| // Copyright 2017 The Go Authors. All rights reserved. |
| // Use of this source code is governed by a BSD-style |
| // license that can be found in the LICENSE file. |
| |
| //go:generate go run make_tables.go |
| |
| // Package bits implements bit counting and manipulation |
| // functions for the predeclared unsigned integer types. |
| // |
| // Functions in this package may be implemented directly by |
| // the compiler, for better performance. For those functions |
| // the code in this package will not be used. Which |
| // functions are implemented by the compiler depends on the |
| // architecture and the Go release. |
| package bits |
| |
| const uintSize = 32 << (^uint(0) >> 63) // 32 or 64 |
| |
| // UintSize is the size of a uint in bits. |
| const UintSize = uintSize |
| |
| // --- LeadingZeros --- |
| |
| // LeadingZeros returns the number of leading zero bits in x; the result is UintSize for x == 0. |
| func LeadingZeros(x uint) int { return UintSize - Len(x) } |
| |
| // LeadingZeros8 returns the number of leading zero bits in x; the result is 8 for x == 0. |
| func LeadingZeros8(x uint8) int { return 8 - Len8(x) } |
| |
| // LeadingZeros16 returns the number of leading zero bits in x; the result is 16 for x == 0. |
| func LeadingZeros16(x uint16) int { return 16 - Len16(x) } |
| |
| // LeadingZeros32 returns the number of leading zero bits in x; the result is 32 for x == 0. |
| func LeadingZeros32(x uint32) int { return 32 - Len32(x) } |
| |
| // LeadingZeros64 returns the number of leading zero bits in x; the result is 64 for x == 0. |
| func LeadingZeros64(x uint64) int { return 64 - Len64(x) } |
| |
| // --- TrailingZeros --- |
| |
| // See http://supertech.csail.mit.edu/papers/debruijn.pdf |
| const deBruijn32 = 0x077CB531 |
| |
| var deBruijn32tab = [32]byte{ |
| 0, 1, 28, 2, 29, 14, 24, 3, 30, 22, 20, 15, 25, 17, 4, 8, |
| 31, 27, 13, 23, 21, 19, 16, 7, 26, 12, 18, 6, 11, 5, 10, 9, |
| } |
| |
| const deBruijn64 = 0x03f79d71b4ca8b09 |
| |
| var deBruijn64tab = [64]byte{ |
| 0, 1, 56, 2, 57, 49, 28, 3, 61, 58, 42, 50, 38, 29, 17, 4, |
| 62, 47, 59, 36, 45, 43, 51, 22, 53, 39, 33, 30, 24, 18, 12, 5, |
| 63, 55, 48, 27, 60, 41, 37, 16, 46, 35, 44, 21, 52, 32, 23, 11, |
| 54, 26, 40, 15, 34, 20, 31, 10, 25, 14, 19, 9, 13, 8, 7, 6, |
| } |
| |
| // TrailingZeros returns the number of trailing zero bits in x; the result is UintSize for x == 0. |
| func TrailingZeros(x uint) int { |
| if UintSize == 32 { |
| return TrailingZeros32(uint32(x)) |
| } |
| return TrailingZeros64(uint64(x)) |
| } |
| |
| // TrailingZeros8 returns the number of trailing zero bits in x; the result is 8 for x == 0. |
| func TrailingZeros8(x uint8) int { |
| return int(ntz8tab[x]) |
| } |
| |
| // TrailingZeros16 returns the number of trailing zero bits in x; the result is 16 for x == 0. |
| func TrailingZeros16(x uint16) int { |
| if x == 0 { |
| return 16 |
| } |
| // see comment in TrailingZeros64 |
| return int(deBruijn32tab[uint32(x&-x)*deBruijn32>>(32-5)]) |
| } |
| |
| // TrailingZeros32 returns the number of trailing zero bits in x; the result is 32 for x == 0. |
| func TrailingZeros32(x uint32) int { |
| if x == 0 { |
| return 32 |
| } |
| // see comment in TrailingZeros64 |
| return int(deBruijn32tab[(x&-x)*deBruijn32>>(32-5)]) |
| } |
| |
| // TrailingZeros64 returns the number of trailing zero bits in x; the result is 64 for x == 0. |
