatZero Natural = *&Natural{};
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NatOne Natural = *&Natural{1};
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NatTwo Natural = *&Natural{2};
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NatTen Natural = *&Natural{10};
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)
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// Creation
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export func Nat(x uint) Natural {
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switch x {
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case 0: return NatZero;
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case 1: return NatOne;
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case 2: return NatTwo;
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case 10: return NatTen;
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}
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assert(Digit(x) < B);
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return *&Natural{Digit(x)}; // TODO(rsc): *&
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}
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// Predicates
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func (x *Natural) IsOdd() bool {
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return len(x) > 0 && x[0]&1 != 0;
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}
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func (x *Natural) IsZero() bool {
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return len(x) == 0;
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}
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// Operations
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func Normalize(x *Natural) Natural {
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n := len(x);
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for n > 0 && x[n - 1] == 0 { n-- }
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if n < len(x) {
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x = x[0 : n]; // trim leading 0's
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}
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return x;
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}
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func (x *Natural) Add(y *Natural) *Natural {
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n := len(x);
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m := len(y);
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if n < m {
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return y.Add(x);
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}
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c := Digit(0);
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z := new(*Natural, n + 1);
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i := 0;
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for i < m {
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t := c + x[i] + y[i];
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c, z[i] = t>>W, t&M;
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i++;
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}
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for i < n {
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t := c + x[i];
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c, z[i] = t>>W, t&M;
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i++;
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}
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if c != 0 {
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z[i] = c;
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i++;
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}
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return z[0 : i];
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}
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func (x *Natural) Sub(y *Natural) *Natural {
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n := len(x);
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m := len(y);
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if n < m {
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panic("underflow")
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}
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c := Digit(0);
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z := new(*Natural, n);
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i := 0;
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for i < m {
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t := c + x[i] - y[i];
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c, z[i] = Digit(int64(t)>>W), t&M; // requires arithmetic shift!
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i++;
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}
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for i < n {
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t := c + x[i];
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c, z[i] = Digit(int64(t)>>W), t&M; // requires arithmetic shift!
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i++;
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}
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for i > 0 && z[i - 1] == 0 { // normalize
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i--;
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}
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return z[0 : i];
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}
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// Returns c = x*y div B, z = x*y mod B.
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func Mul11(x, y Digit) (Digit, Digit) {
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// Split x and y into 2 sub-digits each,
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// multiply the digits separately while avoiding overflow,
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// and return the product as two separate digits.
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// This code also works for non-even bit widths W
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// which is why there are separate constants below
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// for half-digits.
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const W2 = (W + 1)/2;
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const DW = W2*2 - W; // 0 or 1
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const B2 = 1<<W2;
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const M2 = B2 - 1;
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// split x and y into sub-digits
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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*B2^2 + 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 the result digits but avoid overflow
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// z = z1*B + z0 = x*y
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z0 := (t1<<W2 + t0)&M;
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z1 := t2<<DW + (t1 + t0>>W2)>>(W-W2);
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return z1, z0;
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}
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func (x *Natural) Mul(y *Natural) *Natural {
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n := len(x);
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m := len(y);
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z := new(*Natural, n + m);
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for j := 0; j < m; j++ {
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d := y[j];
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if d != 0 {
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c := Digit(0);
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for i := 0; i < n; i++ {
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// z[i+j] += c + x[i]*d;
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z1, z0 := Mul11(x[i], d);
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t := c + z[i+j] + z0;
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c, z[i+j] = t>>W, t&M;
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c += z1;
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}
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z[n+j] = c;
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}
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}
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return Normalize(z);
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}
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// DivMod needs multi-precision division which is not available if Digit
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// is already using the largest uint size. Instead, unpack each operand
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// into operands with twice as many digits of half the size (Digit2), do
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// DivMod, and then pack the results again.
