/* 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