| func TrailingZeros64(x uint64) int { |
| if x == 0 { |
| return 64 |
| } |
| // If popcount is fast, replace code below with return popcount(^x & (x - 1)). |
| // |
| // x & -x leaves only the right-most bit set in the word. Let k be the |
| // index of that bit. Since only a single bit is set, the value is two |
| // to the power of k. Multiplying by a power of two is equivalent to |
| // left shifting, in this case by k bits. The de Bruijn (64 bit) constant |
| // is such that all six bit, consecutive substrings are distinct. |
| // Therefore, if we have a left shifted version of this constant we can |
| // find by how many bits it was shifted by looking at which six bit |
| // substring ended up at the top of the word. |
| // (Knuth, volume 4, section 7.3.1) |
| return int(deBruijn64tab[(x&-x)*deBruijn64>>(64-6)]) |
| } |
| |
| // --- OnesCount --- |
| |
| const m0 = 0x5555555555555555 // 01010101 ... |
| const m1 = 0x3333333333333333 // 00110011 ... |
| const m2 = 0x0f0f0f0f0f0f0f0f // 00001111 ... |
| const m3 = 0x00ff00ff00ff00ff // etc. |
| const m4 = 0x0000ffff0000ffff |
| |
| // OnesCount returns the number of one bits ("population count") in x. |
| func OnesCount(x uint) int { |
| if UintSize == 32 { |
| return OnesCount32(uint32(x)) |
| } |
| return OnesCount64(uint64(x)) |
| } |
| |
| // OnesCount8 returns the number of one bits ("population count") in x. |
| func OnesCount8(x uint8) int { |
| return int(pop8tab[x]) |
| } |
| |
| // OnesCount16 returns the number of one bits ("population count") in x. |
| func OnesCount16(x uint16) int { |
| return int(pop8tab[x>>8] + pop8tab[x&0xff]) |
| } |
| |
| // OnesCount32 returns the number of one bits ("population count") in x. |
| func OnesCount32(x uint32) int { |
| return int(pop8tab[x>>24] + pop8tab[x>>16&0xff] + pop8tab[x>>8&0xff] + pop8tab[x&0xff]) |
| } |
| |
| // OnesCount64 returns the number of one bits ("population count") in x. |
| func OnesCount64(x uint64) int { |
| // Implementation: Parallel summing of adjacent bits. |
| // See "Hacker's Delight", Chap. 5: Counting Bits. |
| // The following pattern shows the general approach: |
| // |
| // x = x>>1&(m0&m) + x&(m0&m) |
| // x = x>>2&(m1&m) + x&(m1&m) |
| // x = x>>4&(m2&m) + x&(m2&m) |
| // x = x>>8&(m3&m) + x&(m3&m) |
| // x = x>>16&(m4&m) + x&(m4&m) |
| // x = x>>32&(m5&m) + x&(m5&m) |
| // return int(x) |
| // |
| // Masking (& operations) can be left away when there's no |
| // danger that a field's sum will carry over into the next |
| // field: Since the result cannot be > 64, 8 bits is enough |
| // and we can ignore the masks for the shifts by 8 and up. |
| // Per "Hacker's Delight", the first line can be simplified |
| // more, but it saves at best one instruction, so we leave |
| // it alone for clarity. |
| const m = 1<<64 - 1 |
| x = x>>1&(m0&m) + x&(m0&m) |
| x = x>>2&(m1&m) + x&(m1&m) |
| x = (x>>4 + x) & (m2 & m) |
| x += x >> 8 |
| x += x >> 16 |
| x += x >> 32 |
| return int(x) & (1<<7 - 1) |
| } |
| |
| // --- RotateLeft --- |
| |
| // RotateLeft returns the value of x rotated left by (k mod UintSize) bits. |
| // To rotate x right by k bits, call RotateLeft(x, -k). |
| // |
| // This function's execution time does not depend on the inputs. |
| func RotateLeft(x uint, k int) uint { |
| if UintSize == 32 { |
| return uint(RotateLeft32(uint32(x), k)) |
