summaryrefslogtreecommitdiffstats
path: root/src/math/big/arith.go
blob: 750ce8aa398df4ba6d32244ea9e211c592e38821 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
// 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 Go implementations of elementary multi-precision
// arithmetic operations on word vectors. These have the suffix _g.
// These are needed for platforms without assembly implementations of these routines.
// This file also contains elementary operations that can be implemented
// sufficiently efficiently in Go.

package big

import "math/bits"

// A Word represents a single digit of a multi-precision unsigned integer.
type Word uint

const (
	_S = _W / 8 // word size in bytes

	_W = bits.UintSize // word size in bits
	_B = 1 << _W       // digit base
	_M = _B - 1        // digit mask
)

// Many of the loops in this file are of the form
//   for i := 0; i < len(z) && i < len(x) && i < len(y); i++
// i < len(z) is the real condition.
// However, checking i < len(x) && i < len(y) as well is faster than
// having the compiler do a bounds check in the body of the loop;
// remarkably it is even faster than hoisting the bounds check
// out of the loop, by doing something like
//   _, _ = x[len(z)-1], y[len(z)-1]
// There are other ways to hoist the bounds check out of the loop,
// but the compiler's BCE isn't powerful enough for them (yet?).
// See the discussion in CL 164966.

// ----------------------------------------------------------------------------
// Elementary operations on words
//
// These operations are used by the vector operations below.

// z1<<_W + z0 = x*y
func mulWW_g(x, y Word) (z1, z0 Word) {
	hi, lo := bits.Mul(uint(x), uint(y))
	return Word(hi), Word(lo)
}

// z1<<_W + z0 = x*y + c
func mulAddWWW_g(x, y, c Word) (z1, z0 Word) {
	hi, lo := bits.Mul(uint(x), uint(y))
	var cc uint
	lo, cc = bits.Add(lo, uint(c), 0)
	return Word(hi + cc), Word(lo)
}

// nlz returns the number of leading zeros in x.
// Wraps bits.LeadingZeros call for convenience.
func nlz(x Word) uint {
	return uint(bits.LeadingZeros(uint(x)))
}

// The resulting carry c is either 0 or 1.
func addVV_g(z, x, y []Word) (c Word) {
	// The comment near the top of this file discusses this for loop condition.
	for i := 0; i < len(z) && i < len(x) && i < len(y); i++ {
		zi, cc := bits.Add(uint(x[i]), uint(y[i]), uint(c))
		z[i] = Word(zi)
		c = Word(cc)
	}
	return
}

// The resulting carry c is either 0 or 1.
func subVV_g(z, x, y []Word) (c Word) {
	// The comment near the top of this file discusses this for loop condition.
	for i := 0; i < len(z) && i < len(x) && i < len(y); i++ {
		zi, cc := bits.Sub(uint(x[i]), uint(y[i]), uint(c))
		z[i] = Word(zi)
		c = Word(cc)
	}
	return
}

// The resulting carry c is either 0 or 1.
func addVW_g(z, x []Word, y Word) (c Word) {
	c = y
	// The comment near the top of this file discusses this for loop condition.
	for i := 0; i < len(z) && i < len(x); i++ {
		zi, cc := bits.Add(uint(x[i]), uint(c), 0)
		z[i] = Word(zi)
		c = Word(cc)
	}
	return
}

// addVWlarge is addVW, but intended for large z.
// The only difference is that we check on every iteration
// whether we are done with carries,
// and if so, switch to a much faster copy instead.
// This is only a good idea for large z,
// because the overhead of the check and the function call
// outweigh the benefits when z is small.
func addVWlarge(z, x []Word, y Word) (c Word) {
	c = y
	// The comment near the top of this file discusses this for loop condition.
	for i := 0; i < len(z) && i < len(x); i++ {
		if c == 0 {
			copy(z[i:], x[i:])
			return
		}
		zi, cc := bits.Add(uint(x[i]), uint(c), 0)
		z[i] = Word(zi)
		c = Word(cc)
	}
	return
}

func subVW_g(z, x []Word, y Word) (c Word) {
	c = y
	// The comment near the top of this file discusses this for loop condition.
	for i := 0; i < len(z) && i < len(x); i++ {
		zi, cc := bits.Sub(uint(x[i]), uint(c), 0)
		z[i] = Word(zi)
		c = Word(cc)
	}
	return
}

// subVWlarge is to subVW as addVWlarge is to addVW.
func subVWlarge(z, x []Word, y Word) (c Word) {
	c = y
	// The comment near the top of this file discusses this for loop condition.
	for i := 0; i < len(z) && i < len(x); i++ {
		if c == 0 {
			copy(z[i:], x[i:])
			return
		}
		zi, cc := bits.Sub(uint(x[i]), uint(c), 0)
		z[i] = Word(zi)
		c = Word(cc)
	}
	return
}

func shlVU_g(z, x []Word, s uint) (c Word) {
	if s == 0 {
		copy(z, x)
		return
	}
	if len(z) == 0 {
		return
	}
	s &= _W - 1 // hint to the compiler that shifts by s don't need guard code
	ŝ := _W - s
	ŝ &= _W - 1 // ditto
	c = x[len(z)-1] >> ŝ
	for i := len(z) - 1; i > 0; i-- {
		z[i] = x[i]<<s | x[i-1]>>ŝ
	}
	z[0] = x[0] << s
	return
}

func shrVU_g(z, x []Word, s uint) (c Word) {
	if s == 0 {
		copy(z, x)
		return
	}
	if len(z) == 0 {
		return
	}
	s &= _W - 1 // hint to the compiler that shifts by s don't need guard code
	ŝ := _W - s
	ŝ &= _W - 1 // ditto
	c = x[0] << ŝ
	for i := 0; i < len(z)-1; i++ {
		z[i] = x[i]>>s | x[i+1]<<ŝ
	}
	z[len(z)-1] = x[len(z)-1] >> s
	return
}

