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# Source file src/math/bits/bits.go

## Documentation: math/bits

```     1  // Copyright 2017 The Go Authors. All rights reserved.
2  // Use of this source code is governed by a BSD-style
3  // license that can be found in the LICENSE file.
4
5  //go:generate go run make_tables.go
6
7  // Package bits implements bit counting and manipulation
8  // functions for the predeclared unsigned integer types.
9  package bits
10
11  const uintSize = 32 << (^uint(0) >> 32 & 1) // 32 or 64
12
13  // UintSize is the size of a uint in bits.
14  const UintSize = uintSize
15
16  // --- LeadingZeros ---
17
18  // LeadingZeros returns the number of leading zero bits in x; the result is UintSize for x == 0.
19  func LeadingZeros(x uint) int { return UintSize - Len(x) }
20
21  // LeadingZeros8 returns the number of leading zero bits in x; the result is 8 for x == 0.
22  func LeadingZeros8(x uint8) int { return 8 - Len8(x) }
23
24  // LeadingZeros16 returns the number of leading zero bits in x; the result is 16 for x == 0.
25  func LeadingZeros16(x uint16) int { return 16 - Len16(x) }
26
27  // LeadingZeros32 returns the number of leading zero bits in x; the result is 32 for x == 0.
28  func LeadingZeros32(x uint32) int { return 32 - Len32(x) }
29
30  // LeadingZeros64 returns the number of leading zero bits in x; the result is 64 for x == 0.
31  func LeadingZeros64(x uint64) int { return 64 - Len64(x) }
32
33  // --- TrailingZeros ---
34
35  // See http://supertech.csail.mit.edu/papers/debruijn.pdf
36  const deBruijn32 = 0x077CB531
37
38  var deBruijn32tab = [32]byte{
39  	0, 1, 28, 2, 29, 14, 24, 3, 30, 22, 20, 15, 25, 17, 4, 8,
40  	31, 27, 13, 23, 21, 19, 16, 7, 26, 12, 18, 6, 11, 5, 10, 9,
41  }
42
43  const deBruijn64 = 0x03f79d71b4ca8b09
44
45  var deBruijn64tab = [64]byte{
46  	0, 1, 56, 2, 57, 49, 28, 3, 61, 58, 42, 50, 38, 29, 17, 4,
47  	62, 47, 59, 36, 45, 43, 51, 22, 53, 39, 33, 30, 24, 18, 12, 5,
48  	63, 55, 48, 27, 60, 41, 37, 16, 46, 35, 44, 21, 52, 32, 23, 11,
49  	54, 26, 40, 15, 34, 20, 31, 10, 25, 14, 19, 9, 13, 8, 7, 6,
50  }
51
52  // TrailingZeros returns the number of trailing zero bits in x; the result is UintSize for x == 0.
53  func TrailingZeros(x uint) int {
54  	if UintSize == 32 {
55  		return TrailingZeros32(uint32(x))
56  	}
57  	return TrailingZeros64(uint64(x))
58  }
59
60  // TrailingZeros8 returns the number of trailing zero bits in x; the result is 8 for x == 0.
61  func TrailingZeros8(x uint8) int {
62  	return int(ntz8tab[x])
63  }
64
65  // TrailingZeros16 returns the number of trailing zero bits in x; the result is 16 for x == 0.
66  func TrailingZeros16(x uint16) int {
67  	if x == 0 {
68  		return 16
69  	}
70  	// see comment in TrailingZeros64
71  	return int(deBruijn32tab[uint32(x&-x)*deBruijn32>>(32-5)])
72  }
73
74  // TrailingZeros32 returns the number of trailing zero bits in x; the result is 32 for x == 0.
75  func TrailingZeros32(x uint32) int {
76  	if x == 0 {
77  		return 32
78  	}
79  	// see comment in TrailingZeros64
80  	return int(deBruijn32tab[(x&-x)*deBruijn32>>(32-5)])
81  }
82
83  // TrailingZeros64 returns the number of trailing zero bits in x; the result is 64 for x == 0.
84  func TrailingZeros64(x uint64) int {
85  	if x == 0 {
86  		return 64
87  	}
88  	// If popcount is fast, replace code below with return popcount(^x & (x - 1)).