| } |
| return uint(RotateLeft64(uint64(x), k)) |
| } |
| |
| // RotateLeft8 returns the value of x rotated left by (k mod 8) bits. |
| // To rotate x right by k bits, call RotateLeft8(x, -k). |
| // |
| // This function's execution time does not depend on the inputs. |
| func RotateLeft8(x uint8, k int) uint8 { |
| const n = 8 |
| s := uint(k) & (n - 1) |
| return x<<s | x>>(n-s) |
| } |
| |
| // RotateLeft16 returns the value of x rotated left by (k mod 16) bits. |
| // To rotate x right by k bits, call RotateLeft16(x, -k). |
| // |
| // This function's execution time does not depend on the inputs. |
| func RotateLeft16(x uint16, k int) uint16 { |
| const n = 16 |
| s := uint(k) & (n - 1) |
| return x<<s | x>>(n-s) |
| } |
| |
| // RotateLeft32 returns the value of x rotated left by (k mod 32) bits. |
| // To rotate x right by k bits, call RotateLeft32(x, -k). |
| // |
| // This function's execution time does not depend on the inputs. |
| func RotateLeft32(x uint32, k int) uint32 { |
| const n = 32 |
| s := uint(k) & (n - 1) |
| return x<<s | x>>(n-s) |
| } |
| |
| // RotateLeft64 returns the value of x rotated left by (k mod 64) bits. |
| // To rotate x right by k bits, call RotateLeft64(x, -k). |
| // |
| // This function's execution time does not depend on the inputs. |
| func RotateLeft64(x uint64, k int) uint64 { |
| const n = 64 |
| s := uint(k) & (n - 1) |
| return x<<s | x>>(n-s) |
| } |
| |
| // --- Reverse --- |
| |
| // Reverse returns the value of x with its bits in reversed order. |
| func Reverse(x uint) uint { |
| if UintSize == 32 { |
| return uint(Reverse32(uint32(x))) |
| } |
| return uint(Reverse64(uint64(x))) |
| } |
| |
| // Reverse8 returns the value of x with its bits in reversed order. |
| func Reverse8(x uint8) uint8 { |
| return rev8tab[x] |
| } |
| |
| // Reverse16 returns the value of x with its bits in reversed order. |
| func Reverse16(x uint16) uint16 { |
| return uint16(rev8tab[x>>8]) | uint16(rev8tab[x&0xff])<<8 |
| } |
| |
| // Reverse32 returns the value of x with its bits in reversed order. |
| func Reverse32(x uint32) uint32 { |
| const m = 1<<32 - 1 |
| x = x>>1&(m0&m) | x&(m0&m)<<1 |
| x = x>>2&(m1&m) | x&(m1&m)<<2 |
| x = x>>4&(m2&m) | x&(m2&m)<<4 |
| return ReverseBytes32(x) |
| } |
| |
| // Reverse64 returns the value of x with its bits in reversed order. |
| func Reverse64(x uint64) uint64 { |
| const m = 1<<64 - 1 |
| x = x>>1&(m0&m) | x&(m0&m)<<1 |
| x = x>>2&(m1&m) | x&(m1&m)<<2 |
| x = x>>4&(m2&m) | x&(m2&m)<<4 |
| return ReverseBytes64(x) |
| } |
| |
| // --- ReverseBytes --- |
| |
| // ReverseBytes returns the value of x with its bytes in reversed order. |
| // |
| // This function's execution time does not depend on the inputs. |
| func ReverseBytes(x uint) uint { |
| if UintSize == 32 { |
| return uint(ReverseBytes32(uint32(x))) |
| } |
| return uint(ReverseBytes64(uint64(x))) |
| } |
| |
| // ReverseBytes16 returns the value of x with its bytes in reversed order. |
| // |
| // This function's execution time does not depend on the inputs. |
| func ReverseBytes16(x uint16) uint16 { |
| return x>>8 | x<<8 |
| } |
| |
| // ReverseBytes32 returns the value of x with its bytes in reversed order. |
| // |
| // This function's execution time does not depend on the inputs. |
| func ReverseBytes32(x uint32) uint32 { |
| const m = 1<<32 - 1 |
| x = x>>8&(m3&m) | x&(m3&m)<<8 |
