cbrt.c

/* origin: FreeBSD /usr/src/lib/msun/src/s_cbrt.c */
/*
 * ====================================================
 * 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.
 */
/* cbrt(x)
 * Return cube root of x
 */

#include <math.h>
#include <stdint.h>

static const uint32_t
B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */
B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */

/* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */
static const double
P0 =  1.87595182427177009643,  /* 0x3ffe03e6, 0x0f61e692 */
P1 = -1.88497979543377169875,  /* 0xbffe28e0, 0x92f02420 */
P2 =  1.621429720105354466140, /* 0x3ff9f160, 0x4a49d6c2 */
P3 = -0.758397934778766047437, /* 0xbfe844cb, 0xbee751d9 */
P4 =  0.145996192886612446982; /* 0x3fc2b000, 0xd4e4edd7 */

double cbrt(double x)
{
	union {double f; uint64_t i;} u = {x};
	double_t r,s,t,w;
	uint32_t hx = u.i>>32 & 0x7fffffff;

	if (hx >= 0x7ff00000)  /* cbrt(NaN,INF) is itself */
		return x+x;

	/*
	 * 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) { /* zero or subnormal? */
		u.f = x*0x1p54;
		hx = u.i>>32 & 0x7fffffff;
		if (hx == 0)
			return x;  /* cbrt(0) is itself */
		hx = hx/3 + B2;
	} else
		hx = hx/3 + B1;
	u.i &= 1ULL<<63;
	u.i |= (uint64_t)hx << 32;
	t = u.f;

	/*
	 * New cbrt to 23 bits:
	 *    cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
	 * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
	 * to within 2**-23.5 when |r - 1| < 1/10.  The rough approximation
	 * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
	 * gives us bounds for r = t**3/x.
	 *
	 * Try to optimize for parallel evaluation as in __tanf.c.
	 */
	r = (t*t)*(t/x);
	t = t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4));

	/*
	 * Round t away from zero to 23 bits (sloppily except for ensuring that
	 * the result is larger in magnitude than cbrt(x) but not much more than
	 * 2 23-bit ulps larger).  With rounding towards zero, the error bound
	 * would be ~5/6 instead of ~4/6.  With a maximum error of 2 23-bit ulps
	 * in the rounded t, the infinite-precision error in the Newton
	 * approximation barely affects third digit in the final error
	 * 0.667; the error in the rounded t can be up to about 3 23-bit ulps
	 * before the final error is larger than 0.667 ulps.
	 */
	u.f = t;
	u.i = (u.i + 0x80000000) & 0xffffffffc0000000ULL;
	t = u.f;

	/* 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;
}