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/*-
* ====================================================
* 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 <sys/cdefs.h>
#if 0
__FBSDID("$FreeBSD: src/lib/msun/src/s_cbrtl.c,v 1.1 2011/03/12 19:37:35 kargl Exp $");
#endif
#include <float.h>
#include <ieeefp.h>
#include <math.h>
#include "math_private.h"
#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)
{
long double v, r, s, t, w;
double dr, dt, dx;
float ft, fx;
uint32_t hx, lx;
uint16_t expsign, es;
int k;
volatile double vd1, vd2;
GET_LDOUBLE_EXP(expsign,x);
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. */
GET_LDOUBLE_WORDS(es,hx,lx,x);
if ((hx|lx) == 0) {
return (x);
}
/* Adjust subnormal numbers. */
x *= 0x1.0p514;
GET_LDOUBLE_EXP(k,x);
k &= 0x7fff;
k -= BIAS + 514;
} else
k -= BIAS;
SET_LDOUBLE_EXP(x,BIAS);
v = 1;
switch (k % 3) {
case 1:
case -2:
x = 2*x;
k--;
break;
case 2:
case -1:
x = 4*x;
k -= 2;
break;
}
SET_LDOUBLE_EXP(v, (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);
/*
* 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).
*/
vd2 = 0x1.0p32;
vd1 = 0x1.0p-31;
#define vd ((long double)vd2 + vd1)
t = dt + vd - 0x1.0p32;
/*
* 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;
return (t);
}
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