libgo/go/math/log.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 /*
   8         Floating-point logarithm.
   9 */
  10
  11 // The original C code, the long comment, and the constants
  12 // below are from FreeBSD's /usr/src/lib/msun/src/e_log.c
  13 // and came with this notice. The go code is a simpler
  14 // version of the original C.
  15 //
  16 // ====================================================
  17 // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  18 //
  19 // Developed at SunPro, a Sun Microsystems, Inc. business.
  20 // Permission to use, copy, modify, and distribute this
  21 // software is freely granted, provided that this notice
  22 // is preserved.
  23 // ====================================================
  24 //
  25 // __ieee754_log(x)
  26 // Return the logarithm of x
  27 //
  28 // Method :
  29 //   1. Argument Reduction: find k and f such that
  30 //                      x = 2**k * (1+f),
  31 //         where  sqrt(2)/2 < 1+f < sqrt(2) .
  32 //
  33 //   2. Approximation of log(1+f).
  34 //      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  35 //               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  36 //               = 2s + s*R
  37 //      We use a special Reme algorithm on [0,0.1716] to generate
  38 //      a polynomial of degree 14 to approximate R.  The maximum error
  39 //      of this polynomial approximation is bounded by 2**-58.45. In
  40 //      other words,
  41 //                      2      4      6      8      10      12      14
  42 //          R(z) ~ L1*s +L2*s +L3*s +L4*s +L5*s  +L6*s  +L7*s
  43 //      (the values of L1 to L7 are listed in the program) and
  44 //          |      2          14          |     -58.45
  45 //          | L1*s +...+L7*s    -  R(z) | <= 2
  46 //          |                             |
  47 //      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  48 //      In order to guarantee error in log below 1ulp, we compute log by
  49 //              log(1+f) = f - s*(f - R)                (if f is not too large)
  50 //              log(1+f) = f - (hfsq - s*(hfsq+R)).     (better accuracy)
  51 //
  52 //      3. Finally,  log(x) = k*Ln2 + log(1+f).
  53 //                          = k*Ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*Ln2_lo)))
  54 //         Here Ln2 is split into two floating point number:
  55 //                      Ln2_hi + Ln2_lo,
  56 //         where n*Ln2_hi is always exact for |n| < 2000.
  57 //
  58 // Special cases:
  59 //      log(x) is NaN with signal if x < 0 (including -INF) ;
  60 //      log(+INF) is +INF; log(0) is -INF with signal;
  61 //      log(NaN) is that NaN with no signal.
  62 //
  63 // Accuracy:
  64 //      according to an error analysis, the error is always less than
  65 //      1 ulp (unit in the last place).
  66 //
  67 // Constants:
  68 // The hexadecimal values are the intended ones for the following
  69 // constants. The decimal values may be used, provided that the
  70 // compiler will convert from decimal to binary accurately enough
  71 // to produce the hexadecimal values shown.
  72
  73 // Log returns the natural logarithm of x.
  74 //
  75 // Special cases are:
  76 //      Log(+Inf) = +Inf
  77 //      Log(0) = -Inf
  78 //      Log(x < 0) = NaN
  79 //      Log(NaN) = NaN
  80
  81 //extern log
  82 func libc_log(float64) float64
  83
  84 func Log(x float64) float64 {
  85         return libc_log(x)
  86 }
  87
  88 func log(x float64) float64 {
  89         const (
  90                 Ln2Hi = 6.93147180369123816490e-01 /* 3fe62e42 fee00000 */
  91                 Ln2Lo = 1.90821492927058770002e-10 /* 3dea39ef 35793c76 */
  92                 L1    = 6.666666666666735130e-01   /* 3FE55555 55555593 */
  93                 L2    = 3.999999999940941908e-01   /* 3FD99999 9997FA04 */
  94                 L3    = 2.857142874366239149e-01   /* 3FD24924 94229359 */
  95                 L4    = 2.222219843214978396e-01   /* 3FCC71C5 1D8E78AF */
  96                 L5    = 1.818357216161805012e-01   /* 3FC74664 96CB03DE */
  97                 L6    = 1.531383769920937332e-01   /* 3FC39A09 D078C69F */
  98                 L7    = 1.479819860511658591e-01   /* 3FC2F112 DF3E5244 */
  99         )
 100
 101         // special cases
 102         switch {
 103         case IsNaN(x) || IsInf(x, 1):
 104                 return x
 105         case x < 0:
 106                 return NaN()
 107         case x == 0:
 108                 return Inf(-1)
 109         }
 110
 111         // reduce
 112         f1, ki := Frexp(x)
 113         if f1 < Sqrt2/2 {
 114                 f1 *= 2
 115                 ki--
 116         }
 117         f := f1 - 1
 118         k := float64(ki)
 119
 120         // compute
 121         s := f / (2 + f)
 122         s2 := s * s
 123         s4 := s2 * s2
 124         t1 := s2 * (L1 + s4*(L3+s4*(L5+s4*L7)))
 125         t2 := s4 * (L2 + s4*(L4+s4*L6))
 126         R := t1 + t2
 127         hfsq := 0.5 * f * f
 128         return k*Ln2Hi - ((hfsq - (s*(hfsq+R) + k*Ln2Lo)) - f)
 129 }