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