89  	//
90  	// x & -x leaves only the right-most bit set in the word. Let k be the
91  	// index of that bit. Since only a single bit is set, the value is two
92  	// to the power of k. Multiplying by a power of two is equivalent to
93  	// left shifting, in this case by k bits. The de Bruijn (64 bit) constant
94  	// is such that all six bit, consecutive substrings are distinct.
95  	// Therefore, if we have a left shifted version of this constant we can
96  	// find by how many bits it was shifted by looking at which six bit
97  	// substring ended up at the top of the word.
98  	// (Knuth, volume 4, section 7.3.1)
99  	return int(deBruijn64tab[(x&-x)*deBruijn64>>(64-6)])
100  }
101
102  // --- OnesCount ---
103
104  const m0 = 0x5555555555555555 // 01010101 ...
105  const m1 = 0x3333333333333333 // 00110011 ...
106  const m2 = 0x0f0f0f0f0f0f0f0f // 00001111 ...
107  const m3 = 0x00ff00ff00ff00ff // etc.
108  const m4 = 0x0000ffff0000ffff
109
110  // OnesCount returns the number of one bits ("population count") in x.
111  func OnesCount(x uint) int {
112  	if UintSize == 32 {
113  		return OnesCount32(uint32(x))
114  	}
115  	return OnesCount64(uint64(x))
116  }
117
118  // OnesCount8 returns the number of one bits ("population count") in x.
119  func OnesCount8(x uint8) int {
120  	return int(pop8tab[x])
121  }
122
123  // OnesCount16 returns the number of one bits ("population count") in x.
124  func OnesCount16(x uint16) int {
125  	return int(pop8tab[x>>8] + pop8tab[x&0xff])
126  }
127
128  // OnesCount32 returns the number of one bits ("population count") in x.
129  func OnesCount32(x uint32) int {
130  	return int(pop8tab[x>>24] + pop8tab[x>>16&0xff] + pop8tab[x>>8&0xff] + pop8tab[x&0xff])
131  }
132
133  // OnesCount64 returns the number of one bits ("population count") in x.
134  func OnesCount64(x uint64) int {
135  	// Implementation: Parallel summing of adjacent bits.
136  	// See "Hacker's Delight", Chap. 5: Counting Bits.
137  	// The following pattern shows the general approach:
138  	//
139  	//   x = x>>1&(m0&m) + x&(m0&m)
140  	//   x = x>>2&(m1&m) + x&(m1&m)
141  	//   x = x>>4&(m2&m) + x&(m2&m)
142  	//   x = x>>8&(m3&m) + x&(m3&m)
143  	//   x = x>>16&(m4&m) + x&(m4&m)
144  	//   x = x>>32&(m5&m) + x&(m5&m)
145  	//   return int(x)
146  	//
147  	// Masking (& operations) can be left away when there's no
148  	// danger that a field's sum will carry over into the next
149  	// field: Since the result cannot be > 64, 8 bits is enough
150  	// and we can ignore the masks for the shifts by 8 and up.
151  	// Per "Hacker's Delight", the first line can be simplified
152  	// more, but it saves at best one instruction, so we leave
153  	// it alone for clarity.
154  	const m = 1<<64 - 1
155  	x = x>>1&(m0&m) + x&(m0&m)
156  	x = x>>2&(m1&m) + x&(m1&m)
157  	x = (x>>4 + x) & (m2 & m)
158  	x += x >> 8
159  	x += x >> 16
160  	x += x >> 32
161  	return int(x) & (1<<7 - 1)
162  }
163
164  // --- RotateLeft ---
165
166  // RotateLeft returns the value of x rotated left by (k mod UintSize) bits.
167  // To rotate x right by k bits, call RotateLeft(x, -k).
168  //
169  // This function's execution time does not depend on the inputs.
170  func RotateLeft(x uint, k int) uint {
171  	if UintSize == 32 {
172  		return uint(RotateLeft32(uint32(x), k))
173  	}
174  	return uint(RotateLeft64(uint64(x), k))
175  }
176
177  // RotateLeft8 returns the value of x rotated left by (k mod 8) bits.
178  // To rotate x right by k bits, call RotateLeft8(x, -k).
