From 554fd8c5195424bdbcabf5de30fdc183aba391bd Mon Sep 17 00:00:00 2001 From: upstream source tree Date: Sun, 15 Mar 2015 20:14:05 -0400 Subject: obtained gcc-4.6.4.tar.bz2 from upstream website; verified gcc-4.6.4.tar.bz2.sig; imported gcc-4.6.4 source tree from verified upstream tarball. downloading a git-generated archive based on the 'upstream' tag should provide you with a source tree that is binary identical to the one extracted from the above tarball. if you have obtained the source via the command 'git clone', however, do note that line-endings of files in your working directory might differ from line-endings of the respective files in the upstream repository. --- libgo/go/big/nat.go | 1067 +++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 1067 insertions(+) create mode 100644 libgo/go/big/nat.go (limited to 'libgo/go/big/nat.go') diff --git a/libgo/go/big/nat.go b/libgo/go/big/nat.go new file mode 100644 index 000000000..a308f69e8 --- /dev/null +++ b/libgo/go/big/nat.go @@ -0,0 +1,1067 @@ +// 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 contains operations on unsigned multi-precision integers. +// These are the building blocks for the operations on signed integers +// and rationals. + +// This package implements multi-precision arithmetic (big numbers). +// The following numeric types are supported: +// +// - Int signed integers +// - Rat rational numbers +// +// All methods on Int take the result as the receiver; if it is one +// of the operands it may be overwritten (and its memory reused). +// To enable chaining of operations, the result is also returned. +// +package big + +import "rand" + +// An unsigned integer x 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 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 or nil slice (length = 0). + +type nat []Word + +var ( + natOne = nat{1} + natTwo = nat{2} + natTen = nat{10} +) + + +func (z nat) clear() { + for i := range z { + z[i] = 0 + } +} + + +func (z nat) norm() nat { + i := len(z) + for i > 0 && z[i-1] == 0 { + i-- + } + return z[0:i] +} + + +func (z nat) make(n int) nat { + if n <= cap(z) { + return z[0:n] // reuse z + } + // Choosing a good value for e has significant performance impact + // because it increases the chance that a value can be reused. + const e = 4 // extra capacity + return make(nat, n, n+e) +} + + +func (z nat) setWord(x Word) nat { + if x == 0 { + return z.make(0) + } + z = z.make(1) + z[0] = x + return z +} + + +func (z nat) setUint64(x uint64) nat { + // single-digit values + if w := Word(x); uint64(w) == x { + return z.setWord(w) + } + + // compute number of words n required to represent x + n := 0 + for t := x; t > 0; t >>= _W { + n++ + } + + // split x into n words + z = z.make(n) + for i := range z { + z[i] = Word(x & _M) + x >>= _W + } + + return z +} + + +func (z nat) set(x nat) nat { + z = z.make(len(x)) + copy(z, x) + return z +} + + +func (z nat) add(x, y nat) nat { + m := len(x) + n := len(y) + + switch { + case m < n: + return z.add(y, x) + case m == 0: + // n == 0 because m >= n; result is 0 + return z.make(0) + case n == 0: + // result is x + return z.set(x) + } + // m > 0 + + z = z.make(m + 1) + c := addVV(z[0:n], x, y) + if m > n { + c = addVW(z[n:m], x[n:], c) + } + z[m] = c + + return