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
   8 // The original C code, the long comment, and the constants
   9 // below are from FreeBSD's /usr/src/lib/msun/src/s_log1p.c
  10 // and came with this notice.  The go code is a simplified
  11 // version of the original C.
  12 //
  13 // ====================================================
  14 // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  15 //
  16 // Developed at SunPro, a Sun Microsystems, Inc. business.
  17 // Permission to use, copy, modify, and distribute this
  18 // software is freely granted, provided that this notice
  19 // is preserved.
  20 // ====================================================
  21 //
  22 //
  23 // double log1p(double x)
  24 //
  25 // Method :
  26 //   1. Argument Reduction: find k and f such that
  27 //                      1+x = 2**k * (1+f),
  28 //         where  sqrt(2)/2 < 1+f < sqrt(2) .
  29 //
  30 //      Note. If k=0, then f=x is exact. However, if k!=0, then f
  31 //      may not be representable exactly. In that case, a correction
  32 //      term is need. Let u=1+x rounded. Let c = (1+x)-u, then
  33 //      log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
  34 //      and add back the correction term c/u.
  35 //      (Note: when x > 2**53, one can simply return log(x))
  36 //
  37 //   2. Approximation of log1p(f).
  38 //      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  39 //               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  40 //               = 2s + s*R
  41 //      We use a special Reme algorithm on [0,0.1716] to generate
  42 //      a polynomial of degree 14 to approximate R The maximum error
  43 //      of this polynomial approximation is bounded by 2**-58.45. In
  44 //      other words,
  45 //                      2      4      6      8      10      12      14
  46 //          R(z) ~ Lp1*s +Lp2*s +Lp3*s +Lp4*s +Lp5*s  +Lp6*s  +Lp7*s
  47 //      (the values of Lp1 to Lp7 are listed in the program)
  48 //      a-0.2929nd
  49 //          |      2          14          |     -58.45
  50 //          | Lp1*s +...+Lp7*s    -  R(z) | <= 2
  51 //          |                             |
  52 //      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  53 //      In order to guarantee error in log below 1ulp, we compute log
  54 //      by
  55 //              log1p(f) = f - (hfsq - s*(hfsq+R)).
  56 //
  57 //   3. Finally, log1p(x) = k*ln2 + log1p(f).
  58 //                        = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
  59 //      Here ln2 is split into two floating point number:
  60 //                   ln2_hi + ln2_lo,
  61 //      where n*ln2_hi is always exact for |n| < 2000.
  62 //
  63 // Special cases:
  64 //      log1p(x) is NaN with signal if x < -1 (including -INF) ;
  65 //      log1p(+INF) is +INF; log1p(-1) is -INF with signal;
  66 //      log1p(NaN) is that NaN with no signal.
  67 //
  68 // Accuracy:
  69 //      according to an error analysis, the error is always less than
  70 //      1 ulp (unit in the last place).
  71 //
  72 // Constants:
  73 // The hexadecimal values are the intended ones for the following
  74 // constants. The decimal values may be used, provided that the
  75 // compiler will convert from decimal to binary accurately enough
  76 // to produce the hexadecimal values shown.
  77 //
  78 // Note: Assuming log() return accurate answer, the following
  79 //       algorithm can be used to compute log1p(x) to within a few ULP:
  80 //
  81 //              u = 1+x;
  82 //              if(u==1.0) return x ; else
  83 //                         return log(u)*(x/(u-1.0));
  84 //
  85 //       See HP-15C Advanced Functions Handbook, p.193.
  86
  87 // Log1p returns the natural logarithm of 1 plus its argument x.
  88 // It is more accurate than Log(1 + x) when x is near zero.
