modules/fdlibm/src/k_log.h

   1
   2 /* @(#)e_log.c 1.3 95/01/18 */
   3 /*
   4  * ====================================================
   5  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   6  *
   7  * Developed at SunSoft, a Sun Microsystems, Inc. business.
   8  * Permission to use, copy, modify, and distribute this
   9  * software is freely granted, provided that this notice
  10  * is preserved.
  11  * ====================================================
  12  */
  13
  14 //#include <sys/cdefs.h>
  15 //__FBSDID("$FreeBSD$");
  16
  17 /*
  18  * k_log1p(f):
  19  * Return log(1+f) - f for 1+f in ~[sqrt(2)/2, sqrt(2)].
  20  *
  21  * The following describes the overall strategy for computing
  22  * logarithms in base e.  The argument reduction and adding the final
  23  * term of the polynomial are done by the caller for increased accuracy
  24  * when different bases are used.
  25  *
  26  * Method :
  27  *   1. Argument Reduction: find k and f such that
  28  *                      x = 2^k * (1+f),
  29  *         where  sqrt(2)/2 < 1+f < sqrt(2) .
  30  *
  31  *   2. Approximation of log(1+f).
  32  *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  33  *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  34  *               = 2s + s*R
  35  *      We use a special Reme algorithm on [0,0.1716] to generate
  36  *      a polynomial of degree 14 to approximate R The maximum error
  37  *      of this polynomial approximation is bounded by 2**-58.45. In
  38  *      other words,
  39  *                      2      4      6      8      10      12      14
  40  *          R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
  41  *      (the values of Lg1 to Lg7 are listed in the program)
  42  *      and
  43  *          |      2          14          |     -58.45
  44  *          | Lg1*s +...+Lg7*s    -  R(z) | <= 2
  45  *          |                             |
  46  *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  47  *      In order to guarantee error in log below 1ulp, we compute log
  48  *      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
  74 static const double
  75 Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
  76 Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
  77 Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
  78 Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
  79 Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
  80 Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
  81 Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
  82
  83 /*
  84  * We always inline k_log1p(), since doing so produces a
  85  * substantial performance improvement (~40% on amd64).
  86  */
  87 static inline double
  88 k_log1p(double f)
  89 {
  90         double hfsq,s,z,R,w,t1,t2;
  91
  92         s = f/(2.0+f);
  93         z = s*s;
  94         w = z*z;
  95         t1= w*(Lg2+w*(Lg4+w*Lg6));
  96         t2= z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
  97         R = t2+t1;
  98         hfsq=0.5*f*f;
  99         return s*(hfsq+R);
 100 }