/* 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 "libm.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)
{
int32_t hx;
union dshape u;
double r,s,t=0.0,w;
uint32_t sign;
uint32_t high,low;
EXTRACT_WORDS(hx, low, x);
sign = hx & 0x80000000;
hx ^= sign;
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? */
if ((hx|low) == 0)
return x; /* cbrt(0) is itself */
SET_HIGH_WORD(t, 0x43500000); /* set t = 2**54 */
t *= x;
GET_HIGH_WORD(high, t);
INSERT_WORDS(t, sign|((high&0x7fffffff)/3+B2), 0);
} else
INSERT_WORDS(t, sign|(hx/3+B1), 0);
/*
* 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 k_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.value = t;
u.bits = (u.bits + 0x80000000) & 0xffffffffc0000000ULL;
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;
}