src/math/log.c

   1 /* origin: FreeBSD /usr/src/lib/msun/src/e_log.c */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunSoft, a Sun Microsystems, Inc. business.
   7  * Permission to use, copy, modify, and distribute this
   8  * software is freely granted, provided that this notice
   9  * is preserved.
  10  * ====================================================
  11  */
  12 /* log(x)
  13  * Return the logarithm of x
  14  *
  15  * Method :
  16  *   1. Argument Reduction: find k and f such that
  17  *                      x = 2^k * (1+f),
  18  *         where  sqrt(2)/2 < 1+f < sqrt(2) .
  19  *
  20  *   2. Approximation of log(1+f).
  21  *      Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
  22  *               = 2s + 2/3 s**3 + 2/5 s**5 + .....,
  23  *               = 2s + s*R
  24  *      We use a special Remez algorithm on [0,0.1716] to generate
  25  *      a polynomial of degree 14 to approximate R The maximum error
  26  *      of this polynomial approximation is bounded by 2**-58.45. In
  27  *      other words,
  28  *                      2      4      6      8      10      12      14
  29  *          R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
  30  *      (the values of Lg1 to Lg7 are listed in the program)
  31  *      and
  32  *          |      2          14          |     -58.45
  33  *          | Lg1*s +...+Lg7*s    -  R(z) | <= 2
  34  *          |                             |
  35  *      Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
  36  *      In order to guarantee error in log below 1ulp, we compute log
  37  *      by
  38  *              log(1+f) = f - s*(f - R)        (if f is not too large)
  39  *              log(1+f) = f - (hfsq - s*(hfsq+R)).     (better accuracy)
  40  *
  41  *      3. Finally,  log(x) = k*ln2 + log(1+f).
  42  *                          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
  43  *         Here ln2 is split into two floating point number:
  44  *                      ln2_hi + ln2_lo,
  45  *         where n*ln2_hi is always exact for |n| < 2000.
  46  *
  47  * Special cases:
  48  *      log(x) is NaN with signal if x < 0 (including -INF) ;
  49  *      log(+INF) is +INF; log(0) is -INF with signal;
  50  *      log(NaN) is that NaN with no signal.
  51  *
  52  * Accuracy:
  53  *      according to an error analysis, the error is always less than
  54  *      1 ulp (unit in the last place).
  55  *
  56  * Constants:
  57  * The hexadecimal values are the intended ones for the following
  58  * constants. The decimal values may be used, provided that the
  59  * compiler will convert from decimal to binary accurately enough
  60  * to produce the hexadecimal values shown.
  61  */
  62
  63 #include <math.h>
  64 #include <stdint.h>
  65
  66 static const double
  67 ln2_hi = 6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
  68 ln2_lo = 1.90821492927058770002e-10,  /* 3dea39ef 35793c76 */
  69 Lg1 = 6.666666666666735130e-01,  /* 3FE55555 55555593 */
  70 Lg2 = 3.999999999940941908e-01,  /* 3FD99999 9997FA04 */
  71 Lg3 = 2.857142874366239149e-01,  /* 3FD24924 94229359 */
  72 Lg4 = 2.222219843214978396e-01,  /* 3FCC71C5 1D8E78AF */
  73 Lg5 = 1.818357216161805012e-01,  /* 3FC74664 96CB03DE */
  74 Lg6 = 1.531383769920937332e-01,  /* 3FC39A09 D078C69F */
  75 Lg7 = 1.479819860511658591e-01;  /* 3FC2F112 DF3E5244 */
  76
  77 double log(double x)
  78 {
  79         union {double f; uint64_t i;} u = {x};
  80         double_t hfsq,f,s,z,R,w,t1,t2,dk;
  81         uint32_t hx;
  82         int k;
  83
  84         hx = u.i>>32;
  85         k = 0;
  86         if (hx < 0x00100000 || hx>>31) {
  87                 if (u.i<<1 == 0)
  88                         return -1/(x*x);  /* log(+-0)=-inf */
  89                 if (hx>>31)
  90                         return (x-x)/0.0; /* log(-#) = NaN */
  91                 /* subnormal number, scale x up */
  92                 k -= 54;
  93                 x *= 0x1p54;
  94                 u.f = x;
  95                 hx = u.i>>32;
  96         } else if (hx >= 0x7ff00000) {
  97                 return x;
  98         } else if (hx == 0x3ff00000 && u.i<<32 == 0)
  99                 return 0;
 100
 101         /* reduce x into [sqrt(2)/2, sqrt(2)] */
 102         hx += 0x3ff00000 - 0x3fe6a09e;
 103         k += (int)(hx>>20) - 0x3ff;
 104         hx = (hx&0x000fffff) + 0x3fe6a09e;
 105         u.i = (uint64_t)hx<<32 | (u.i&0xffffffff);
 106         x = u.f;
 107
 108         f = x - 1.0;
 109         hfsq = 0.5*f*f;
 110         s = f/(2.0+f);
 111         z = s*s;
 112         w = z*z;
 113         t1 = w*(Lg2+w*(Lg4+w*Lg6));
 114         t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
 115         R = t2 + t1;
 116         dk = k;
 117         return s*(hfsq+R) + dk*ln2_lo - hfsq + f + dk*ln2_hi;
 118 }