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func Unpack(x *Natural) []Digit2 {
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n := len(x);
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z := new([]Digit2, n*2 + 1); // add space for extra digit (used by DivMod)
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for i := 0; i < n; i++ {
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t := x[i];
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z[i*2] = Digit2(t & M2);
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z[i*2 + 1] = Digit2(t >> W2 & M2);
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}
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// normalize result
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k := 2*n;
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for k > 0 && z[k - 1] == 0 { k-- }
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return z[0 : k]; // trim leading 0's
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}
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func Pack(x []Digit2) *Natural {
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n := (len(x) + 1) / 2;
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z := new(*Natural, n);
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if len(x) & 1 == 1 {
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// handle odd len(x)
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n--;
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z[n] = Digit(x[n*2]);
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}
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for i := 0; i < n; i++ {
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z[i] = Digit(x[i*2 + 1]) << W2 | Digit(x[i*2]);
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}
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return Normalize(z);
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}
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func Mul1(z, x []Digit2, y Digit2) Digit2 {
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n := len(x);
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c := Digit(0);
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f := Digit(y);
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for i := 0; i < n; i++ {
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t := c + Digit(x[i])*f;
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c, z[i] = t>>W2, Digit2(t&M2);
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}
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return Digit2(c);
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}
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func Div1(z, x []Digit2, y Digit2) Digit2 {
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n := len(x);
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c := Digit(0);
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d := Digit(y);
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for i := n-1; i >= 0; i-- {
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t := c*B2 + Digit(x[i]);
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c, z[i] = t%d, Digit2(t/d);
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}
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return Digit2(c);
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}
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// DivMod returns q and r with x = y*q + r and 0 <= r < y.
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// x and y are destroyed in the process.
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//
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// The algorithm used here is based on 1). 2) describes the same algorithm
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// in C. A discussion and summary of the relevant theorems can be found in
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// 3). 3) also describes an easier way to obtain the trial digit - however
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// it relies on tripple-precision arithmetic which is why Knuth's method is
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// used here.
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//
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// 1) D. Knuth, "The Art of Computer Programming. Volume 2. Seminumerical
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// Algorithms." Addison-Wesley, Reading, 1969.
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// (Algorithm D, Sec. 4.3.1)
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//
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// 2) Henry S. Warren, Jr., "A Hacker's Delight". Addison-Wesley, 2003.
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// (9-2 Multiword Division, p.140ff)
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//
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// 3) P. Brinch Hansen, Multiple-length division revisited: A tour of the
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// minefield. "Software - Practice and Experience 24", (June 1994),
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// 579-601. John Wiley & Sons, Ltd.
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func DivMod(x, y []Digit2) ([]Digit2, []Digit2) {
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n := len(x);
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m := len(y);
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if m == 0 {
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panic("division by zero");
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}
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assert(n+1 <= cap(x)); // space for one extra digit
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x = x[0 : n + 1];
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assert(x[n] == 0);
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if m == 1 {
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// division by single digit
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// result is shifted left by 1 in place!
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x[0] = Div1(x[1 : n+1], x[0 : n], y[0]);
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} else if m > n {
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// y > x => quotient = 0, remainder = x
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// TODO in this case we shouldn't even unpack x and y
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m = n;
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} else {
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// general case
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assert(2 <= m && m <= n);
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// normalize x and y
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// TODO Instead of multiplying, it would be sufficient to
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// shift y such that the normalization condition is
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// satisfied (as done in "Hacker's Delight").
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f := B2 / (Digit(y[m-1]) + 1);
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if f != 1 {
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Mul1(x, x, Digit2(f));
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Mul1(y, y, Digit2(f));
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}
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assert(B2/2 <= y[m-1] && y[m-1] < B2); // incorrect scaling
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y1, y2 := Digit(y[m-1]), Digit(y[m-2]);
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d2 := Digit(y1)<<W2 + Digit(y2);
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for i := n-m; i >= 0; i-- {
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k := i+m;
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// compute trial digit (Knuth)
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var q Digit;
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{ x0, x1, x2 := Digit(x[k]), Digit(x[k-1]), Digit(x[k-2]);
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if x0 != y1 {
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q = (x0<<W2 + x1)/y1;
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} else {
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q = B2 - 1;
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}
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for y2*q > (x0<<W2 + x1 - y1*q)<<W2 + x2 {
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q--
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}
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}
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// subtract y*q
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c := Digit(0);
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for j := 0; j < m; j++ {
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t := c + Digit(x[i+j]) - Digit(y[j])*q;
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c, x[i+j] = Digit(int64(t)>>W2), Digit2(t&M2); // requires arithmetic shift!