179  //
180  // This function's execution time does not depend on the inputs.
181  func RotateLeft8(x uint8, k int) uint8 {
182  	const n = 8
183  	s := uint(k) & (n - 1)
184  	return x<<s | x>>(n-s)
185  }
186
187  // RotateLeft16 returns the value of x rotated left by (k mod 16) bits.
188  // To rotate x right by k bits, call RotateLeft16(x, -k).
189  //
190  // This function's execution time does not depend on the inputs.
191  func RotateLeft16(x uint16, k int) uint16 {
192  	const n = 16
193  	s := uint(k) & (n - 1)
194  	return x<<s | x>>(n-s)
195  }
196
197  // RotateLeft32 returns the value of x rotated left by (k mod 32) bits.
198  // To rotate x right by k bits, call RotateLeft32(x, -k).
199  //
200  // This function's execution time does not depend on the inputs.
201  func RotateLeft32(x uint32, k int) uint32 {
202  	const n = 32
203  	s := uint(k) & (n - 1)
204  	return x<<s | x>>(n-s)
205  }
206
207  // RotateLeft64 returns the value of x rotated left by (k mod 64) bits.
208  // To rotate x right by k bits, call RotateLeft64(x, -k).
209  //
210  // This function's execution time does not depend on the inputs.
211  func RotateLeft64(x uint64, k int) uint64 {
212  	const n = 64
213  	s := uint(k) & (n - 1)
214  	return x<<s | x>>(n-s)
215  }
216
217  // --- Reverse ---
218
219  // Reverse returns the value of x with its bits in reversed order.
220  func Reverse(x uint) uint {
221  	if UintSize == 32 {
222  		return uint(Reverse32(uint32(x)))
223  	}
224  	return uint(Reverse64(uint64(x)))
225  }
226
227  // Reverse8 returns the value of x with its bits in reversed order.
228  func Reverse8(x uint8) uint8 {
229  	return rev8tab[x]
230  }
231
232  // Reverse16 returns the value of x with its bits in reversed order.
233  func Reverse16(x uint16) uint16 {
234  	return uint16(rev8tab[x>>8]) | uint16(rev8tab[x&0xff])<<8
235  }
236
237  // Reverse32 returns the value of x with its bits in reversed order.
238  func Reverse32(x uint32) uint32 {
239  	const m = 1<<32 - 1
240  	x = x>>1&(m0&m) | x&(m0&m)<<1
241  	x = x>>2&(m1&m) | x&(m1&m)<<2
242  	x = x>>4&(m2&m) | x&(m2&m)<<4
243  	return ReverseBytes32(x)
244  }
245
246  // Reverse64 returns the value of x with its bits in reversed order.
247  func Reverse64(x uint64) uint64 {
248  	const m = 1<<64 - 1
249  	x = x>>1&(m0&m) | x&(m0&m)<<1
250  	x = x>>2&(m1&m) | x&(m1&m)<<2
251  	x = x>>4&(m2&m) | x&(m2&m)<<4
252  	return ReverseBytes64(x)
253  }
254
255  // --- ReverseBytes ---
256
257  // ReverseBytes returns the value of x with its bytes in reversed order.
258  //
259  // This function's execution time does not depend on the inputs.
260  func ReverseBytes(x uint) uint {
261  	if UintSize == 32 {
262  		return uint(ReverseBytes32(uint32(x)))
263  	}
264  	return uint(ReverseBytes64(uint64(x)))
265  }
266
267  // ReverseBytes16 returns the value of x with its bytes in reversed order.
268  //
269  // This function's execution time does not depend on the inputs.
270  func ReverseBytes16(x uint16) uint16 {
271  	return x>>8 | x<<8
272  }
273
274  // ReverseBytes32 returns the value of x with its bytes in reversed order.
275  //
276  // This function's execution time does not depend on the inputs.
277  func ReverseBytes32(x uint32) uint32 {
278  	const m = 1<<32 - 1
279  	x = x>>8&(m3&m) | x&(m3&m)<<8
280  	return x>>16 | x<<16
281  }
282
283  // ReverseBytes64 returns the value of x with its bytes in reversed order.
