// 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.
package bits
const uintSize = 32 << (^uint(0) >> 32 & 1) // 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) {
const (
two32 = 1 << 32
mask32 = two32 - 1
)
if y == 0 {
panic(divideError)
}
if y <= hi {
panic(overflowError)
}
s := uint(LeadingZeros64(y))
y <<= s
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
}
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