1/* origin: FreeBSD /usr/src/lib/msun/src/s_cbrt.c */
  2/*
  3 * ====================================================
  4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  5 *
  6 * Developed at SunPro, a Sun Microsystems, Inc. business.
  7 * Permission to use, copy, modify, and distribute this
  8 * software is freely granted, provided that this notice
  9 * is preserved.
 10 * ====================================================
 11 *
 12 * Optimized by Bruce D. Evans.
 13 */
 14/* cbrt(x)
 15 * Return cube root of x
 16 */
 17
 18#include <math.h>
 19#include <stdint.h>
 20
 21static const uint32_t
 22B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */
 23B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */
 24
 25/* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */
 26static const double
 27P0 =  1.87595182427177009643,  /* 0x3ffe03e6, 0x0f61e692 */
 28P1 = -1.88497979543377169875,  /* 0xbffe28e0, 0x92f02420 */
 29P2 =  1.621429720105354466140, /* 0x3ff9f160, 0x4a49d6c2 */
 30P3 = -0.758397934778766047437, /* 0xbfe844cb, 0xbee751d9 */
 31P4 =  0.145996192886612446982; /* 0x3fc2b000, 0xd4e4edd7 */
 32
 33double cbrt(double x)
 34{
 35	union {double f; uint64_t i;} u = {x};
 36	double_t r,s,t,w;
 37	uint32_t hx = u.i>>32 & 0x7fffffff;
 38
 39	if (hx >= 0x7ff00000)  /* cbrt(NaN,INF) is itself */
 40		return x+x;
 41
 42	/*
 43	 * Rough cbrt to 5 bits:
 44	 *    cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
 45	 * where e is integral and >= 0, m is real and in [0, 1), and "/" and
 46	 * "%" are integer division and modulus with rounding towards minus
 47	 * infinity.  The RHS is always >= the LHS and has a maximum relative
 48	 * error of about 1 in 16.  Adding a bias of -0.03306235651 to the
 49	 * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
 50	 * floating point representation, for finite positive normal values,
 51	 * ordinary integer divison of the value in bits magically gives
 52	 * almost exactly the RHS of the above provided we first subtract the
 53	 * exponent bias (1023 for doubles) and later add it back.  We do the
 54	 * subtraction virtually to keep e >= 0 so that ordinary integer
 55	 * division rounds towards minus infinity; this is also efficient.
 56	 */
 57	if (hx < 0x00100000) { /* zero or subnormal? */
 58		u.f = x*0x1p54;
 59		hx = u.i>>32 & 0x7fffffff;
 60		if (hx == 0)
 61			return x;  /* cbrt(0) is itself */
 62		hx = hx/3 + B2;
 63	} else
 64		hx = hx/3 + B1;
 65	u.i &= 1ULL<<63;
 66	u.i |= (uint64_t)hx << 32;
 67	t = u.f;
 68
 69	/*
 70	 * New cbrt to 23 bits:
 71	 *    cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
 72	 * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
 73	 * to within 2**-23.5 when |r - 1| < 1/10.  The rough approximation
 74	 * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
 75	 * gives us bounds for r = t**3/x.
 76	 *
 77	 * Try to optimize for parallel evaluation as in __tanf.c.
 78	 */
 79	r = (t*t)*(t/x);
 80	t = t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4));
 81
 82	/*
 83	 * Round t away from zero to 23 bits (sloppily except for ensuring that
 84	 * the result is larger in magnitude than cbrt(x) but not much more than
 85	 * 2 23-bit ulps larger).  With rounding towards zero, the error bound
 86	 * would be ~5/6 instead of ~4/6.  With a maximum error of 2 23-bit ulps
 87	 * in the rounded t, the infinite-precision error in the Newton
 88	 * approximation barely affects third digit in the final error
 89	 * 0.667; the error in the rounded t can be up to about 3 23-bit ulps
 90	 * before the final error is larger than 0.667 ulps.
 91	 */
 92	u.f = t;
 93	u.i = (u.i + 0x80000000) & 0xffffffffc0000000ULL;
 94	t = u.f;
 95
 96	/* one step Newton iteration to 53 bits with error < 0.667 ulps */
 97	s = t*t;         /* t*t is exact */
 98	r = x/s;         /* error <= 0.5 ulps; |r| < |t| */
 99	w = t+t;         /* t+t is exact */
100	r = (r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */
101	t = t+t*r;       /* error <= 0.5 + 0.5/3 + epsilon */
102	return t;
103}