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) >> 63) // 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