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/* 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
#define BIAS (LDBL_MAX_EXP - 1)
static const unsigned B1 = 709958130; /* B1 = (127-127.0/3-0.03306235651)*2**23 */
long double cbrtl(long double x)
{
union IEEEl2bits u, v;
long double r, s, t, w;
double dr, dt, dx;
float ft, fx;
uint32_t hx;
uint16_t expsign;
int k;
u.e = x;
expsign = u.xbits.expsign;
k = expsign & 0x7fff;
/*
* If x = +-Inf, then cbrt(x) = +-Inf.
* If x = NaN, then cbrt(x) = NaN.
*/
if (k == BIAS + LDBL_MAX_EXP)
return x + x;
if (k == 0) {
/* If x = +-0, then cbrt(x) = +-0. */
if ((u.bits.manh | u.bits.manl) == 0)
return x;
/* Adjust subnormal numbers. */
u.e *= 0x1.0p514;
k = u.bits.exp;
k -= BIAS + 514;
} else
k -= BIAS;
u.xbits.expsign = BIAS;
v.e = 1;
x = u.e;
switch (k % 3) {
case 1:
case -2:
x = 2*x;
k--;
break;
case 2:
case -1:
x = 4*x;
k -= 2;
break;
}
v.xbits.expsign = (expsign & 0x8000) | (BIAS + k / 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: */
fx = x;
GET_FLOAT_WORD(hx, fx);
SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1));
/* ~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.e;
return t;
}
#endif
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