libgo/go/math/log1p.go

   1 // Copyright 2010 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 // The original C code, the long comment, and the constants
   8 // below are from FreeBSD's /usr/src/lib/msun/src/s_log1p.c
   9 // and came with this notice.  The go code is a simplified
  10 // version of the original C.
  11 //
  12 // ====================================================
  13 // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  14 //
  15 // Developed at SunPro, a Sun Microsystems, Inc. business.
  16 // Permission to use, copy, modify, and distribute this
  17 // software is freely granted, provided that this notice
  18 // is preserved.
  19 // ====================================================
  20 //
  21 //
  22 // double log1p(double x)
  23 //
  24 // Method :
  25 //   1. Argument Reduction: find k and f such that
  26 //                      1+x = 2**k * (1+f),
  27 //         where  sqrt(2)/2 < 1+f < sqrt(2) .
  28 //
  29 //      Note. If k=0, then f=x is exact. However, if k!=0, then f
  30 //      may not be representable exactly. In that case, a correction
  31 //      term is need. Let u=1+x rounded. Let c = (1+x)-u, then
  32 //      log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
  33 //      and add back the correction term c/u.
  34 //      (Note: when x > 2**53, one can simply return log(x))
  35 //
  36 //   2. Approximation of log1p(f).
  37 //      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  38 //               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  39 //               = 2s + s*R
  40 //      We use a special Reme algorithm on [0,0.1716] to generate
  41 //      a polynomial of degree 14 to approximate R The maximum error
  42 //      of this polynomial approximation is bounded by 2**-58.45. In
  43 //      other words,
  44 //                      2      4      6      8      10      12      14
  45 //          R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
  46 //      (the values of Lp1 to Lp7 are listed in the program)
  47 //      and
  48 //          |      2          14          |     -58.45
  49 //          | Lp1*s +...+Lp7*s    -  R(z) | <= 2
  50 //          |                             |
  51 //      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  52 //      In order to guarantee error in log below 1ulp, we compute log
  53 //      by
  54 //              log1p(f) = f - (hfsq - s*(hfsq+R)).
  55 //
  56 //   3. Finally, log1p(x) = k*ln2 + log1p(f).
  57 //                        = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
  58 //      Here ln2 is split into two floating point number:
  59 //                   ln2_hi + ln2_lo,
  60 //      where n*ln2_hi is always exact for |n| < 2000.
  61 //
  62 // Special cases:
  63 //      log1p(x) is NaN with signal if x < -1 (including -INF) ;
  64 //      log1p(+INF) is +INF; log1p(-1) is -INF with signal;
  65 //      log1p(NaN) is that NaN with no signal.
  66 //
  67 // Accuracy:
  68 //      according to an error analysis, the error is always less than
  69 //      1 ulp (unit in the last place).
  70 //
  71 // Constants:
  72 // The hexadecimal values are the intended ones for the following
  73 // constants. The decimal values may be used, provided that the
  74 // compiler will convert from decimal to binary accurately enough
  75 // to produce the hexadecimal values shown.
  76 //
  77 // Note: Assuming log() return accurate answer, the following
  78 //       algorithm can be used to compute log1p(x) to within a few ULP:
  79 //
  80 //              u = 1+x;
  81 //              if(u==1.0) return x ; else
  82 //                         return log(u)*(x/(u-1.0));
  83 //
  84 //       See HP-15C Advanced Functions Handbook, p.193.
  85
  86 // Log1p returns the natural logarithm of 1 plus its argument x.
  87 // It is more accurate than Log(1 + x) when x is near zero.
