libgo/go/math/exp.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 // The original C code, the long comment, and the constants
   9 // below are from FreeBSD's /usr/src/lib/msun/src/e_exp.c
  10 // and came with this notice.  The go code is a simplified
  11 // version of the original C.
  12 //
  13 // ====================================================
  14 // Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
  15 //
  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 // exp(x)
  23 // Returns the exponential of x.
  24 //
  25 // Method
  26 //   1. Argument reduction:
  27 //      Reduce x to an r so that |r| <= 0.5*ln2 ~ 0.34658.
  28 //      Given x, find r and integer k such that
  29 //
  30 //               x = k*ln2 + r,  |r| <= 0.5*ln2.
  31 //
  32 //      Here r will be represented as r = hi-lo for better
  33 //      accuracy.
  34 //
  35 //   2. Approximation of exp(r) by a special rational function on
  36 //      the interval [0,0.34658]:
  37 //      Write
  38 //          R(r**2) = r*(exp(r)+1)/(exp(r)-1) = 2 + r*r/6 - r**4/360 + ...
  39 //      We use a special Remes algorithm on [0,0.34658] to generate
  40 //      a polynomial of degree 5 to approximate R. The maximum error
  41 //      of this polynomial approximation is bounded by 2**-59. In
  42 //      other words,
  43 //          R(z) ~ 2.0 + P1*z + P2*z**2 + P3*z**3 + P4*z**4 + P5*z**5
  44 //      (where z=r*r, and the values of P1 to P5 are listed below)
  45 //      and
  46 //          |                  5          |     -59
  47 //          | 2.0+P1*z+...+P5*z   -  R(z) | <= 2
  48 //          |                             |
  49 //      The computation of exp(r) thus becomes
  50 //                             2*r
  51 //              exp(r) = 1 + -------
  52 //                            R - r
  53 //                                 r*R1(r)
  54 //                     = 1 + r + ----------- (for better accuracy)
  55 //                                2 - R1(r)
  56 //      where
  57 //                               2       4             10
  58 //              R1(r) = r - (P1*r  + P2*r  + ... + P5*r   ).
  59 //
  60 //   3. Scale back to obtain exp(x):
  61 //      From step 1, we have
  62 //         exp(x) = 2**k * exp(r)
  63 //
  64 // Special cases:
  65 //      exp(INF) is INF, exp(NaN) is NaN;
  66 //      exp(-INF) is 0, and
  67 //      for finite argument, only exp(0)=1 is exact.
  68 //
  69 // Accuracy:
  70 //      according to an error analysis, the error is always less than
  71 //      1 ulp (unit in the last place).
  72 //
  73 // Misc. info.
  74 //      For IEEE double
  75 //          if x >  7.09782712893383973096e+02 then exp(x) overflow
  76 //          if x < -7.45133219101941108420e+02 then exp(x) underflow
  77 //
  78 // Constants:
  79 // The hexadecimal values are the intended ones for the following
  80 // constants. The decimal values may be used, provided that the
  81 // compiler will convert from decimal to binary accurately enough
  82 // to produce the hexadecimal values shown.
  83
  84 // Exp returns e**x, the base-e exponential of x.
  85 //
  86 // Special cases are:
  87 //      Exp(+Inf) = +Inf
  88 //      Exp(NaN) = NaN
  89 // Very large values overflow to 0 or +Inf.
  90 // Very small values underflow to 1.
  91 func Exp(x float64) float64 {
  92         const (
  93                 Ln2Hi = 6.93147180369123816490e-01
  94                 Ln2Lo = 1.90821492927058770002e-10
  95                 Log2e = 1.44269504088896338700e+00
  96                 P1    = 1.66666666666666019037e-01  /* 0x3FC55555; 0x5555553E */
  97                 P2    = -2.77777777770155933842e-03 /* 0xBF66C16C; 0x16BEBD93 */
  98                 P3    = 6.61375632143793436117e-05  /* 0x3F11566A; 0xAF25DE2C */
  99                 P4    = -1.65339022054652515390e-06 /* 0xBEBBBD41; 0xC5D26BF1 */
 100                 P5    = 4.13813679705723846039e-08  /* 0x3E663769; 0x72BEA4D0 */
 101
 102                 Overflow  = 7.09782712893383973096e+02
 103                 Underflow = -7.45133219101941108420e+02
 104                 NearZero  = 1.0 / (1 << 28) // 2**-28
 105         )
 106
 107         // TODO(rsc): Remove manual inlining of IsNaN, IsInf
 108         // when compiler does it for us
 109         // special cases
 110         switch {
 111         case x != x || x > MaxFloat64: // IsNaN(x) || IsInf(x, 1):
 112                 return x
 113         case x < -MaxFloat64: // IsInf(x, -1):
 114                 return 0
 115         case x > Overflow:
 116                 return Inf(1)
 117         case x < Underflow:
 118                 return 0
 119         case -NearZero < x && x < NearZero:
 120                 return 1
 121         }
 122
 123         // reduce; computed as r = hi - lo for extra precision.
 124         var k int
 125         switch {
 126         case x < 0:
 127                 k = int(Log2e*x - 0.5)
 128         case x > 0:
 129                 k = int(Log2e*x + 0.5)
 130         }
 131         hi := x - float64(k)*Ln2Hi
 132         lo := float64(k) * Ln2Lo
 133         r := hi - lo
 134
 135         // compute
 136         t := r * r
 137         c := r - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))))
 138         y := 1 - ((lo - (r*c)/(2-c)) - hi)
 139         // TODO(rsc): make sure Ldexp can handle boundary k
 140         return Ldexp(y, k)
 141 }