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}
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// correct if trial digit was too large
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if c + Digit(x[k]) != 0 {
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// add y
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c := Digit(0);
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for j := 0; j < m; j++ {
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t := c + Digit(x[i+j]) + Digit(y[j]);
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c, x[i+j] = t >> W2, Digit2(t & M2)
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}
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assert(c + Digit(x[k]) == 0);
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// correct trial digit
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q--;
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}
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x[k] = Digit2(q);
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}
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// undo normalization for remainder
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if f != 1 {
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c := Div1(x[0 : m], x[0 : m], Digit2(f));
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assert(c == 0);
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}
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}
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return x[m : n+1], x[0 : m];
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}
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func (x *Natural) Div(y *Natural) *Natural {
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q, r := DivMod(Unpack(x), Unpack(y));
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return Pack(q);
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}
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func (x *Natural) Mod(y *Natural) *Natural {
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q, r := DivMod(Unpack(x), Unpack(y));
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return Pack(r);
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}
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func (x *Natural) DivMod(y *Natural) (*Natural, *Natural) {
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q, r := DivMod(Unpack(x), Unpack(y));
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return Pack(q), Pack(r);
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}
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func Shl(z, x []Digit, s uint) Digit {
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assert(s <= W);
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n := len(x);
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c := Digit(0);
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for i := 0; i < n; i++ {
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c, z[i] = x[i] >> (W-s), x[i] << s & M | c;
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}
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return c;
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}
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func (x *Natural) Shl(s uint) *Natural {
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n := uint(len(x));
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m := n + s/W;
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z := new(*Natural, m+1);
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z[m] = Shl(z[m-n : m], x, s%W);
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return Normalize(z);
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}
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func Shr(z, x []Digit, s uint) Digit {
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assert(s <= W);
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n := len(x);
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c := Digit(0);
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for i := n - 1; i >= 0; i-- {
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c, z[i] = x[i] << (W-s) & M, x[i] >> s | c;
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}
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return c;
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}
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func (x *Natural) Shr(s uint) *Natural {
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n := uint(len(x));
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m := n - s/W;
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if m > n { // check for underflow
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m = 0;
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}
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z := new(*Natural, m);
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Shr(z, x[n-m : n], s%W);
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return Normalize(z);
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}
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func (x *Natural) And(y *Natural) *Natural {
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n := len(x);
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m := len(y);
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if n < m {
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return y.And(x);
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}
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z := new(*Natural, m);
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for i := 0; i < m; i++ {
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z[i] = x[i] & y[i];
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}
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// upper bits are 0
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return Normalize(z);
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}
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func Copy(z, x []Digit) {
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for i, e := range x {
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z[i] = e
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}
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}
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func (x *Natural) Or(y *Natural) *Natural {
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n := len(x);
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m := len(y);
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if n < m {
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return y.Or(x);
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}
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z := new(*Natural, n);
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for i := 0; i < m; i++ {
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z[i] = x[i] | y[i];
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}
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Copy(z[m : n], x[m : n]);
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return z;
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}
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func (x *Natural) Xor(y *Natural) *Natural {
|
|
n := len(x);
|
|
m := len(y);
|
|
if n < m {
|
|
return y.Xor(x);
|
|
}
|
|
|
|
z := new(*Natural, n);
|
|
for i := 0; i < m; i++ {
|
|
z[i] = x[i] ^ y[i];
|
|
}
|
|
Copy(z[m : n], x[m : n]);
|
|
|
|
return Normalize(z);
|
|
}
|
|
|
|
|
|
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;
|
|
}
|
|
|
|
|
|
func Log2(x Digit) uint {
|
|
assert(x > 0);
|
|
n := uint(0);
|
|
for x > 0 {
|
|
x >>= 1;
|
|
n++;
|
|
}
|
|
return n - 1;
|
|
}
|
|
|
|
|
|
func (x *Natural) Log2() uint {
|
|
n := len(x);
|
|
if n > 0 {
|
|
return (uint(n) - 1)*W + Log2(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;
|
|
}
|
|
|
|
|
|
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(Digit(base)) + 1; // +1: round up
|
|
s := new([]byte, n);
|
|
|
|
// don't destroy x
|
|
t := new(*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]);
|
|
}
|
|
|
|
|
|
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;
|
|
}
|
|
|
|
|
|
func (x *Natural) Format(h Fmt.Formatter, c int) {
|
|
t := x.ToString(FmtBase(c)); // BUG in 6g
|
|
Fmt.fprintf(h, "%s", t);
|
|
}
|
|
|
|
|
|
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;
|
|
}
|
|
|
|
|
|
// 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 := new(*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);
|
|
}
|
|
|
|
|
|
// Determines base (octal, decimal, hexadecimal) if base == 0.