z.norm() +} + + +func (z nat) sub(x, y nat) nat { + m := len(x) + n := len(y) + + switch { + case m < n: + panic("underflow") + case m == 0: + // n == 0 because m >= n; result is 0 + return z.make(0) + case n == 0: + // result is x + return z.set(x) + } + // m > 0 + + z = z.make(m) + c := subVV(z[0:n], x, y) + if m > n { + c = subVW(z[n:], x[n:], c) + } + if c != 0 { + panic("underflow") + } + + return z.norm() +} + + +func (x nat) cmp(y nat) (r int) { + m := len(x) + n := len(y) + if m != n || m == 0 { + switch { + case m < n: + r = -1 + case m > n: + r = 1 + } + return + } + + i := m - 1 + for i > 0 && x[i] == y[i] { + i-- + } + + switch { + case x[i] < y[i]: + r = -1 + case x[i] > y[i]: + r = 1 + } + return +} + + +func (z nat) mulAddWW(x nat, y, r Word) nat { + m := len(x) + if m == 0 || y == 0 { + return z.setWord(r) // result is r + } + // m > 0 + + z = z.make(m + 1) + z[m] = mulAddVWW(z[0:m], x, y, r) + + return z.norm() +} + + +// basicMul multiplies x and y and leaves the result in z. +// The (non-normalized) result is placed in z[0 : len(x) + len(y)]. +func basicMul(z, x, y nat) { + z[0 : len(x)+len(y)].clear() // initialize z + for i, d := range y { + if d != 0 { + z[len(x)+i] = addMulVVW(z[i:i+len(x)], x, d) + } + } +} + + +// Fast version of z[0:n+n>>1].add(z[0:n+n>>1], x[0:n]) w/o bounds checks. +// Factored out for readability - do not use outside karatsuba. +func karatsubaAdd(z, x nat, n int) { + if c := addVV(z[0:n], z, x); c != 0 { + addVW(z[n:n+n>>1], z[n:], c) + } +} + + +// Like karatsubaAdd, but does subtract. +func karatsubaSub(z, x nat, n int) { + if c := subVV(z[0:n], z, x); c != 0 { + subVW(z[n:n+n>>1], z[n:], c) + } +} + + +// Operands that are shorter than karatsubaThreshold are multiplied using +// "grade school" multiplication; for longer operands the Karatsuba algorithm +// is used. +var karatsubaThreshold int = 32 // computed by calibrate.go + +// karatsuba multiplies x and y and leaves the result in z. +// Both x and y must have the same length n and n must be a +// power of 2. The result vector z must have len(z) >= 6*n. +// The (non-normalized) result is placed in z[0 : 2*n]. +func karatsuba(z, x, y nat) { + n := len(y) + + // Switch to basic multiplication if numbers are odd or small. + // (n is always even if karatsubaThreshold is even, but be + // conservative) + if n&1 != 0 || n < karatsubaThreshold || n < 2 { + basicMul(z, x, y) + return + } + // n&1 == 0 && n >= karatsubaThreshold && n >= 2 + + // Karatsuba multiplication is based on the observation that + // for two numbers x and y with: + // + // x = x1*b + x0 + // y = y1*b + y0 + // + // the product x*y can be obtained with 3 products z2, z1, z0 + // instead of 4: + // + // x*y = x1*y1*b*b + (x1*y0 + x0*y1)*b + x0*y0 + // = z2*b*b + z1*b + z0 + // + // with: + // + // xd = x1 - x0 + // yd = y0 - y1 + // + // z1 = xd*yd + z1 + z0 + // = (x1-x0)*(y0 - y1) + z1 + z0 + // = x1*y0 - x1*y1 - x0*y0 + x0*y1 + z1 + z0 + // = x1*y0 - z1 - z0 + x0*y1 + z1 + z0 + // = x1*y0 + x0*y1 + + // split x, y into "digits" + n2 := n >> 1 // n2 >= 1 + x1, x0 := x[n2:], x[0:n2] // x = x1*b + y0 + y1, y0 := y[n2:], y[0:n2] // y = y1*b + y0 + + // z is used for the result and temporary storage: + // + // 6*n 5*n 4*n 3*n 2*n 1*n 0*n + // z = [z2 copy|z0 copy| xd*yd | yd:xd | x1*y1 | x0*y0 ] + // + // For each recursive call of karatsuba, an unused slice of + // z is passed in that has (at least) half the length of the + // caller's z. + + // compute z0 and z2 with the result "in place" in z + karatsuba(z, x0, y0) // z0 = x0*y0 + karatsuba(z[n:], x1, y1) // z2 = x1*y1 + + // compute xd (or the negative value if underflow occurs) + s := 1 // sign of product xd*yd + xd := z[2*n : 2*n+n2] + if subVV(xd, x1, x0) != 0 { // x1-x0 + s = -s + subVV(xd, x0, x1) // x0-x1 + } + + // compute yd (or the negative value if underflow occurs) + yd := z[2*n+n2 : 3*n] + if subVV(yd, y0, y1) != 0 { // y0-y1 + s = -s + subVV(yd, y1, y0) // y1-y0 + } + + // p = (x1-x0)*(y0-y1) == x1*y0 - x1*y1 - x0*y0 + x0*y1 for s > 0 + // p = (x0-x1)*(y0-y1) == x0*y0 - x0*y1 - x1*y0 + x1*y1 for s < 0 + p := z[n*3:] + karatsuba(p, xd, yd) + + // save original z2:z0 + // (ok to use upper half of z since we're done recursing) + r := z[n*4:] + copy(r, z) + + // add up all partial products + // + // 2*n n 0 + // z = [ z2 | z0 ] + // + [ z0 ] + // + [ z2 ] + // + [ p ] + // + karatsubaAdd(z[n2:], r, n) + karatsubaAdd(z[n2:], r[n:], n) + if s > 0 { + karatsubaAdd(z[n2:], p, n) + } else { + karatsubaSub(z[n2:], p, n) + } +} + + +// alias returns true if x and y share the same base array. +func alias(x, y nat) bool { + return cap(x) > 0 && cap(y) > 0 && &x[0:cap(x)][cap(x)-1] == &y[0:cap(y)][cap(y)-1] +} + + +// addAt implements z += x*(1<<(_W*i)); z must be long enough. +// (we don't use nat.add because we need z to stay the same +// slice, and we don't need to normalize z after each addition) +func addAt(z, x nat, i int) { + if n := len(x); n > 0 { + if c := addVV(z[i:i+n], z[i:], x); c != 0 { + j := i + n + if j < len(z) { + addVW(z[j:], z[j:], c) + } + } + } +} + + +func max(x, y int) int { + if x > y { + return x + } + return y +} + + +// karatsubaLen computes an approximation to the maximum k <= n such that +// k = p<= 0. Thus, the +// result is the largest number that can be divided repeatedly by 2 before +// becoming about the value of karatsubaThreshold. +func karatsubaLen(n int) int { + i := uint(0) + for n > karatsubaThreshold { + n >>= 1 + i++ + } + return n << i +} + + +func (z nat) mul(x, y nat) nat { + m := len(x) + n := len(y) + + switch { + case m < n: + return z.mul(y, x) + case m == 0 || n == 0: + return z.make(0) + case n == 1: + return z.mulAddWW(x, y[0], 0) + } + // m >= n > 1 + + // determine if z can be reused + if alias(z, x) || alias(z, y) { + z = nil // z is an alias for x or y - cannot reuse + } + + // use basic multiplication if the numbers are small + if n < karatsubaThreshold || n < 2 { + z = z.make(m + n) + basicMul(z, x, y) + return z.norm() + } + // m >= n && n >= karatsubaThreshold && n >= 2 + + // determine Karatsuba length k such that + // + // x = x1*b + x0 + // y = y1*b + y0 (and k <= len(y), which implies k <= len(x)) + // b = 1<<(_W*k) ("base" of digits xi, yi) + // + k := karatsubaLen(n) + // k <= n + + // multiply x0 and y0 via Karatsuba + x0 := x[0:k] // x0 is not normalized + y0 := y[0:k] // y0 is not normalized + z = z.make(max(6*k, m+n)) // enough space for karatsuba of x0*y0 and full result of x*y + karatsuba(z, x0, y0) + z = z[0 : m+n] // z has final length but may be incomplete, upper