284  //
285  // This function's execution time does not depend on the inputs.
286  func ReverseBytes64(x uint64) uint64 {
287  	const m = 1<<64 - 1
288  	x = x>>8&(m3&m) | x&(m3&m)<<8
289  	x = x>>16&(m4&m) | x&(m4&m)<<16
290  	return x>>32 | x<<32
291  }
292
293  // --- Len ---
294
295  // Len returns the minimum number of bits required to represent x; the result is 0 for x == 0.
296  func Len(x uint) int {
297  	if UintSize == 32 {
298  		return Len32(uint32(x))
299  	}
300  	return Len64(uint64(x))
301  }
302
303  // Len8 returns the minimum number of bits required to represent x; the result is 0 for x == 0.
304  func Len8(x uint8) int {
305  	return int(len8tab[x])
306  }
307
308  // Len16 returns the minimum number of bits required to represent x; the result is 0 for x == 0.
309  func Len16(x uint16) (n int) {
310  	if x >= 1<<8 {
311  		x >>= 8
312  		n = 8
313  	}
314  	return n + int(len8tab[x])
315  }
316
317  // Len32 returns the minimum number of bits required to represent x; the result is 0 for x == 0.
318  func Len32(x uint32) (n int) {
319  	if x >= 1<<16 {
320  		x >>= 16
321  		n = 16
322  	}
323  	if x >= 1<<8 {
324  		x >>= 8
325  		n += 8
326  	}
327  	return n + int(len8tab[x])
328  }
329
330  // Len64 returns the minimum number of bits required to represent x; the result is 0 for x == 0.
331  func Len64(x uint64) (n int) {
332  	if x >= 1<<32 {
333  		x >>= 32
334  		n = 32
335  	}
336  	if x >= 1<<16 {
337  		x >>= 16
338  		n += 16
339  	}
340  	if x >= 1<<8 {
341  		x >>= 8
342  		n += 8
343  	}
344  	return n + int(len8tab[x])
345  }
346
347  // --- Add with carry ---
348
349  // Add returns the sum with carry of x, y and carry: sum = x + y + carry.
350  // The carry input must be 0 or 1; otherwise the behavior is undefined.
351  // The carryOut output is guaranteed to be 0 or 1.
352  //
353  // This function's execution time does not depend on the inputs.
354  func Add(x, y, carry uint) (sum, carryOut uint) {
355  	if UintSize == 32 {
356  		s32, c32 := Add32(uint32(x), uint32(y), uint32(carry))
357  		return uint(s32), uint(c32)
358  	}
359  	s64, c64 := Add64(uint64(x), uint64(y), uint64(carry))
360  	return uint(s64), uint(c64)
361  }
362
363  // Add32 returns the sum with carry of x, y and carry: sum = x + y + carry.
364  // The carry input must be 0 or 1; otherwise the behavior is undefined.
365  // The carryOut output is guaranteed to be 0 or 1.
366  //
367  // This function's execution time does not depend on the inputs.
368  func Add32(x, y, carry uint32) (sum, carryOut uint32) {
369  	sum64 := uint64(x) + uint64(y) + uint64(carry)
370  	sum = uint32(sum64)
371  	carryOut = uint32(sum64 >> 32)
372  	return
373  }
374
375  // Add64 returns the sum with carry of x, y and carry: sum = x + y + carry.
376  // The carry input must be 0 or 1; otherwise the behavior is undefined.
377  // The carryOut output is guaranteed to be 0 or 1.
378  //
379  // This function's execution time does not depend on the inputs.
380  func Add64(x, y, carry uint64) (sum, carryOut uint64) {
381  	sum = x + y + carry
382  	// The sum will overflow if both top bits are set (x & y) or if one of them
383  	// is (x | y), and a carry from the lower place happened. If such a carry
384  	// happens, the top bit will be 1 + 0 + 1 = 0 (&^ sum).
385  	carryOut = ((x & y) | ((x | y) &^ sum)) >> 63
386  	return
387  }
388
389  // --- Subtract with borrow ---
390
391  // Sub returns the difference of x, y and borrow: diff = x - y - borrow.
392  // The borrow input must be 0 or 1; otherwise the behavior is undefined.
