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.
9package bits
10
11import _ "unsafe" // for go:linkname
12
13const uintSize = 32 << (^uint(0) >> 32 & 1) // 32 or 64
14
15// UintSize is the size of a uint in bits.
16const UintSize = uintSize
17
18// --- LeadingZeros ---
19
20// LeadingZeros returns the number of leading zero bits in x; the result is UintSize for x == 0.
21func 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.
24func 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.
27func 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.
30func 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.
33func LeadingZeros64(x uint64) int { return 64 - Len64(x) }
34
35// --- TrailingZeros ---
36
37// See http://supertech.csail.mit.edu/papers/debruijn.pdf
38const deBruijn32 = 0x077CB531
39
40var 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
45const deBruijn64 = 0x03f79d71b4ca8b09
46
47var 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.
55func 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.
63func 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.
68func 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.
77func 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.
86func 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
106const m0 = 0x5555555555555555 // 01010101 ...
107const m1 = 0x3333333333333333 // 00110011 ...
108const m2 = 0x0f0f0f0f0f0f0f0f // 00001111 ...
109const m3 = 0x00ff00ff00ff00ff // etc.
110const m4 = 0x0000ffff0000ffff
111
112// OnesCount returns the number of one bits ("population count") in x.
113func 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.
121func OnesCount8(x uint8) int {
122	return int(pop8tab[x])
123}
124
125// OnesCount16 returns the number of one bits ("population count") in x.
126func 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.
131func 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.
136func 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).
170func 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).
179func 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).
187func 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).
195func 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).
203func 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.
212func 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.
220func Reverse8(x uint8) uint8 {
221	return rev8tab[x]
222}
223
224// Reverse16 returns the value of x with its bits in reversed order.
225func 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.
230func 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.
240func 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.
253func 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.
261func 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.
266func 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.
273func 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.
283func 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.
291func 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.
296func 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.
305func 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.
318func 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.
339func 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.
351func 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.
363func 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.
377func 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.
389func 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.
401func 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.
415func 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.
427func 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.
436func 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).
458func 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).
471func Div32(hi, lo, y uint32) (quo, rem uint32) {
472	if y != 0 && y <= hi {
473		panic(getOverflowError())
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).
484func 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(getDivideError())
491	}
492	if y <= hi {
493		panic(getOverflowError())
494	}
495
496	s := uint(LeadingZeros64(y))
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
531//go:linkname getOverflowError runtime.getOverflowError
532func getOverflowError() error
533
534//go:linkname getDivideError runtime.getDivideError
535func getDivideError() error
536