  88 //
  89 // Special cases are:
  90 //      Log1p(+Inf) = +Inf
  91 //      Log1p(±0) = ±0
  92 //      Log1p(-1) = -Inf
  93 //      Log1p(x < -1) = NaN
  94 //      Log1p(NaN) = NaN
  95
  96 //extern log1p
  97 func libc_log1p(float64) float64
  98
  99 func Log1p(x float64) float64 {
 100         return libc_log1p(x)
 101 }
 102
 103 func log1p(x float64) float64 {
 104         const (
 105                 Sqrt2M1     = 4.142135623730950488017e-01  // Sqrt(2)-1 = 0x3fda827999fcef34
 106                 Sqrt2HalfM1 = -2.928932188134524755992e-01 // Sqrt(2)/2-1 = 0xbfd2bec333018866
 107                 Small       = 1.0 / (1 << 29)              // 2**-29 = 0x3e20000000000000
 108                 Tiny        = 1.0 / (1 << 54)              // 2**-54
 109                 Two53       = 1 << 53                      // 2**53
 110                 Ln2Hi       = 6.93147180369123816490e-01   // 3fe62e42fee00000
 111                 Ln2Lo       = 1.90821492927058770002e-10   // 3dea39ef35793c76
 112                 Lp1         = 6.666666666666735130e-01     // 3FE5555555555593
 113                 Lp2         = 3.999999999940941908e-01     // 3FD999999997FA04
 114                 Lp3         = 2.857142874366239149e-01     // 3FD2492494229359
 115                 Lp4         = 2.222219843214978396e-01     // 3FCC71C51D8E78AF
 116                 Lp5         = 1.818357216161805012e-01     // 3FC7466496CB03DE
 117                 Lp6         = 1.531383769920937332e-01     // 3FC39A09D078C69F
 118                 Lp7         = 1.479819860511658591e-01     // 3FC2F112DF3E5244
 119         )
 120
 121         // special cases
 122         switch {
 123         case x < -1 || IsNaN(x): // includes -Inf
 124                 return NaN()
 125         case x == -1:
 126                 return Inf(-1)
 127         case IsInf(x, 1):
 128                 return Inf(1)
 129         }
 130
 131         absx := x
 132         if absx < 0 {
 133                 absx = -absx
 134         }
 135
 136         var f float64
 137         var iu uint64
 138         k := 1
 139         if absx < Sqrt2M1 { //  |x| < Sqrt(2)-1
 140                 if absx < Small { // |x| < 2**-29
 141                         if absx < Tiny { // |x| < 2**-54
 142                                 return x
 143                         }
 144                         return x - x*x*0.5
 145                 }
 146                 if x > Sqrt2HalfM1 { // Sqrt(2)/2-1 < x
 147                         // (Sqrt(2)/2-1) < x < (Sqrt(2)-1)
 148                         k = 0
 149                         f = x
 150                         iu = 1
 151                 }
 152         }
 153         var c float64
 154         if k != 0 {
 155                 var u float64
 156                 if absx < Two53 { // 1<<53
 157                         u = 1.0 + x
 158                         iu = Float64bits(u)
 159                         k = int((iu >> 52) - 1023)
 160                         if k > 0 {
 161                                 c = 1.0 - (u - x)
 162                         } else {
 163                                 c = x - (u - 1.0) // correction term
 164                                 c /= u
 165                         }
 166                 } else {
 167                         u = x
 168                         iu = Float64bits(u)
 169                         k = int((iu >> 52) - 1023)
 170                         c = 0
 171                 }
 172                 iu &= 0x000fffffffffffff
 173                 if iu < 0x0006a09e667f3bcd { // mantissa of Sqrt(2)
 174                         u = Float64frombits(iu | 0x3ff0000000000000) // normalize u
 175                 } else {
 176                         k += 1
 177                         u = Float64frombits(iu | 0x3fe0000000000000) // normalize u/2
 178                         iu = (0x0010000000000000 - iu) >> 2
 179                 }
 180                 f = u - 1.0 // Sqrt(2)/2 < u < Sqrt(2)
 181         }
 182         hfsq := 0.5 * f * f
 183         var s, R, z float64
 184         if iu == 0 { // |f| < 2**-20
 185                 if f == 0 {
 186                         if k == 0 {
 187                                 return 0
 188                         } else {
 189                                 c += float64(k) * Ln2Lo
 190                                 return float64(k)*Ln2Hi + c
 191                         }
 192                 }
 193                 R = hfsq * (1.0 - 0.66666666666666666*f) // avoid division
 194                 if k == 0 {
 195                         return f - R
 196                 }
 197                 return float64(k)*Ln2Hi - ((R - (float64(k)*Ln2Lo + c)) - f)
 198         }
 199         s = f / (2.0 + f)
 200         z = s * s
 201         R = z * (Lp1 + z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))))
 202         if k == 0 {
 203                 return f - (hfsq - s*(hfsq+R))
 204         }
 205         return float64(k)*Ln2Hi - ((hfsq - (s*(hfsq+R) + (float64(k)*Ln2Lo + c))) - f)
 206 }