|
|
// Returns the number and base.
|
|
export func NatFromString(s string, base uint, slen *int) (*Natural, uint) {
|
|
// 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;
|
|
}
|
|
}
|
|
|
|
// provide number of string bytes consumed if necessary
|
|
if slen != nil {
|
|
*slen = i;
|
|
}
|
|
|
|
return x, base;
|
|
}
|
|
|
|
|
|
// Natural number functions
|
|
|
|
func Pop1(x Digit) uint {
|
|
n := uint(0);
|
|
for x != 0 {
|
|
x &= x-1;
|
|
n++;
|
|
}
|
|
return n;
|
|
}
|
|
|
|
|
|
func (x *Natural) Pop() uint {
|
|
n := uint(0);
|
|
for i := len(x) - 1; i >= 0; i-- {
|
|
n += Pop1(x[i]);
|
|
}
|
|
return n;
|
|
}
|
|
|
|
|
|
func (x *Natural) Pow(n uint) *Natural {
|
|
z := Nat(1);
|
|
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;
|
|
}
|
|
|
|
|
|
export func MulRange(a, b uint) *Natural {
|
|
switch {
|
|
case a > b: return Nat(1);
|
|
case a == b: return Nat(a);
|
|
case a + 1 == b: return Nat(a).Mul(Nat(b));
|
|
}
|
|
m := (a + b)>>1;
|
|
assert(a <= m && m < b);
|
|
return MulRange(a, m).Mul(MulRange(m + 1, b));
|
|
}
|
|
|
|
|
|
export func Fact(n uint) *Natural {
|
|
// Using MulRange() instead of the basic for-loop
|
|
// lead to faster factorial computation.
|
|
return MulRange(2, n);
|
|
}
|
|
|
|
|
|
export func Binomial(n, k uint) *Natural {
|
|
return MulRange(n-k+1, n).Div(MulRange(1, k));
|
|
}
|
|
|
|
|
|
func (x *Natural) Gcd(y *Natural) *Natural {
|
|
// Euclidean algorithm.
|
|
for !y.IsZero() {
|
|
x, y = y, x.Mod(y);
|
|
}
|
|
return x;
|
|
}
|
|
|
|
|
|
// ----------------------------------------------------------------------------
|
|
// Integer numbers
|
|
//
|
|
// Integers are normalized if the mantissa is normalized and the sign is
|
|
// false for mant == 0. Use MakeInt to create normalized Integers.
|
|
|
|
export type Integer struct {
|
|
sign bool;
|
|
mant *Natural;
|
|
}
|
|
|
|
|
|
// Creation
|
|
|
|
export func MakeInt(sign bool, mant *Natural) *Integer {
|
|
if mant.IsZero() {
|
|
sign = false; // normalize
|
|
}
|
|
return &Integer{sign, mant};
|
|
}
|
|
|
|
|
|
export func Int(x int) *Integer {
|
|
sign := false;
|
|
var ux uint;
|
|
if x < 0 {
|
|
sign = true;
|
|
if -x == x {
|
|
// smallest negative integer
|
|
t := ^0;
|
|
ux = ^(uint(t) >> 1);
|
|
} else {
|
|
ux = uint(-x);
|
|
}
|
|
} else {
|
|
ux = uint(x);
|
|
}
|
|
return MakeInt(sign, Nat(ux));
|
|
}
|
|
|
|
|
|
// Predicates
|
|
|
|
func (x *Integer) IsOdd() bool {
|
|
return x.mant.IsOdd();
|
|
}
|
|
|
|
|
|
func (x *Integer) IsZero() bool {
|
|
return x.mant.IsZero();
|
|
}
|
|
|
|
|
|
func (x *Integer) IsNeg() bool {
|
|
return x.sign && !x.mant.IsZero()
|
|
}
|
|
|
|
|
|
func (x *Integer) IsPos() bool {
|
|
return !x.sign && !x.mant.IsZero()
|
|
}
|
|
|
|
|
|
// Operations
|
|
|
|
func (x *Integer) Neg() *Integer {
|
|
return MakeInt(!x.sign, x.mant);
|
|
}
|
|
|
|
|
|
func (x *Integer) Add(y *Integer) *Integer {
|
|
var z *Integer;
|
|
if x.sign == y.sign {
|
|
// x + y == x + y
|
|
// (-x) + (-y) == -(x + y)
|
|
z = MakeInt(x.sign, x.mant.Add(y.mant));
|
|
} else {
|
|