| return x>>16 | x<<16 |
| } |
| |
| // ReverseBytes64 returns the value of x with its bytes in reversed order. |
| // |
| // This function's execution time does not depend on the inputs. |
| func ReverseBytes64(x uint64) uint64 { |
| const m = 1<<64 - 1 |
| x = x>>8&(m3&m) | x&(m3&m)<<8 |
| x = x>>16&(m4&m) | x&(m4&m)<<16 |
| return x>>32 | x<<32 |
| } |
| |
| // --- Len --- |
| |
| // Len returns the minimum number of bits required to represent x; the result is 0 for x == 0. |
| func Len(x uint) int { |
| if UintSize == 32 { |
| return Len32(uint32(x)) |
| } |
| return Len64(uint64(x)) |
| } |
| |
| // Len8 returns the minimum number of bits required to represent x; the result is 0 for x == 0. |
| func Len8(x uint8) int { |
| return int(len8tab[x]) |
| } |
| |
| // Len16 returns the minimum number of bits required to represent x; the result is 0 for x == 0. |
| func Len16(x uint16) (n int) { |
| if x >= 1<<8 { |
| x >>= 8 |
| n = 8 |
| } |
| return n + int(len8tab[x]) |
| } |
| |
| // Len32 returns the minimum number of bits required to represent x; the result is 0 for x == 0. |
| func Len32(x uint32) (n int) { |
| if x >= 1<<16 { |
| x >>= 16 |
| n = 16 |
| } |
| if x >= 1<<8 { |
| x >>= 8 |
| n += 8 |
| } |
| return n + int(len8tab[x]) |
| } |
| |
| // Len64 returns the minimum number of bits required to represent x; the result is 0 for x == 0. |
| func Len64(x uint64) (n int) { |
| if x >= 1<<32 { |
| x >>= 32 |
| n = 32 |
| } |
| if x >= 1<<16 { |
| x >>= 16 |
| n += 16 |
| } |
| if x >= 1<<8 { |
| x >>= 8 |
| n += 8 |
| } |
| return n + int(len8tab[x]) |
| } |
| |
| // --- Add with carry --- |
| |
| // Add returns the sum with carry of x, y and carry: sum = x + y + carry. |
| // The carry input must be 0 or 1; otherwise the behavior is undefined. |
| // The carryOut output is guaranteed to be 0 or 1. |
| // |
| // This function's execution time does not depend on the inputs. |
| func Add(x, y, carry uint) (sum, carryOut uint) { |
| if UintSize == 32 { |
| s32, c32 := Add32(uint32(x), uint32(y), uint32(carry)) |
| return uint(s32), uint(c32) |
| } |
| s64, c64 := Add64(uint64(x), uint64(y), uint64(carry)) |
| return uint(s64), uint(c64) |
| } |
| |
| // Add32 returns the sum with carry of x, y and carry: sum = x + y + carry. |
| // The carry input must be 0 or 1; otherwise the behavior is undefined. |
| // The carryOut output is guaranteed to be 0 or 1. |
| // |
| // This function's execution time does not depend on the inputs. |
| func Add32(x, y, carry uint32) (sum, carryOut uint32) { |
| sum64 := uint64(x) + uint64(y) + uint64(carry) |
| sum = uint32(sum64) |
| carryOut = uint32(sum64 >> 32) |
| return |
| } |
| |
| // Add64 returns the sum with carry of x, y and carry: sum = x + y + carry. |
| // The carry input must be 0 or 1; otherwise the behavior is undefined. |
| // The carryOut output is guaranteed to be 0 or 1. |
| // |
| // This function's execution time does not depend on the inputs. |
| func Add64(x, y, carry uint64) (sum, carryOut uint64) { |
| sum = x + y + carry |
| // The sum will overflow if both top bits are set (x & y) or if one of them |
| // is (x | y), and a carry from the lower place happened. If such a carry |
| // happens, the top bit will be 1 + 0 + 1 = 0 (&^ sum). |
| carryOut = ((x & y) | ((x | y) &^ sum)) >> 63 |
| return |
| } |
| |
| // --- Subtract with borrow --- |
| |
