e_log.c

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/* @(#)e_log.c 5.1 93/09/24 */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */

#if defined(LIBM_SCCS) && !defined(lint)
static const char rcsid[] =
    "$NetBSD: e_log.c,v 1.8 1995/05/10 20:45:49 jtc Exp $";
#endif

/* __ieee754_log(x)
 * Return the logrithm of x
 *
 * Method :
 *   1. Argument Reduction: find k and f such that
 *      x = 2^k * (1+f),
 *     where  sqrt(2)/2 < 1+f < sqrt(2) .
 *
 *   2. Approximation of log(1+f).
 *  Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
 *     = 2s + 2/3 s**3 + 2/5 s**5 + .....,
 *         = 2s + s*R
 *      We use a special Reme algorithm on [0,0.1716] to generate
 *  a polynomial of degree 14 to approximate R The maximum error
 *  of this polynomial approximation is bounded by 2**-58.45. In
 *  other words,
 *            2      4      6      8      10      12      14
 *      R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
 *    (the values of Lg1 to Lg7 are listed in the program)
 *  and
 *      |      2          14          |     -58.45
 *      | Lg1*s +...+Lg7*s    -  R(z) | <= 2
 *      |                             |
 *  Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2.
 *  In order to guarantee error in log below 1ulp, we compute log
 *  by
 *    log(1+f) = f - s*(f - R)  (if f is not too large)
 *    log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)
 *
 *  3. Finally,  log(x) = k*ln2 + log(1+f).
 *          = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))
 *     Here ln2 is split into two floating point number:
 *      ln2_hi + ln2_lo,
 *     where n*ln2_hi is always exact for |n| < 2000.
 *
 * Special cases:
 *  log(x) is NaN with signal if x < 0 (including -INF) ;
 *  log(+INF) is +INF; log(0) is -INF with signal;
 *  log(NaN) is that NaN with no signal.
 *
 * Accuracy:
 *  according to an error analysis, the error is always less than
 *  1 ulp (unit in the last place).
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 */

#include "math_libm.h"
#include "math_private.h"

#ifdef __STDC__
static const double
#else
static double
#endif
  ln2_hi = 6.93147180369123816490e-01,  /* 3fe62e42 fee00000 */
    ln2_lo = 1.90821492927058770002e-10,        /* 3dea39ef 35793c76 */
    two54 = 1.80143985094819840000e+16, /* 43500000 00000000 */
    Lg1 = 6.666666666666735130e-01,     /* 3FE55555 55555593 */
    Lg2 = 3.999999999940941908e-01,     /* 3FD99999 9997FA04 */
    Lg3 = 2.857142874366239149e-01,     /* 3FD24924 94229359 */
    Lg4 = 2.222219843214978396e-01,     /* 3FCC71C5 1D8E78AF */
    Lg5 = 1.818357216161805012e-01,     /* 3FC74664 96CB03DE */
    Lg6 = 1.531383769920937332e-01,     /* 3FC39A09 D078C69F */
    Lg7 = 1.479819860511658591e-01;     /* 3FC2F112 DF3E5244 */

#ifdef __STDC__
static const double zero = 0.0;
#else
static double zero = 0.0;
#endif

#ifdef __STDC__
double attribute_hidden
__ieee754_log(double x)
#else
double attribute_hidden
__ieee754_log(x)
     double x;
#endif
{
    double hfsq, f, s, z, R, w, t1, t2, dk;
    int32_t k, hx, i, j;
    u_int32_t lx;

    EXTRACT_WORDS(hx, lx, x);

    k = 0;
    if (hx < 0x00100000) {      /* x < 2**-1022  */
        if (((hx & 0x7fffffff) | lx) == 0)
            return -two54 / zero;       /* log(+-0)=-inf */
        if (hx < 0)
            return (x - x) / zero;      /* log(-#) = NaN */
        k -= 54;
        x *= two54;             /* subnormal number, scale up x */
        GET_HIGH_WORD(hx, x);
    }
    if (hx >= 0x7ff00000)
        return x + x;
    k += (hx >> 20) - 1023;
    hx &= 0x000fffff;
    i = (hx + 0x95f64) & 0x100000;
    SET_HIGH_WORD(x, hx | (i ^ 0x3ff00000));    /* normalize x or x/2 */
    k += (i >> 20);
    f = x - 1.0;
    if ((0x000fffff & (2 + hx)) < 3) {  /* |f| < 2**-20 */
        if (f == zero) {
            if (k == 0)
                return zero;
            else {
                dk = (double) k;
                return dk * ln2_hi + dk * ln2_lo;
            }
        }
        R = f * f * (0.5 - 0.33333333333333333 * f);
        if (k == 0)
            return f - R;
        else {
            dk = (double) k;
            return dk * ln2_hi - ((R - dk * ln2_lo) - f);
        }
    }
    s = f / (2.0 + f);
    dk = (double) k;
    z = s * s;
    i = hx - 0x6147a;
    w = z * z;
    j = 0x6b851 - hx;
    t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
    t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
    i |= j;
    R = t2 + t1;
    if (i > 0) {
        hfsq = 0.5 * f * f;
        if (k == 0)
            return f - (hfsq - s * (hfsq + R));
        else
            return dk * ln2_hi - ((hfsq - (s * (hfsq + R) + dk * ln2_lo)) -
                                  f);
    } else {
        if (k == 0)
            return f - s * (f - R);
        else
            return dk * ln2_hi - ((s * (f - R) - dk * ln2_lo) - f);
    }
}