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 11const uintSize = 32 << (^uint(0) >> 32 & 1) // 32 or 64 12 13// UintSize is the size of a uint in bits. 14const UintSize = uintSize 15 16// --- LeadingZeros --- 17 18// LeadingZeros returns the number of leading zero bits in x; the result is UintSize for x == 0. 19func 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. 22func 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. 25func 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. 28func 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. 31func LeadingZeros64(x uint64) int { return 64 - Len64(x) } 32 33// --- TrailingZeros --- 34 35// See http://supertech.csail.mit.edu/papers/debruijn.pdf 36const deBruijn32 = 0x077CB531 37 38var 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 43const deBruijn64 = 0x03f79d71b4ca8b09 44 45var 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. 53func 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. 61func 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. 66func 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. 75func 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. 84func 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 104const m0 = 0x5555555555555555 // 01010101 ... 105const m1 = 0x3333333333333333 // 00110011 ... 106const m2 = 0x0f0f0f0f0f0f0f0f // 00001111 ... 107const m3 = 0x00ff00ff00ff00ff // etc. 108const m4 = 0x0000ffff0000ffff 109 110// OnesCount returns the number of one bits ("population count") in x. 111func 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. 119func OnesCount8(x uint8) int { 120 return int(pop8tab[x]) 121} 122 123// OnesCount16 returns the number of one bits ("population count") in x. 124func 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. 129func 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. 134func 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. 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). 179// 180// This function's execution time does not depend on the inputs. 181func 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. 191func 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. 201func 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. 211func 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. 220func 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. 228func Reverse8(x uint8) uint8 { 229 return rev8tab[x] 230} 231 232// Reverse16 returns the value of x with its bits in reversed order. 233func 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. 238func 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. 247func 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. 260func 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. 270func 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. 277func 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. 286func 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. 296func 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. 304func 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. 309func 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. 318func 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. 331func 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. 354func 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. 368func 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. 380func 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. 396func 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. 410func 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. 425func 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. 439func 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. 453func 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. 464func 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). 486func 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). 499func Div32(hi, lo, y uint32) (quo, rem uint32) { 500 if y != 0 && y <= hi { 501 panic(getOverflowError()) 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). 512func 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(getDivideError()) 519 } 520 if y <= hi { 521 panic(getOverflowError()) 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. 562func 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. 572func 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. 579func 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