| // Sub returns the difference of x, y and borrow: diff = x - y - borrow. |
| // The borrow input must be 0 or 1; otherwise the behavior is undefined. |
| // The borrowOut output is guaranteed to be 0 or 1. |
| // |
| // This function's execution time does not depend on the inputs. |
| func Sub(x, y, borrow uint) (diff, borrowOut uint) { |
| if UintSize == 32 { |
| d32, b32 := Sub32(uint32(x), uint32(y), uint32(borrow)) |
| return uint(d32), uint(b32) |
| } |
| d64, b64 := Sub64(uint64(x), uint64(y), uint64(borrow)) |
| return uint(d64), uint(b64) |
| } |
| |
| // Sub32 returns the difference of x, y and borrow, diff = x - y - borrow. |
| // The borrow input must be 0 or 1; otherwise the behavior is undefined. |
| // The borrowOut output is guaranteed to be 0 or 1. |
| // |
| // This function's execution time does not depend on the inputs. |
| func Sub32(x, y, borrow uint32) (diff, borrowOut uint32) { |
| diff = x - y - borrow |
| // The difference will underflow if the top bit of x is not set and the top |
| // bit of y is set (^x & y) or if they are the same (^(x ^ y)) and a borrow |
| // from the lower place happens. If that borrow happens, the result will be |
| // 1 - 1 - 1 = 0 - 0 - 1 = 1 (& diff). |
| borrowOut = ((^x & y) | (^(x ^ y) & diff)) >> 31 |
| return |
| } |
| |
| // Sub64 returns the difference of x, y and borrow: diff = x - y - borrow. |
| // The borrow input must be 0 or 1; otherwise the behavior is undefined. |
| // The borrowOut output is guaranteed to be 0 or 1. |
| // |
| // This function's execution time does not depend on the inputs. |
| func Sub64(x, y, borrow uint64) (diff, borrowOut uint64) { |
| diff = x - y - borrow |
| // See Sub32 for the bit logic. |
| borrowOut = ((^x & y) | (^(x ^ y) & diff)) >> 63 |
| return |
| } |
| |
| // --- Full-width multiply --- |
| |
| // Mul returns the full-width product of x and y: (hi, lo) = x * y |
| // with the product bits' upper half returned in hi and the lower |
| // half returned in lo. |
| // |
| // This function's execution time does not depend on the inputs. |
| func Mul(x, y uint) (hi, lo uint) { |
| if UintSize == 32 { |
| h, l := Mul32(uint32(x), uint32(y)) |
| return uint(h), uint(l) |
| } |
| h, l := Mul64(uint64(x), uint64(y)) |
| return uint(h), uint(l) |
| } |
| |
| // Mul32 returns the 64-bit product of x and y: (hi, lo) = x * y |
| // with the product bits' upper half returned in hi and the lower |
| // half returned in lo. |
| // |
| // This function's execution time does not depend on the inputs. |
| func Mul32(x, y uint32) (hi, lo uint32) { |
| tmp := uint64(x) * uint64(y) |
| hi, lo = uint32(tmp>>32), uint32(tmp) |
| return |
| } |
| |
| // Mul64 returns the 128-bit product of x and y: (hi, lo) = x * y |
| // with the product bits' upper half returned in hi and the lower |
| // half returned in lo. |
| // |
| // This function's execution time does not depend on the inputs. |
| func Mul64(x, y uint64) (hi, lo uint64) { |
| const mask32 = 1<<32 - 1 |
| x0 := x & mask32 |
| x1 := x >> 32 |
| y0 := y & mask32 |
| y1 := y >> 32 |
| w0 := x0 * y0 |
| t := x1*y0 + w0>>32 |
| w1 := t & mask32 |
| w2 := t >> 32 |
| w1 += x0 * y1 |
| hi = x1*y1 + w2 + w1>>32 |
| lo = x * y |
| return |
| } |
| |
| // --- Full-width divide --- |
| |
| // Div returns the quotient and remainder of (hi, lo) divided by y: |
| // quo = (hi, lo)/y, rem = (hi, lo)%y with the dividend bits' upper |