// x + (-y) == x - y == -(y - x)
|
|
// (-x) + y == y - x == -(x - y)
|
|
if x.mant.Cmp(y.mant) >= 0 {
|
|
z = MakeInt(false, x.mant.Sub(y.mant));
|
|
} else {
|
|
z = MakeInt(true, y.mant.Sub(x.mant));
|
|
}
|
|
}
|
|
if x.sign {
|
|
z.sign = !z.sign;
|
|
}
|
|
return z;
|
|
}
|
|
|
|
|
|
func (x *Integer) Sub(y *Integer) *Integer {
|
|
var z *Integer;
|
|
if x.sign != y.sign {
|
|
// x - (-y) == x + y
|
|
// (-x) - y == -(x + y)
|
|
z = MakeInt(false, x.mant.Add(y.mant));
|
|
} else {
|
|
// x - y == x - y == -(y - x)
|
|
// (-x) - (-y) == y - x == -(x - y)
|
|
if x.mant.Cmp(y.mant) >= 0 {
|
|
z = MakeInt(false, x.mant.Sub(y.mant));
|
|
} else {
|
|
z = MakeInt(true, y.mant.Sub(x.mant));
|
|
}
|
|
}
|
|
if x.sign {
|
|
z.sign = !z.sign;
|
|
}
|
|
return z;
|
|
}
|
|
|
|
|
|
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));
|
|
}
|
|
|
|
|
|
func (x *Integer) MulNat(y *Natural) *Integer {
|
|
// x * y == x * y
|
|
// (-x) * y == -(x * y)
|
|
return MakeInt(x.sign, x.mant.Mul(y));
|
|
}
|
|
|
|
|
|
// 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));
|
|
}
|
|
|
|
|
|
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));
|
|
}
|
|
|
|
|
|
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 and Mod implement Euclidian division and modulus:
|
|
//
|
|
// d = x.Div(y)
|
|
// m = x.Mod(y) with: 0 <= m < |d| and: y = x*d + m
|
|
//
|
|
// ( 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;
|
|
}
|
|
|
|
|
|
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;
|
|
}
|
|
|
|
|
|
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;
|
|
}
|
|
|
|
|
|
func (x *Integer) Shl(s uint) *Integer {
|
|
return MakeInt(x.sign, x.mant.Shl(s));
|
|
}
|
|
|
|
|
|
func (x *Integer) Shr(s uint) *Integer {
|
|
z := MakeInt(x.sign, x.mant.Shr(s));
|
|
if x.IsNeg() {
|
|
panic("UNIMPLEMENTED Integer.Shr() of negative values");
|
|
}
|
|
return z;
|
|
}
|
|
|
|
|
|
func (x *Integer) And(y *Integer) *Integer {
|
|
var z *Integer;
|
|
if !x.sign && !y.sign {
|
|
z = MakeInt(false, x.mant.And(y.mant));
|
|
} else {
|
|
panic("UNIMPLEMENTED Integer.And() of negative values");
|
|
}
|
|
return z;
|
|
}
|
|
|
|
|
|
func (x *Integer) Or(y *Integer) *Integer {
|
|
var z *Integer;
|
|
if !x.sign && !y.sign {
|
|
z = MakeInt(false, x.mant.Or(y.mant));
|
|
} else {
|
|
panic("UNIMPLEMENTED Integer.Or() of negative values");
|
|
}
|
|
return z;
|
|
}
|
|
|
|
|
|
func (x *Integer) Xor(y *Integer) *Integer {
|
|
var z *Integer;
|
|
if !x.sign && !y.sign {
|
|
z = MakeInt(false, x.mant.Xor(y.mant));
|
|
} else {
|
|
panic("UNIMPLEMENTED Integer.Xor() of negative values");
|
|
}
|
|
return z;
|
|
}
|
|
|
|
|
|
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;
|
|
}
|
|
|
|
|
|
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);
|
|
}
|
|
|
|
|
|
func (x *Integer) String() string {
|
|
return x.ToString(10);
|
|
}
|
|
|
|
|
|
func (x *Integer) Format(h Fmt.Formatter, c int) {
|
|
t := x.ToString(FmtBase(c)); // BUG in 6g
|
|
Fmt.fprintf(h, "%s", t);
|
|
}
|
|
|
|
|
|
// Determines base (octal, decimal, hexadecimal) if base == 0.