func mulAddVWW_g(z, x []Word, y, r Word) (c Word) {
	c = r
	// The comment near the top of this file discusses this for loop condition.
	for i := 0; i < len(z) && i < len(x); i++ {
		c, z[i] = mulAddWWW_g(x[i], y, c)
	}
	return
}

func addMulVVW_g(z, x []Word, y Word) (c Word) {
	// The comment near the top of this file discusses this for loop condition.
	for i := 0; i < len(z) && i < len(x); i++ {
		z1, z0 := mulAddWWW_g(x[i], y, z[i])
		lo, cc := bits.Add(uint(z0), uint(c), 0)
		c, z[i] = Word(cc), Word(lo)
		c += z1
	}
	return
}

// q = ( x1 << _W + x0 - r)/y. m = floor(( _B^2 - 1 ) / d - _B). Requiring x1<y.
// An approximate reciprocal with a reference to "Improved Division by Invariant Integers
// (IEEE Transactions on Computers, 11 Jun. 2010)"
func divWW(x1, x0, y, m Word) (q, r Word) {
	s := nlz(y)
	if s != 0 {
		x1 = x1<<s | x0>>(_W-s)
		x0 <<= s
		y <<= s
	}
	d := uint(y)
	// We know that
	//   m = ⎣(B^2-1)/d⎦-B
	//   ⎣(B^2-1)/d⎦ = m+B
	//   (B^2-1)/d = m+B+delta1    0 <= delta1 <= (d-1)/d
	//   B^2/d = m+B+delta2        0 <= delta2 <= 1
	// The quotient we're trying to compute is
	//   quotient = ⎣(x1*B+x0)/d⎦
	//            = ⎣(x1*B*(B^2/d)+x0*(B^2/d))/B^2⎦
	//            = ⎣(x1*B*(m+B+delta2)+x0*(m+B+delta2))/B^2⎦
	//            = ⎣(x1*m+x1*B+x0)/B + x0*m/B^2 + delta2*(x1*B+x0)/B^2⎦
	// The latter two terms of this three-term sum are between 0 and 1.
	// So we can compute just the first term, and we will be low by at most 2.
	t1, t0 := bits.Mul(uint(m), uint(x1))
	_, c := bits.Add(t0, uint(x0), 0)
	t1, _ = bits.Add(t1, uint(x1), c)
	// The quotient is either t1, t1+1, or t1+2.
	// We'll try t1 and adjust if needed.
	qq := t1
	// compute remainder r=x-d*q.
	dq1, dq0 := bits.Mul(d, qq)
	r0, b := bits.Sub(uint(x0), dq0, 0)
	r1, _ := bits.Sub(uint(x1), dq1, b)
	// The remainder we just computed is bounded above by B+d:
	// r = x1*B + x0 - d*q.
	//   = x1*B + x0 - d*⎣(x1*m+x1*B+x0)/B⎦
	//   = x1*B + x0 - d*((x1*m+x1*B+x0)/B-alpha)                                   0 <= alpha < 1
	//   = x1*B + x0 - x1*d/B*m                         - x1*d - x0*d/B + d*alpha
	//   = x1*B + x0 - x1*d/B*⎣(B^2-1)/d-B⎦             - x1*d - x0*d/B + d*alpha
	//   = x1*B + x0 - x1*d/B*⎣(B^2-1)/d-B⎦             - x1*d - x0*d/B + d*alpha
	//   = x1*B + x0 - x1*d/B*((B^2-1)/d-B-beta)        - x1*d - x0*d/B + d*alpha   0 <= beta < 1
	//   = x1*B + x0 - x1*B + x1/B + x1*d + x1*d/B*beta - x1*d - x0*d/B + d*alpha
	//   =        x0        + x1/B        + x1*d/B*beta        - x0*d/B + d*alpha
	//   = x0*(1-d/B) + x1*(1+d*beta)/B + d*alpha
	//   <  B*(1-d/B) +  d*B/B          + d          because x0<B (and 1-d/B>0), x1<d, 1+d*beta<=B, alpha<1
	//   =  B - d     +  d              + d
	//   = B+d
	// So r1 can only be 0 or 1. If r1 is 1, then we know q was too small.
	// Add 1 to q and subtract d from r. That guarantees that r is <B, so
	// we no longer need to keep track of r1.
	if r1 != 0 {
		qq++
		r0 -= d
	}
	// If the remainder is still too large, increment q one more time.
	if r0 >= d {
		qq++
		r0 -= d
	}
	return Word(qq), Word(r0 >> s)
}

func divWVW(z []Word, xn Word, x []Word, y Word) (r Word) {
	r = xn
	if len(x) == 1 {
		qq, rr := bits.Div(uint(r), uint(x[0]), uint(y))
		z[0] = Word(qq)
		return Word(rr)
	}
	rec := reciprocalWord(y)
	for i := len(z) - 1; i >= 0; i-- {
		z[i], r = divWW(r, x[i], y, rec)
	}
	return r
}

// reciprocalWord return the reciprocal of the divisor. rec = floor(( _B^2 - 1 ) / u - _B). u = d1 << nlz(d1).
func reciprocalWord(d1 Word) Word {
	u := uint(d1 << nlz(d1))
	x1 := ^u
	x0 := uint(_M)
	rec, _ := bits.Div(x1, x0, u) // (_B^2-1)/U-_B = (_B*(_M-C)+_M)/U
	return Word(rec)
}