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 // Sqrt returns the square root of x.
   8 //
   9 // Special cases are:
  10 //      Sqrt(+Inf) = +Inf
  11 //      Sqrt(±0) = ±0
  12 //      Sqrt(x < 0) = NaN
  13 //      Sqrt(NaN) = NaN
  14
  15 //extern sqrt
  16 func libc_sqrt(float64) float64
  17
  18 func Sqrt(x float64) float64 {
  19         return libc_sqrt(x)
  20 }
  21
  22 // The original C code and the long comment below are
  23 // from FreeBSD's /usr/src/lib/msun/src/e_sqrt.c and
  24 // came with this notice.  The go code is a simplified
  25 // version of the original C.
  26 //
  27 // ====================================================
  28 // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  29 //
  30 // Developed at SunPro, a Sun Microsystems, Inc. business.
  31 // Permission to use, copy, modify, and distribute this
  32 // software is freely granted, provided that this notice
  33 // is preserved.
  34 // ====================================================
  35 //
  36 // __ieee754_sqrt(x)
  37 // Return correctly rounded sqrt.
  38 //           -----------------------------------------
  39 //           | Use the hardware sqrt if you have one |
  40 //           -----------------------------------------
  41 // Method:
  42 //   Bit by bit method using integer arithmetic. (Slow, but portable)
  43 //   1. Normalization
  44 //      Scale x to y in [1,4) with even powers of 2:
  45 //      find an integer k such that  1 <= (y=x*2**(2k)) < 4, then
  46 //              sqrt(x) = 2**k * sqrt(y)
  47 //   2. Bit by bit computation
  48 //      Let q  = sqrt(y) truncated to i bit after binary point (q = 1),
  49 //           i                                                   0
  50 //                                     i+1         2
  51 //          s  = 2*q , and      y  =  2   * ( y - q  ).          (1)
  52 //           i      i            i                 i
  53 //
  54 //      To compute q    from q , one checks whether
  55 //                  i+1       i
  56 //
  57 //                            -(i+1) 2
  58 //                      (q + 2      )  <= y.                     (2)
  59 //                        i
  60 //                                                            -(i+1)
  61 //      If (2) is false, then q   = q ; otherwise q   = q  + 2      .
  62 //                             i+1   i             i+1   i
  63 //
  64 //      With some algebraic manipulation, it is not difficult to see
  65 //      that (2) is equivalent to
  66 //                             -(i+1)
  67 //                      s  +  2       <= y                       (3)
  68 //                       i                i
  69 //
  70 //      The advantage of (3) is that s  and y  can be computed by
  71 //                                    i      i
  72 //      the following recurrence formula:
  73 //          if (3) is false
  74 //
  75 //          s     =  s  ,       y    = y   ;                     (4)
  76 //           i+1      i          i+1    i
  77 //
  78 //      otherwise,
  79 //                         -i                      -(i+1)
  80 //          s     =  s  + 2  ,  y    = y  -  s  - 2              (5)
  81 //           i+1      i          i+1    i     i
  82 //
  83 //      One may easily use induction to prove (4) and (5).
  84 //      Note. Since the left hand side of (3) contain only i+2 bits,
  85 //            it does not necessary to do a full (53-bit) comparison
  86 //            in (3).
  87 //   3. Final rounding
  88 //      After generating the 53 bits result, we compute one more bit.
  89 //      Together with the remainder, we can decide whether the
  90 //      result is exact, bigger than 1/2ulp, or less than 1/2ulp
  91 //      (it will never equal to 1/2ulp).
  92 //      The rounding mode can be detected by checking whether
  93 //      huge + tiny is equal to huge, and whether huge - tiny is
  94 //      equal to huge for some floating point number "huge" and "tiny".
  95 //
  96 //
  97 // Notes:  Rounding mode detection omitted.  The constants "mask", "shift",
  98 // and "bias" are found in src/pkg/math/bits.go
  99
 100 // Sqrt returns the square root of x.
 101 //
 102 // Special cases are:
 103 //      Sqrt(+Inf) = +Inf
 104 //      Sqrt(±0) = ±0
 105 //      Sqrt(x < 0) = NaN
 106 //      Sqrt(NaN) = NaN
 107 func sqrt(x float64) float64 {
 108         // special cases
 109         switch {
 110         case x == 0 || IsNaN(x) || IsInf(x, 1):
 111                 return x
 112         case x < 0:
 113                 return NaN()
 114         }
 115         ix := Float64bits(x)
 116         // normalize x
 117         exp := int((ix >> shift) & mask)
 118         if exp == 0 { // subnormal x
 119                 for ix&1<<shift == 0 {
 120                         ix <<= 1
 121                         exp--
 122                 }
 123                 exp++
 124         }
 125         exp -= bias // unbias exponent
 126         ix &^= mask << shift
 127         ix |= 1 << shift
 128         if exp&1 == 1 { // odd exp, double x to make it even
 129                 ix <<= 1
 130         }
 131         exp >>= 1 // exp = exp/2, exponent of square root
 132         // generate sqrt(x) bit by bit
 133         ix <<= 1
 134         var q, s uint64               // q = sqrt(x)
 135         r := uint64(1 << (shift + 1)) // r = moving bit from MSB to LSB
 136         for r != 0 {
 137                 t := s + r
 138                 if t <= ix {
 139                         s = t + r
 140                         ix -= t
 141                         q += r
 142                 }
 143                 ix <<= 1
 144                 r >>= 1
 145         }
 146         // final rounding
 147         if ix != 0 { // remainder, result not exact
 148                 q += q & 1 // round according to extra bit
 149         }
 150         ix = q>>1 + uint64(exp-1+bias)<<shift // significand + biased exponent
 151         return Float64frombits(ix)
 152 }
 153
 154 func sqrtC(f float64, r *float64) {
 155         *r = sqrt(f)
 156 }