|
|
// Returns the number and base.
|
|
export func IntFromString(s string, base uint, slen *int) (*Integer, uint) {
|
|
// get sign, if any
|
|
sign := false;
|
|
if len(s) > 0 && (s[0] == '-' || s[0] == '+') {
|
|
sign = s[0] == '-';
|
|
s = s[1 : len(s)];
|
|
}
|
|
|
|
var mant *Natural;
|
|
mant, base = NatFromString(s, base, slen);
|
|
|
|
// correct slen if necessary
|
|
if slen != nil && sign {
|
|
*slen++;
|
|
}
|
|
|
|
return MakeInt(sign, mant), base;
|
|
}
|
|
|
|
|
|
// ----------------------------------------------------------------------------
|
|
// Rational numbers
|
|
|
|
export type Rational struct {
|
|
a *Integer; // numerator
|
|
b *Natural; // denominator
|
|
}
|
|
|
|
|
|
// Creation
|
|
|
|
export 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);
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}
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return &Rational{a, b};
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}
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export func Rat(a0 int, b0 int) *Rational {
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a, b := Int(a0), Int(b0);
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if b.sign {
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a = a.Neg();
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}
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return MakeRat(a, b.mant);
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}
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// Predicates
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|
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func (x *Rational) IsZero() bool {
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return x.a.IsZero();
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}
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|
|
|
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func (x *Rational) IsNeg() bool {
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|
return x.a.IsNeg();
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|
}
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|
|
|
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|
func (x *Rational) IsPos() bool {
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|
return x.a.IsPos();
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|
}
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|
|
|
|
|
func (x *Rational) IsInt() bool {
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|
return x.b.Cmp(Nat(1)) == 0;
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|
}
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|
|
|
|
|
// Operations
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|
|
|
func (x *Rational) Neg() *Rational {
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|
return MakeRat(x.a.Neg(), x.b);
|
|
}
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|
|
|
|
|
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));
|
|
}
|
|
|
|
|
|
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));
|
|
}
|
|
|
|
|
|
func (x *Rational) Mul(y *Rational) *Rational {
|
|
return MakeRat(x.a.Mul(y.a), x.b.Mul(y.b));
|
|
}
|
|
|
|
|
|
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);
|
|
}
|
|
|
|
|
|
func (x *Rational) Cmp(y *Rational) int {
|
|
return (x.a.MulNat(y.b)).Cmp(y.a.MulNat(x.b));
|
|
}
|
|
|
|
|
|
func (x *Rational) ToString(base uint) string {
|
|
s := x.a.ToString(base);
|
|
if !x.IsInt() {
|
|
s += "/" + x.b.ToString(base);
|
|
}
|
|
return s;
|
|
}
|
|
|
|
|
|
func (x *Rational) String() string {
|
|
return x.ToString(10);
|
|
}
|
|
|
|
|
|
func (x *Rational) Format(h Fmt.Formatter, c int) {
|
|
t := x.ToString(FmtBase(c)); // BUG in 6g
|
|
Fmt.fprintf(h, "%s", t);
|
|
}
|
|
|
|
|
|
// Determines base (octal, decimal, hexadecimal) if base == 0.
|
|
// Returns the number and base of the nominator.
|
|
export func RatFromString(s string, base uint, slen *int) (*Rational, uint) {
|
|
// read nominator
|
|
var alen, blen int;
|
|
a, abase := IntFromString(s, base, &alen);
|
|
b := Nat(1);
|
|
|
|
// read denominator or fraction, if any
|
|
if alen < len(s) {
|
|
ch := s[alen];
|
|
if ch == '/' {
|
|
alen++;
|
|
b, base = NatFromString(s[alen : len(s)], base, &blen);
|
|
} else if ch == '.' {
|
|
alen++;
|
|
b, base = NatFromString(s[alen : len(s)], abase, &blen);
|
|
assert(base == abase);
|
|
f := Nat(base).Pow(uint(blen));
|
|
a = MakeInt(a.sign, a.mant.Mul(f).Add(b));
|
|
b = f;
|
|
}
|
|
}
|
|
|
|
// provide number of string bytes consumed if necessary
|
|
if slen != nil {
|
|
*slen = alen + blen;
|
|
}
|
|
|
|
return MakeRat(a, b), abase;
|
|
}
|