source: azure_iot_hub/trunk/musl-1.1.18/src/math/log10.c@ 389

/* origin: FreeBSD /usr/src/lib/msun/src/e_log10.c */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunSoft, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */
/*
 * Return the base 10 logarithm of x.  See log.c for most comments.
 *
 * Reduce x to 2^k (1+f) and calculate r = log(1+f) - f + f*f/2
 * as in log.c, then combine and scale in extra precision:
 *    log10(x) = (f - f*f/2 + r)/log(10) + k*log10(2)
 */

#include <math.h>
#include <stdint.h>

static const double
ivln10hi  = 4.34294481878168880939e-01, /* 0x3fdbcb7b, 0x15200000 */
ivln10lo  = 2.50829467116452752298e-11, /* 0x3dbb9438, 0xca9aadd5 */
log10_2hi = 3.01029995663611771306e-01, /* 0x3FD34413, 0x509F6000 */
log10_2lo = 3.69423907715893078616e-13, /* 0x3D59FEF3, 0x11F12B36 */
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 */

double log10(double x)
{
        union {double f; uint64_t i;} u = {x};
        double_t hfsq,f,s,z,R,w,t1,t2,dk,y,hi,lo,val_hi,val_lo;
        uint32_t hx;
        int k;

        hx = u.i>>32;
        k = 0;
        if (hx < 0x00100000 || hx>>31) {
                if (u.i<<1 == 0)
                        return -1/(x*x);  /* log(+-0)=-inf */
                if (hx>>31)
                        return (x-x)/0.0; /* log(-#) = NaN */
                /* subnormal number, scale x up */
                k -= 54;
                x *= 0x1p54;
                u.f = x;
                hx = u.i>>32;
        } else if (hx >= 0x7ff00000) {
                return x;
        } else if (hx == 0x3ff00000 && u.i<<32 == 0)
                return 0;

        /* reduce x into [sqrt(2)/2, sqrt(2)] */
        hx += 0x3ff00000 - 0x3fe6a09e;
        k += (int)(hx>>20) - 0x3ff;
        hx = (hx&0x000fffff) + 0x3fe6a09e;
        u.i = (uint64_t)hx<<32 | (u.i&0xffffffff);
        x = u.f;

        f = x - 1.0;
        hfsq = 0.5*f*f;
        s = f/(2.0+f);
        z = s*s;
        w = z*z;
        t1 = w*(Lg2+w*(Lg4+w*Lg6));
        t2 = z*(Lg1+w*(Lg3+w*(Lg5+w*Lg7)));
        R = t2 + t1;

        /* See log2.c for details. */
        /* hi+lo = f - hfsq + s*(hfsq+R) ~ log(1+f) */
        hi = f - hfsq;
        u.f = hi;
        u.i &= (uint64_t)-1<<32;
        hi = u.f;
        lo = f - hi - hfsq + s*(hfsq+R);

        /* val_hi+val_lo ~ log10(1+f) + k*log10(2) */
        val_hi = hi*ivln10hi;
        dk = k;
        y = dk*log10_2hi;
        val_lo = dk*log10_2lo + (lo+hi)*ivln10lo + lo*ivln10hi;

        /*
         * Extra precision in for adding y is not strictly needed
         * since there is no very large cancellation near x = sqrt(2) or
         * x = 1/sqrt(2), but we do it anyway since it costs little on CPUs
         * with some parallelism and it reduces the error for many args.
         */
        w = y + val_hi;
        val_lo += (y - w) + val_hi;
        val_hi = w;

        return val_lo + val_hi;
}