portion is garbage + + // If x1 and/or y1 are not 0, add missing terms to z explicitly: + // + // m+n 2*k 0 + // z = [ ... | x0*y0 ] + // + [ x1*y1 ] + // + [ x1*y0 ] + // + [ x0*y1 ] + // + if k < n || m != n { + x1 := x[k:] // x1 is normalized because x is + y1 := y[k:] // y1 is normalized because y is + var t nat + t = t.mul(x1, y1) + copy(z[2*k:], t) + z[2*k+len(t):].clear() // upper portion of z is garbage + t = t.mul(x1, y0.norm()) + addAt(z, t, k) + t = t.mul(x0.norm(), y1) + addAt(z, t, k) + } + + return z.norm() +} + + +// mulRange computes the product of all the unsigned integers in the +// range [a, b] inclusively. If a > b (empty range), the result is 1. +func (z nat) mulRange(a, b uint64) nat { + switch { + case a == 0: + // cut long ranges short (optimization) + return z.setUint64(0) + case a > b: + return z.setUint64(1) + case a == b: + return z.setUint64(a) + case a+1 == b: + return z.mul(nat(nil).setUint64(a), nat(nil).setUint64(b)) + } + m := (a + b) / 2 + return z.mul(nat(nil).mulRange(a, m), nat(nil).mulRange(m+1, b)) +} + + +// q = (x-r)/y, with 0 <= r < y +func (z nat) divW(x nat, y Word) (q nat, r Word) { + m := len(x) + switch { + case y == 0: + panic("division by zero") + case y == 1: + q = z.set(x) // result is x + return + case m == 0: + q = z.make(0) // result is 0 + return + } + // m > 0 + z = z.make(m) + r = divWVW(z, 0, x, y) + q = z.norm() + return +} + + +func (z nat) div(z2, u, v nat) (q, r nat) { + if len(v) == 0 { + panic("division by zero") + } + + if u.cmp(v) < 0 { + q = z.make(0) + r = z2.set(u) + return + } + + if len(v) == 1 { + var rprime Word + q, rprime = z.divW(u, v[0]) + if rprime > 0 { + r = z2.make(1) + r[0] = rprime + } else { + r = z2.make(0) + } + return + } + + q, r = z.divLarge(z2, u, v) + return +} + + +// q = (uIn-r)/v, with 0 <= r < y +// Uses z as storage for q, and u as storage for r if possible. +// See Knuth, Volume 2, section 4.3.1, Algorithm D. +// Preconditions: +// len(v) >= 2 +// len(uIn) >= len(v) +func (z nat) divLarge(u, uIn, v nat) (q, r nat) { + n := len(v) + m := len(uIn) - n + + // determine if z can be reused + // TODO(gri) should find a better solution - this if statement + // is very costly (see e.g. time pidigits -s -n 10000) + if alias(z, uIn) || alias(z, v) { + z = nil // z is an alias for uIn or v - cannot reuse + } + q = z.make(m + 1) + + qhatv := make(nat, n+1) + if alias(u, uIn) || alias(u, v) { + u = nil // u is an alias for uIn or v - cannot reuse + } + u = u.make(len(uIn) + 1) + u.clear() + + // D1. + shift := Word(leadingZeros(v[n-1])) + shlVW(v, v, shift) + u[len(uIn)] = shlVW(u[0:len(uIn)], uIn, shift) + + // D2. + for j := m; j >= 0; j-- { + // D3. + qhat := Word(_M) + if u[j+n] != v[n-1] { + var rhat Word + qhat, rhat = divWW(u[j+n], u[j+n-1], v[n-1]) + + // x1 | x2 = q̂v_{n-2} + x1, x2 := mulWW(qhat, v[n-2]) + // test if q̂v_{n-2} > br̂ + u_{j+n-2} + for greaterThan(x1, x2, rhat, u[j+n-2]) { + qhat-- + prevRhat := rhat + rhat += v[n-1] + // v[n-1] >= 0, so this tests for overflow. + if rhat < prevRhat { + break + } + x1, x2 = mulWW(qhat, v[n-2]) + } + } + + // D4. + qhatv[n] = mulAddVWW(qhatv[0:n], v, qhat, 0) + + c := subVV(u[j:j+len(qhatv)], u[j:], qhatv) + if c != 0 { + c := addVV(u[j:j+n], u[j:], v) + u[j+n] += c + qhat-- + } + + q[j] = qhat + } + + q = q.norm() + shrVW(u, u, shift) + shrVW(v, v, shift) + r = u.norm() + + return q, r +} + + +// Length of x in bits. x must be normalized. +func (x nat) bitLen() int { + if i := len(x) - 1; i >= 0 { + return i*_W + bitLen(x[i]) + } + return 0 +} + + +func hexValue(ch byte) int { + var d byte + switch { + case '0' <= ch && ch <= '9': + d = ch - '0' + case 'a' <= ch && ch <= 'f': + d = ch - 'a' + 10 + case 'A' <= ch && ch <= 'F': + d = ch - 'A' + 10 + default: + return -1 + } + return int(d) +} + + +// scan 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, and a ``0b'' or ``0B'' prefix selects +// base 2. Otherwise the selected base is 10. +// +func (z nat) scan(s string, base int) (nat, int, int) { + // determine base if necessary + i, n := 0, len(s) + if base == 0 { + base = 10 + if n > 0 && s[0] == '0' { + base, i = 8, 1 + if n > 1 { + switch s[1] { + case 'x', 'X': + base, i = 16, 2 + case 'b', 'B': + base, i = 2, 2 + } + } + } + } + + // reject illegal bases or strings consisting only of prefix + if base < 2 || 16 < base || (base != 8 && i >= n) { + return z, 0, 0 + } + + // convert string + z = z.make(0) + for ; i < n; i++ { + d := hexValue(s[i]) + if 0 <= d && d < base { + z = z.mulAddWW(z, Word(base), Word(d)) + } else { + break + } + } + + return z.norm(), base, i +} + + +// string converts x to a string for a given base, with 2 <= base <= 16. +// TODO(gri) in the style of the other routines, perhaps this should take +// a []byte buffer and return it +func (x nat) string(base int) string { + if base < 2 || 16 < base { + panic("illegal base") + } + + if len(x) == 0 { + return "0" + } + + // allocate buffer for conversion + i := x.bitLen()/log2(Word(base)) + 1 // +1: round up + s := make([]byte, i) + + // don't destroy x + q := nat(nil).set(x) + + // convert + for len(q) > 0 { + i-- + var r Word + q, r = q.divW(q, Word(base)) + s[i] = "0123456789abcdef"[r] + } + + return string(s[i:]) +} + + +const deBruijn32 = 0x077CB531 + +var deBruijn32Lookup = []byte{ + 0, 1, 28, 2, 29, 14, 24, 3, 30, 22, 20, 15, 25, 17, 4, 8, + 31, 27, 13, 23, 21, 19, 16, 7, 26, 12, 18, 6, 11, 5, 10, 9, +} + +const deBruijn64 = 0x03f79d71b4ca8b09 + +var deBruijn64Lookup = []byte{ + 0, 1, 56, 2, 57, 49, 28, 3, 61, 58, 42, 50, 38, 29, 17, 4, + 62, 47, 59, 36, 45, 43, 51, 22, 53, 39, 33, 30, 24, 18, 12, 5, + 63, 55, 48, 27, 60, 41, 37, 16, 46, 35, 44, 21, 52, 32, 23, 11, + 54, 26, 40, 15, 34, 20, 31, 10, 25, 14, 19, 9, 13, 8, 7, 6, +} + +// trailingZeroBits returns the number of consecutive zero bits on the right +// side of the given Word. +// See Knuth, volume 4, section 7.3.1 +func trailingZeroBits(x Word) int { + // x & -x leaves only the right-most bit set in the word. Let k be the + // index of that bit. Since only a single bit is set, the value is two + // to the power of k. Multipling by a power of two is equivalent to + // left shifting, in this case by k bits. The de Bruijn constant is + // such that all six bit, consecutive substrings are distinct. + // Therefore, if we have a left shifted version of this constant we can + // find by how many bits it was shifted by looking at which six bit + // substring ended up at the top of the word. + switch _W { + case 32: + return int(deBruijn32Lookup[((x&-x)*deBruijn32)>>27]) + case 