Source file src/math/bits/bits.go

Documentation: math/bits

     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) >> 32 & 1) // 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  

View as plain text