393  // The borrowOut output is guaranteed to be 0 or 1.
394  //
395  // This function's execution time does not depend on the inputs.
396  func Sub(x, y, borrow uint) (diff, borrowOut uint) {
397  	if UintSize == 32 {
398  		d32, b32 := Sub32(uint32(x), uint32(y), uint32(borrow))
399  		return uint(d32), uint(b32)
400  	}
401  	d64, b64 := Sub64(uint64(x), uint64(y), uint64(borrow))
402  	return uint(d64), uint(b64)
403  }
404
405  // Sub32 returns the difference of x, y and borrow, diff = x - y - borrow.
406  // The borrow input must be 0 or 1; otherwise the behavior is undefined.
407  // The borrowOut output is guaranteed to be 0 or 1.
408  //
409  // This function's execution time does not depend on the inputs.
410  func Sub32(x, y, borrow uint32) (diff, borrowOut uint32) {
411  	diff = x - y - borrow
412  	// The difference will underflow if the top bit of x is not set and the top
413  	// bit of y is set (^x & y) or if they are the same (^(x ^ y)) and a borrow
414  	// from the lower place happens. If that borrow happens, the result will be
415  	// 1 - 1 - 1 = 0 - 0 - 1 = 1 (& diff).
416  	borrowOut = ((^x & y) | (^(x ^ y) & diff)) >> 31
417  	return
418  }
419
420  // Sub64 returns the difference of x, y and borrow: diff = x - y - borrow.
421  // The borrow input must be 0 or 1; otherwise the behavior is undefined.
422  // The borrowOut output is guaranteed to be 0 or 1.
423  //
424  // This function's execution time does not depend on the inputs.
425  func Sub64(x, y, borrow uint64) (diff, borrowOut uint64) {
426  	diff = x - y - borrow
427  	// See Sub32 for the bit logic.
428  	borrowOut = ((^x & y) | (^(x ^ y) & diff)) >> 63
429  	return
430  }
431
432  // --- Full-width multiply ---
433
434  // Mul returns the full-width product of x and y: (hi, lo) = x * y
435  // with the product bits' upper half returned in hi and the lower
436  // half returned in lo.
437  //
438  // This function's execution time does not depend on the inputs.
439  func Mul(x, y uint) (hi, lo uint) {
440  	if UintSize == 32 {
441  		h, l := Mul32(uint32(x), uint32(y))
442  		return uint(h), uint(l)
443  	}
444  	h, l := Mul64(uint64(x), uint64(y))
445  	return uint(h), uint(l)
446  }
447
448  // Mul32 returns the 64-bit product of x and y: (hi, lo) = x * y
449  // with the product bits' upper half returned in hi and the lower
450  // half returned in lo.
451  //
452  // This function's execution time does not depend on the inputs.
453  func Mul32(x, y uint32) (hi, lo uint32) {
454  	tmp := uint64(x) * uint64(y)
455  	hi, lo = uint32(tmp>>32), uint32(tmp)
456  	return
457  }
458
459  // Mul64 returns the 128-bit product of x and y: (hi, lo) = x * y
460  // with the product bits' upper half returned in hi and the lower
461  // half returned in lo.
462  //
463  // This function's execution time does not depend on the inputs.
464  func Mul64(x, y uint64) (hi, lo uint64) {
465  	const mask32 = 1<<32 - 1
466  	x0 := x & mask32
467  	x1 := x >> 32
468  	y0 := y & mask32
469  	y1 := y >> 32
470  	w0 := x0 * y0
471  	t := x1*y0 + w0>>32
472  	w1 := t & mask32
473  	w2 := t >> 32
474  	w1 += x0 * y1
475  	hi = x1*y1 + w2 + w1>>32
476  	lo = x * y
477  	return
478  }
479
480  // --- Full-width divide ---
481
482  // Div returns the quotient and remainder of (hi, lo) divided by y:
483  // quo = (hi, lo)/y, rem = (hi, lo)%y with the dividend bits' upper
484  // half in parameter hi and the lower half in parameter lo.
485  // Div panics for y == 0 (division by zero) or y <= hi (quotient overflow).
