xref: /freebsd/lib/msun/src/s_cbrt.c (revision ce804bff581bc312583c4487f0167cd12f642a07)

/* @(#)s_cbrt.c 5.1 93/09/24 */
/*
 * ====================================================
 * 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.
 * ====================================================
 *
 * Optimized by Bruce D. Evans.
 */

#ifndef lint
static char rcsid[] = "$FreeBSD$";
#endif

#include "math.h"
#include "math_private.h"

/* cbrt(x)
 * Return cube root of x
 */
static const u_int32_t
	B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */
	B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */

static const double
C =  5.42857142857142815906e-01, /* 19/35     = 0x3FE15F15, 0xF15F15F1 */
D = -7.05306122448979611050e-01, /* -864/1225 = 0xBFE691DE, 0x2532C834 */
E =  1.41428571428571436819e+00, /* 99/70     = 0x3FF6A0EA, 0x0EA0EA0F */
F =  1.60714285714285720630e+00, /* 45/28     = 0x3FF9B6DB, 0x6DB6DB6E */
G =  3.57142857142857150787e-01; /* 5/14      = 0x3FD6DB6D, 0xB6DB6DB7 */

double
cbrt(double x)
{
	int32_t	hx;
	union {
	    double value;
	    uint64_t bits;
	} u;
	double r,s,t=0.0,w;
	uint64_t bits;
	u_int32_t sign;
	u_int32_t high,low;

	GET_HIGH_WORD(hx,x);
	sign=hx&0x80000000; 		/* sign= sign(x) */
	hx  ^=sign;
	if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */
	GET_LOW_WORD(low,x);
	if((hx|low)==0)
	    return(x);			/* cbrt(0) is itself */

    /*
     * Rough cbrt to 5 bits:
     *    cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
     * where e is integral and >= 0, m is real and in [0, 1), and "/" and
     * "%" are integer division and modulus with rounding towards minus
     * infinity.  The RHS is always >= the LHS and has a maximum relative
     * error of about 1 in 16.  Adding a bias of -0.03306235651 to the
     * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
     * floating point representation, for finite positive normal values,
     * ordinary integer divison of the value in bits magically gives
     * almost exactly the RHS of the above provided we first subtract the
     * exponent bias (1023 for doubles) and later add it back.  We do the
     * subtraction virtually to keep e >= 0 so that ordinary integer
     * division rounds towards minus infinity; this is also efficient.
     */
	if(hx<0x00100000) { 		/* subnormal number */
	    SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */
	    t*=x;
	    GET_HIGH_WORD(high,t);
	    SET_HIGH_WORD(t,sign|((high&0x7fffffff)/3+B2));
	} else
	    SET_HIGH_WORD(t,sign|(hx/3+B1));

    /*
     * New cbrt to 25 bits:
     *    cbrt(x) = t*cbrt(x/t**3) ~= t*R(x/t**3)
     * where R(r) = (14*r**2 + 35*r + 5)/(5*r**2 + 35*r + 14) is the
     * (2,2) Pade approximation to cbrt(r) at r = 1.  We replace
     * r = x/t**3 by 1/r = t**3/x since the latter can be evaluated
     * more efficiently, and rearrange the expression for R(r) to use
     * 4 additions and 2 divisions instead of the 4 additions, 4
     * multiplications and 1 division that would be required using
     * Horner's rule on the numerator and denominator.  t being good
     * to 32 bits means that |t/cbrt(x)-1| < 1/32, so |x/t**3-1| < 0.1
     * and for R(r) we can use any approximation to cbrt(r) that is good
     * to 20 bits on [0.9, 1.1].  The (2,2) Pade approximation is not an
     * especially good choice.
     */
	r=t*t/x;
	s=C+r*t;
	t*=G+F/(s+E+D/s);

    /*
     * Round t away from zero to 25 bits (sloppily except for ensuring that
     * the result is larger in magnitude than cbrt(x) but not much more than
     * 2 25-bit ulps larger).  With rounding towards zero, the error bound
     * would be ~5/6 instead of ~4/6.  With a maximum error of 1 25-bit ulps
     * in the rounded t, the infinite-precision error in the Newton
     * approximation barely affects third digit in the the final error
     * 0.667; the error in the rounded t can be up to about 12 25-bit ulps
     * before the final error is larger than 0.667 ulps.
     */
	u.value=t;
	u.bits=(u.bits+0x20000000)&0xfffffffff0000000ULL;
	t=u.value;

    /* one step Newton iteration to 53 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 */

	return(t);
}