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/* s_cbrtf.c -- float version of s_cbrt.c.
* Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
* Debugged and optimized by Bruce D. Evans.
*/
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* 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.
* ====================================================
*/
#ifndef lint
static char rcsid[] = "$FreeBSD$";
#endif
#include "math.h"
#include "math_private.h"
/* cbrtf(x)
* Return cube root of x
*/
static const unsigned
B1 = 709958130, /* B1 = (127-127.0/3-0.03306235651)*2**23 */
B2 = 642849266; /* B2 = (127-127.0/3-24/3-0.03306235651)*2**23 */
/* |1/cbrt(x) - p(x)| < 2**-14.5 (~[-4.37e-4, 4.366e-5]). */
static const float
P0 = 1.5586718321, /* 0x3fc7828f */
P1 = -0.78271341324, /* -0xbf485fe8 */
P2 = 0.22403796017; /* 0x3e656a35 */
float
cbrtf(float x)
{
float r,s,t,w;
int32_t hx;
u_int32_t sign;
u_int32_t high;
GET_FLOAT_WORD(hx,x);
sign=hx&0x80000000; /* sign= sign(x) */
hx ^=sign;
if(hx>=0x7f800000) return(x+x); /* cbrt(NaN,INF) is itself */
if(hx==0)
return(x); /* cbrt(0) is itself */
/* rough cbrt to 5 bits */
if(hx<0x00800000) { /* subnormal number */
SET_FLOAT_WORD(t,0x4b800000); /* set t= 2**24 */
t*=x;
GET_FLOAT_WORD(high,t);
SET_FLOAT_WORD(t,sign|((high&0x7fffffff)/3+B2));
} else
SET_FLOAT_WORD(t,sign|(hx/3+B1));
/* new cbrt to 14 bits */
r=(t*t)*(t/x);
t=t*((P0+r*P1)+(r*r)*P2);
/*
* Round t away from zero to 12 bits (sloppily except for ensuring that
* the result is larger in magnitude than cbrt(x) but not much more than
* 1 12-bit ulp larger). With rounding towards zero, the error bound
* would be ~5/6 instead of ~4/6, and with t 2 12-bit ulps larger the
* infinite-precision error in the Newton approximation would affect
* the second digit instead of the third digit of 4/6 = 0.666..., etc.
*/
GET_FLOAT_WORD(high,t);
SET_FLOAT_WORD(t,(high+0x1800)&0xfffff000);
/* one step Newton iteration to 24 bits with error < 0.669 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 */
return(t);
}
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