libgo/go/math/sqrt.go

   1 // Copyright 2009 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 package math
   6
   7 //extern sqrt
   8 func libc_sqrt(float64) float64
   9
  10 func Sqrt(x float64) float64 {
  11         return libc_sqrt(x)
  12 }
  13
  14 // The original C code and the long comment below are
  15 // from FreeBSD's /usr/src/lib/msun/src/e_sqrt.c and
  16 // came with this notice. The go code is a simplified
  17 // version of the original C.
  18 //
  19 // ====================================================
  20 // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  21 //
  22 // Developed at SunPro, a Sun Microsystems, Inc. business.
  23 // Permission to use, copy, modify, and distribute this
  24 // software is freely granted, provided that this notice
  25 // is preserved.
  26 // ====================================================
  27 //
  28 // __ieee754_sqrt(x)
  29 // Return correctly rounded sqrt.
  30 //           -----------------------------------------
  31 //           | Use the hardware sqrt if you have one |
  32 //           -----------------------------------------
  33 // Method:
  34 //   Bit by bit method using integer arithmetic. (Slow, but portable)
  35 //   1. Normalization
  36 //      Scale x to y in [1,4) with even powers of 2:
  37 //      find an integer k such that  1 <= (y=x*2**(2k)) < 4, then
  38 //              sqrt(x) = 2**k * sqrt(y)
  39 //   2. Bit by bit computation
  40 //      Let q  = sqrt(y) truncated to i bit after binary point (q = 1),
  41 //           i                                                   0
  42 //                                     i+1         2
  43 //          s  = 2*q , and      y  =  2   * ( y - q  ).          (1)
  44 //           i      i            i                 i
  45 //
  46 //      To compute q    from q , one checks whether
  47 //                  i+1       i
  48 //
  49 //                            -(i+1) 2
  50 //                      (q + 2      )  <= y.                     (2)
  51 //                        i
  52 //                                                            -(i+1)
  53 //      If (2) is false, then q   = q ; otherwise q   = q  + 2      .
  54 //                             i+1   i             i+1   i
  55 //
  56 //      With some algebraic manipulation, it is not difficult to see
  57 //      that (2) is equivalent to
  58 //                             -(i+1)
  59 //                      s  +  2       <= y                       (3)
  60 //                       i                i
  61 //
  62 //      The advantage of (3) is that s  and y  can be computed by
  63 //                                    i      i
  64 //      the following recurrence formula:
  65 //          if (3) is false
  66 //
  67 //          s     =  s  ,       y    = y   ;                     (4)
  68 //           i+1      i          i+1    i
  69 //
  70 //      otherwise,
  71 //                         -i                      -(i+1)
  72 //          s     =  s  + 2  ,  y    = y  -  s  - 2              (5)
  73 //           i+1      i          i+1    i     i
  74 //
  75 //      One may easily use induction to prove (4) and (5).
  76 //      Note. Since the left hand side of (3) contain only i+2 bits,
  77 //            it does not necessary to do a full (53-bit) comparison
  78 //            in (3).
  79 //   3. Final rounding
  80 //      After generating the 53 bits result, we compute one more bit.
  81 //      Together with the remainder, we can decide whether the
  82 //      result is exact, bigger than 1/2ulp, or less than 1/2ulp
  83 //      (it will never equal to 1/2ulp).
  84 //      The rounding mode can be detected by checking whether
  85 //      huge + tiny is equal to huge, and whether huge - tiny is
  86 //      equal to huge for some floating point number "huge" and "tiny".
  87 //
  88 //
  89 // Notes:  Rounding mode detection omitted. The constants "mask", "shift",
  90 // and "bias" are found in src/math/bits.go
  91
  92 // Sqrt returns the square root of x.
  93 //
  94 // Special cases are:
  95 //      Sqrt(+Inf) = +Inf
  96 //      Sqrt(±0) = ±0
  97 //      Sqrt(x < 0) = NaN
  98 //      Sqrt(NaN) = NaN
  99
 100 // Note: Sqrt is implemented in assembly on some systems.
 101 // Others have assembly stubs that jump to func sqrt below.
 102 // On systems where Sqrt is a single instruction, the compiler
 103 // may turn a direct call into a direct use of that instruction instead.
 104
 105 func sqrt(x float64) float64 {
 106         // special cases
 107         switch {
 108         case x == 0 || IsNaN(x) || IsInf(x, 1):
 109                 return x
 110         case x < 0:
 111                 return NaN()
 112         }
 113         ix := Float64bits(x)
 114         // normalize x
 115         exp := int((ix >> shift) & mask)
 116         if exp == 0 { // subnormal x
 117                 for ix&(1<<shift) == 0 {
 118                         ix <<= 1
 119                         exp--
 120                 }
 121                 exp++
 122         }
 123         exp -= bias // unbias exponent
 124         ix &^= mask << shift
 125         ix |= 1 << shift
 126         if exp&1 == 1 { // odd exp, double x to make it even
 127                 ix <<= 1
 128         }
 129         exp >>= 1 // exp = exp/2, exponent of square root
 130         // generate sqrt(x) bit by bit
 131         ix <<= 1
 132         var q, s uint64               // q = sqrt(x)
 133         r := uint64(1 << (shift + 1)) // r = moving bit from MSB to LSB
 134         for r != 0 {
 135                 t := s + r
 136                 if t <= ix {
 137                         s = t + r
 138                         ix -= t
 139                         q += r
 140                 }
 141                 ix <<= 1
 142                 r >>= 1
 143         }
 144         // final rounding
 145         if ix != 0 { // remainder, result not exact
 146                 q += q & 1 // round according to extra bit
 147         }
 148         ix = q>>1 + uint64(exp-1+bias)<<shift // significand + biased exponent
 149         return Float64frombits(ix)
 150 }