source: asp3_tinet_ecnl_arm/trunk/musl-1.1.18/src/math/cbrtl.c@ 352

/* origin: FreeBSD /usr/src/lib/msun/src/s_cbrtl.c */
/*-
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 * Copyright (c) 2009-2011, Bruce D. Evans, Steven G. Kargl, David Schultz.
 *
 * 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.
 * ====================================================
 *
 * The argument reduction and testing for exceptional cases was
 * written by Steven G. Kargl with input from Bruce D. Evans
 * and David A. Schultz.
 */

#include "libm.h"

#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
long double cbrtl(long double x)
{
        return cbrt(x);
}
#elif (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
static const unsigned B1 = 709958130; /* B1 = (127-127.0/3-0.03306235651)*2**23 */

long double cbrtl(long double x)
{
        union ldshape u = {x}, v;
        union {float f; uint32_t i;} uft;
        long double r, s, t, w;
        double_t dr, dt, dx;
        float_t ft;
        int e = u.i.se & 0x7fff;
        int sign = u.i.se & 0x8000;

        /*
         * If x = +-Inf, then cbrt(x) = +-Inf.
         * If x = NaN, then cbrt(x) = NaN.
         */
        if (e == 0x7fff)
                return x + x;
        if (e == 0) {
                /* Adjust subnormal numbers. */
                u.f *= 0x1p120;
                e = u.i.se & 0x7fff;
                /* If x = +-0, then cbrt(x) = +-0. */
                if (e == 0)
                        return x;
                e -= 120;
        }
        e -= 0x3fff;
        u.i.se = 0x3fff;
        x = u.f;
        switch (e % 3) {
        case 1:
        case -2:
                x *= 2;
                e--;
                break;
        case 2:
        case -1:
                x *= 4;
                e -= 2;
                break;
        }
        v.f = 1.0;
        v.i.se = sign | (0x3fff + e/3);

        /*
         * The following is the guts of s_cbrtf, with the handling of
         * special values removed and extra care for accuracy not taken,
         * but with most of the extra accuracy not discarded.
         */

        /* ~5-bit estimate: */
        uft.f = x;
        uft.i = (uft.i & 0x7fffffff)/3 + B1;
        ft = uft.f;

        /* ~16-bit estimate: */
        dx = x;
        dt = ft;
        dr = dt * dt * dt;
        dt = dt * (dx + dx + dr) / (dx + dr + dr);

        /* ~47-bit estimate: */
        dr = dt * dt * dt;
        dt = dt * (dx + dx + dr) / (dx + dr + dr);

#if LDBL_MANT_DIG == 64
        /*
         * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8).
         * Round it away from zero to 32 bits (32 so that t*t is exact, and
         * away from zero for technical reasons).
         */
        t = dt + (0x1.0p32L + 0x1.0p-31L) - 0x1.0p32;
#elif LDBL_MANT_DIG == 113
        /*
         * Round dt away from zero to 47 bits.  Since we don't trust the 47,
         * add 2 47-bit ulps instead of 1 to round up.  Rounding is slow and
         * might be avoidable in this case, since on most machines dt will
         * have been evaluated in 53-bit precision and the technical reasons
         * for rounding up might not apply to either case in cbrtl() since
         * dt is much more accurate than needed.
         */
        t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60;
#endif

        /*
         * Final step Newton iteration to 64 or 113 bits with
         * error < 0.667 ulps
         */
        s = t*t;         /* t*t is exact */
        r = x/s;         /* error <= 0.5 ulps; |r| < |t| */
        w = t+t;         /* t+t is exact */
        r = (r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */
        t = t+t*r;       /* error <= 0.5 + 0.5/3 + epsilon */

        t *= v.f;
        return t;
}
#endif