486  func Div(hi, lo, y uint) (quo, rem uint) {
487  	if UintSize == 32 {
488  		q, r := Div32(uint32(hi), uint32(lo), uint32(y))
489  		return uint(q), uint(r)
490  	}
491  	q, r := Div64(uint64(hi), uint64(lo), uint64(y))
492  	return uint(q), uint(r)
493  }
494
495  // Div32 returns the quotient and remainder of (hi, lo) divided by y:
496  // quo = (hi, lo)/y, rem = (hi, lo)%y with the dividend bits' upper
497  // half in parameter hi and the lower half in parameter lo.
498  // Div32 panics for y == 0 (division by zero) or y <= hi (quotient overflow).
499  func Div32(hi, lo, y uint32) (quo, rem uint32) {
500  	if y != 0 && y <= hi {
501  		panic(overflowError)
502  	}
503  	z := uint64(hi)<<32 | uint64(lo)
504  	quo, rem = uint32(z/uint64(y)), uint32(z%uint64(y))
505  	return
506  }
507
508  // Div64 returns the quotient and remainder of (hi, lo) divided by y:
509  // quo = (hi, lo)/y, rem = (hi, lo)%y with the dividend bits' upper
510  // half in parameter hi and the lower half in parameter lo.
511  // Div64 panics for y == 0 (division by zero) or y <= hi (quotient overflow).
512  func Div64(hi, lo, y uint64) (quo, rem uint64) {
513  	const (
514  		two32  = 1 << 32
515  		mask32 = two32 - 1
516  	)
517  	if y == 0 {
518  		panic(divideError)
519  	}
520  	if y <= hi {
521  		panic(overflowError)
522  	}
523
524  	s := uint(LeadingZeros64(y))
525  	y <<= s
526
527  	yn1 := y >> 32
528  	yn0 := y & mask32
529  	un32 := hi<<s | lo>>(64-s)
530  	un10 := lo << s
531  	un1 := un10 >> 32
532  	un0 := un10 & mask32
533  	q1 := un32 / yn1
534  	rhat := un32 - q1*yn1
535
536  	for q1 >= two32 || q1*yn0 > two32*rhat+un1 {
537  		q1--
538  		rhat += yn1
539  		if rhat >= two32 {
540  			break
541  		}
542  	}
543
544  	un21 := un32*two32 + un1 - q1*y
545  	q0 := un21 / yn1
546  	rhat = un21 - q0*yn1
547
548  	for q0 >= two32 || q0*yn0 > two32*rhat+un0 {
549  		q0--
550  		rhat += yn1
551  		if rhat >= two32 {
552  			break
553  		}
554  	}
555
556  	return q1*two32 + q0, (un21*two32 + un0 - q0*y) >> s
557  }
558
559  // Rem returns the remainder of (hi, lo) divided by y. Rem panics for
560  // y == 0 (division by zero) but, unlike Div, it doesn't panic on a
561  // quotient overflow.
562  func Rem(hi, lo, y uint) uint {
563  	if UintSize == 32 {
564  		return uint(Rem32(uint32(hi), uint32(lo), uint32(y)))
565  	}
566  	return uint(Rem64(uint64(hi), uint64(lo), uint64(y)))
567  }
568
569  // Rem32 returns the remainder of (hi, lo) divided by y. Rem32 panics
570  // for y == 0 (division by zero) but, unlike Div32, it doesn't panic
571  // on a quotient overflow.
572  func Rem32(hi, lo, y uint32) uint32 {
573  	return uint32((uint64(hi)<<32 | uint64(lo)) % uint64(y))
574  }
575
576  // Rem64 returns the remainder of (hi, lo) divided by y. Rem64 panics
577  // for y == 0 (division by zero) but, unlike Div64, it doesn't panic
578  // on a quotient overflow.
579  func Rem64(hi, lo, y uint64) uint64 {
580  	// We scale down hi so that hi < y, then use Div64 to compute the
581  	// rem with the guarantee that it won't panic on quotient overflow.
582  	// Given that
583  	//   hi ≡ hi%y    (mod y)
584  	// we have
585  	//   hi<<64 + lo ≡ (hi%y)<<64 + lo    (mod y)
586  	_, rem := Div64(hi%y, lo, y)
587  	return rem
588  }
589
```

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