64: + return int(deBruijn64Lookup[((x&-x)*(deBruijn64&_M))>>58]) + default: + panic("Unknown word size") + } + + return 0 +} + + +// z = x << s +func (z nat) shl(x nat, s uint) nat { + m := len(x) + if m == 0 { + return z.make(0) + } + // m > 0 + + n := m + int(s/_W) + z = z.make(n + 1) + z[n] = shlVW(z[n-m:n], x, Word(s%_W)) + z[0 : n-m].clear() + + return z.norm() +} + + +// z = x >> s +func (z nat) shr(x nat, s uint) nat { + m := len(x) + n := m - int(s/_W) + if n <= 0 { + return z.make(0) + } + // n > 0 + + z = z.make(n) + shrVW(z, x[m-n:], Word(s%_W)) + + return z.norm() +} + + +func (z nat) and(x, y nat) nat { + m := len(x) + n := len(y) + if m > n { + m = n + } + // m <= n + + z = z.make(m) + for i := 0; i < m; i++ { + z[i] = x[i] & y[i] + } + + return z.norm() +} + + +func (z nat) andNot(x, y nat) nat { + m := len(x) + n := len(y) + if n > m { + n = m + } + // m >= n + + z = z.make(m) + for i := 0; i < n; i++ { + z[i] = x[i] &^ y[i] + } + copy(z[n:m], x[n:m]) + + return z.norm() +} + + +func (z nat) or(x, y nat) nat { + m := len(x) + n := len(y) + s := x + if m < n { + n, m = m, n + s = y + } + // m >= n + + z = z.make(m) + for i := 0; i < n; i++ { + z[i] = x[i] | y[i] + } + copy(z[n:m], s[n:m]) + + return z.norm() +} + + +func (z nat) xor(x, y nat) nat { + m := len(x) + n := len(y) + s := x + if m < n { + n, m = m, n + s = y + } + // m >= n + + z = z.make(m) + for i := 0; i < n; i++ { + z[i] = x[i] ^ y[i] + } + copy(z[n:m], s[n:m]) + + return z.norm() +} + + +// greaterThan returns true iff (x1<<_W + x2) > (y1<<_W + y2) +func greaterThan(x1, x2, y1, y2 Word) bool { return x1 > y1 || x1 == y1 && x2 > y2 } + + +// modW returns x % d. +func (x nat) modW(d Word) (r Word) { + // TODO(agl): we don't actually need to store the q value. + var q nat + q = q.make(len(x)) + return divWVW(q, 0, x, d) +} + + +// powersOfTwoDecompose finds q and k such that q * 1<= 0; i-- { + v = y[i] + + for j := 0; j < _W; j++ { + z = z.mul(z, z) + + if v&mask != 0 { + z = z.mul(z, x) + } + + if m != nil { + q, z = q.div(z, z, m) + } + + v <<= 1 + } + } + + return z +} + + +// probablyPrime performs reps Miller-Rabin tests to check whether n is prime. +// If it returns true, n is prime with probability 1 - 1/4^reps. +// If it returns false, n is not prime. +func (n nat) probablyPrime(reps int) bool { + if len(n) == 0 { + return false + } + + if len(n) == 1 { + if n[0] < 2 { + return false + } + + if n[0]%2 == 0 { + return n[0] == 2 + } + + // We have to exclude these cases because we reject all + // multiples of these numbers below. + switch n[0] { + case 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53: + return true + } + } + + const primesProduct32 = 0xC0CFD797 // Π {p ∈ primes, 2 < p <= 29} + const primesProduct64 = 0xE221F97C30E94E1D // Π {p ∈ primes, 2 < p <= 53} + + var r Word + switch _W { + case 32: + r = n.modW(primesProduct32) + case 64: + r = n.modW(primesProduct64 & _M) + default: + panic("Unknown word size") + } + + if r%3 == 0 || r%5 == 0 || r%7 == 0 || r%11 == 0 || + r%13 == 0 || r%17 == 0 || r%19 == 0 || r%23 == 0 || r%29 == 0 { + return false + } + + if _W == 64 && (r%31 == 0 || r%37 == 0 || r%41 == 0 || + r%43 == 0 || r%47 == 0 || r%53 == 0) { + return false + } + + nm1 := nat(nil).sub(n, natOne) + // 1<