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

## Documentation: math/bits

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