src/math/bits/bits.go - go - Git at Google

 // 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
 }

 // 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
 }
	// 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 \|\| q1yn0 > two32rhat+un1 {
	q1--
	rhat += yn1
	if rhat >= two32 {
	break
	}
	}

	un21 := un32two32 + un1 - q1y
	q0 := un21 / yn1
	rhat = un21 - q0*yn1

	for q0 >= two32 \|\| q0yn0 > two32rhat+un0 {
	q0--
	rhat += yn1
	if rhat >= two32 {
	break
	}
	}

	return q1two32 + q0, (un21two32 + 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
	}