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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 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.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 __ieee754_log(double x)

#else

        double __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);

        }

}