| // half in parameter hi and the lower half in parameter lo. |
| // Div panics for y == 0 (division by zero) or y <= hi (quotient overflow). |
| func Div(hi, lo, y uint) (quo, rem uint) { |
| if UintSize == 32 { |
| q, r := Div32(uint32(hi), uint32(lo), uint32(y)) |
| return uint(q), uint(r) |
| } |
| q, r := Div64(uint64(hi), uint64(lo), uint64(y)) |
| return uint(q), uint(r) |
| } |
| |
| // Div32 returns the quotient and remainder of (hi, lo) divided by y: |
| // quo = (hi, lo)/y, rem = (hi, lo)%y with the dividend bits' upper |
| // half in parameter hi and the lower half in parameter lo. |
| // Div32 panics for y == 0 (division by zero) or y <= hi (quotient overflow). |
| func Div32(hi, lo, y uint32) (quo, rem uint32) { |
| if y != 0 && y <= hi { |
| panic(overflowError) |
| } |
| z := uint64(hi)<<32 | uint64(lo) |
| quo, rem = uint32(z/uint64(y)), uint32(z%uint64(y)) |
| return |
| } |
| |
| // Div64 returns the quotient and remainder of (hi, lo) divided by y: |
| // quo = (hi, lo)/y, rem = (hi, lo)%y with the dividend bits' upper |
| // half in parameter hi and the lower half in parameter lo. |
| // Div64 panics for y == 0 (division by zero) or y <= hi (quotient overflow). |
| func Div64(hi, lo, y uint64) (quo, rem uint64) { |
| if y == 0 { |
| panic(divideError) |
| } |
| if y <= hi { |
| panic(overflowError) |
| } |
| |
| // If high part is zero, we can directly return the results. |
| if hi == 0 { |
| return lo / y, lo % y |
| } |
| |
| s := uint(LeadingZeros64(y)) |
| y <<= s |
| |
| const ( |
| two32 = 1 << 32 |
| mask32 = two32 - 1 |
| ) |
| yn1 := y >> 32 |
| yn0 := y & mask32 |
| un32 := hi<<s | lo>>(64-s) |
| un10 := lo << s |
| un1 := un10 >> 32 |
| un0 := un10 & mask32 |
| q1 := un32 / yn1 |
| rhat := un32 - q1*yn1 |
| |
| for q1 >= two32 || q1*yn0 > two32*rhat+un1 { |
| q1-- |
| rhat += yn1 |
| if rhat >= two32 { |
| break |
| } |
| } |
| |
| un21 := un32*two32 + un1 - q1*y |
| q0 := un21 / yn1 |
| rhat = un21 - q0*yn1 |
| |
| for q0 >= two32 || q0*yn0 > two32*rhat+un0 { |
| q0-- |
| rhat += yn1 |
| if rhat >= two32 { |
| break |
| } |
| } |
| |
| return q1*two32 + q0, (un21*two32 + un0 - q0*y) >> s |
| } |
| |
| // Rem returns the remainder of (hi, lo) divided by y. Rem panics for |
| // y == 0 (division by zero) but, unlike Div, it doesn't panic on a |
| // quotient overflow. |
| func Rem(hi, lo, y uint) uint { |
| if UintSize == 32 { |
| return uint(Rem32(uint32(hi), uint32(lo), uint32(y))) |
| } |
| return uint(Rem64(uint64(hi), uint64(lo), uint64(y))) |
| } |
| |
| // Rem32 returns the remainder of (hi, lo) divided by y. Rem32 panics |
| // for y == 0 (division by zero) but, unlike Div32, it doesn't panic |
| // on a quotient overflow. |
| func Rem32(hi, lo, y uint32) uint32 { |
| return uint32((uint64(hi)<<32 | uint64(lo)) % uint64(y)) |
| } |
| |
| // Rem64 returns the remainder of (hi, lo) divided by y. Rem64 panics |
| // for y == 0 (division by zero) but, unlike Div64, it doesn't panic |
| // on a quotient overflow. |
| func Rem64(hi, lo, y uint64) uint64 { |
| // We scale down hi so that hi < y, then use Div64 to compute the |
| // rem with the guarantee that it won't panic on quotient overflow. |
| // Given that |
| // hi ≡ hi%y (mod y) |
| // we have |
| // hi<<64 + lo ≡ (hi%y)<<64 + lo (mod y) |
| _, rem := Div64(hi%y, lo, y) |
| return rem |
| } |