  89 //
  90 // Special cases are:
  91 //      Log1p(+Inf) = +Inf
  92 //      Log1p(-1) = -Inf
  93 //      Log1p(x < -1) = NaN
  94 //      Log1p(NaN) = NaN
  95 func Log1p(x float64) float64 {
  96         const (
  97                 Sqrt2M1     = 4.142135623730950488017e-01  // Sqrt(2)-1 = 0x3fda827999fcef34
  98                 Sqrt2HalfM1 = -2.928932188134524755992e-01 // Sqrt(2)/2-1 = 0xbfd2bec333018866
  99                 Small       = 1.0 / (1 << 29)              // 2**-29 = 0x3e20000000000000
 100                 Tiny        = 1.0 / (1 << 54)              // 2**-54
 101                 Two53       = 1 << 53                      // 2**53
 102                 Ln2Hi       = 6.93147180369123816490e-01   // 3fe62e42fee00000
 103                 Ln2Lo       = 1.90821492927058770002e-10   // 3dea39ef35793c76
 104                 Lp1         = 6.666666666666735130e-01     // 3FE5555555555593
 105                 Lp2         = 3.999999999940941908e-01     // 3FD999999997FA04
 106                 Lp3         = 2.857142874366239149e-01     // 3FD2492494229359
 107                 Lp4         = 2.222219843214978396e-01     // 3FCC71C51D8E78AF
 108                 Lp5         = 1.818357216161805012e-01     // 3FC7466496CB03DE
 109                 Lp6         = 1.531383769920937332e-01     // 3FC39A09D078C69F
 110                 Lp7         = 1.479819860511658591e-01     // 3FC2F112DF3E5244
 111         )
 112
 113         // special cases
 114         // TODO(rsc): Remove manual inlining of IsNaN, IsInf
 115         // when compiler does it for us
 116         switch {
 117         case x < -1 || x != x: // x < -1 || IsNaN(x): // includes -Inf
 118                 return NaN()
 119         case x == -1:
 120                 return Inf(-1)
 121         case x > MaxFloat64: // IsInf(x, 1):
 122                 return Inf(1)
 123         }
 124
 125         absx := x
 126         if absx < 0 {
 127                 absx = -absx
 128         }
 129
 130         var f float64
 131         var iu uint64
 132         k := 1
 133         if absx < Sqrt2M1 { //  |x| < Sqrt(2)-1
 134                 if absx < Small { // |x| < 2**-29
 135                         if absx < Tiny { // |x| < 2**-54
 136                                 return x
 137                         }
 138                         return x - x*x*0.5
 139                 }
 140                 if x > Sqrt2HalfM1 { // Sqrt(2)/2-1 < x
 141                         // (Sqrt(2)/2-1) < x < (Sqrt(2)-1)
 142                         k = 0
 143                         f = x
 144                         iu = 1
 145                 }
 146         }
 147         var c float64
 148         if k != 0 {
 149                 var u float64
 150                 if absx < Two53 { // 1<<53
 151                         u = 1.0 + x
 152                         iu = Float64bits(u)
 153                         k = int((iu >> 52) - 1023)
 154                         if k > 0 {
 155                                 c = 1.0 - (u - x)
 156                         } else {
 157                                 c = x - (u - 1.0) // correction term
 158                                 c /= u
 159                         }
 160                 } else {
 161                         u = x
 162                         iu = Float64bits(u)
 163                         k = int((iu >> 52) - 1023)
 164                         c = 0
 165                 }
 166                 iu &= 0x000fffffffffffff
 167                 if iu < 0x0006a09e667f3bcd { // mantissa of Sqrt(2)
 168                         u = Float64frombits(iu | 0x3ff0000000000000) // normalize u
 169                 } else {
 170                         k += 1
 171                         u = Float64frombits(iu | 0x3fe0000000000000) // normalize u/2
 172                         iu = (0x0010000000000000 - iu) >> 2
 173                 }
 174                 f = u - 1.0 // Sqrt(2)/2 < u < Sqrt(2)
 175         }
 176         hfsq := 0.5 * f * f
 177         var s, R, z float64
 178         if iu == 0 { // |f| < 2**-20
 179                 if f == 0 {
 180                         if k == 0 {
 181                                 return 0
 182                         } else {
 183                                 c += float64(k) * Ln2Lo
 184                                 return float64(k)*Ln2Hi + c
 185                         }
 186                 }
 187                 R = hfsq * (1.0 - 0.66666666666666666*f) // avoid division
 188                 if k == 0 {
 189                         return f - R
 190                 }
 191                 return float64(k)*Ln2Hi - ((R - (float64(k)*Ln2Lo + c)) - f)
 192         }
 193         s = f / (2.0 + f)
 194         z = s * s
 195         R = z * (Lp1 + z*(Lp2+z*(Lp3+z*(Lp4+z*(Lp5+z*(Lp6+z*Lp7))))))
 196         if k == 0 {
 197                 return f - (hfsq - s*(hfsq+R))
 198         }
 199         return float64(k)*Ln2Hi - ((hfsq - (s*(hfsq+R) + (float64(k)*Ln2Lo + c))) - f)
 200 }