1/*-
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 * Copyright (c) 2009-2011, Bruce D. Evans, Steven G. Kargl, David Schultz.
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 * The argument reduction and testing for exceptional cases was
13 * written by Steven G. Kargl with input from Bruce D. Evans
14 * and David A. Schultz.
15 */
16
17#include <float.h>
18#include <ieeefp.h>
19#include <math.h>
20
21#include "math_private.h"
22
23#define	BIAS	(LDBL_MAX_EXP - 1)
24
25static const unsigned
26    B1 = 709958130;	/* B1 = (127-127.0/3-0.03306235651)*2**23 */
27
28long double
29cbrtl(long double x)
30{
31	long double v, r, s, t, w;
32	double dr, dt, dx;
33	float ft, fx;
34	uint64_t hx, lx;
35	uint16_t expsign;
36	int k;
37
38	GET_LDOUBLE_MSW64(hx,x);
39	k = (hx>>48)&0x7fff;
40
41	/*
42	 * If x = +-Inf, then cbrt(x) = +-Inf.
43	 * If x = NaN, then cbrt(x) = NaN.
44	 */
45	if (k == BIAS + LDBL_MAX_EXP)
46		return (x + x);
47
48	if (k == 0) {
49		/* If x = +-0, then cbrt(x) = +-0. */
50		GET_LDOUBLE_WORDS64(hx,lx,x);
51		if (((hx&0x7fffffffffffffffLL)|lx) == 0) {
52			return (x);
53		}
54		/* Adjust subnormal numbers. */
55		x *= 0x1.0p514;
56		GET_LDOUBLE_MSW64(hx,x);
57		k = (hx>>48)&0x7fff;
58		k -= BIAS + 514;
59	} else
60		k -= BIAS;
61	GET_LDOUBLE_MSW64(hx,x);
62	hx = (hx&0x8000ffffffffffffLL)|((uint64_t)BIAS<<48);
63	SET_LDOUBLE_MSW64(x,hx);
64	v = 1;
65
66	switch (k % 3) {
67	case 1:
68	case -2:
69		x = 2*x;
70		k--;
71		break;
72	case 2:
73	case -1:
74		x = 4*x;
75		k -= 2;
76		break;
77	}
78	GET_LDOUBLE_MSW64(hx,x);
79	expsign = ((hx>>48) & 0x8000) | (BIAS + k / 3);
80	hx = (hx&0x8000ffffffffffffLL)|((uint64_t)expsign<<48);
81	SET_LDOUBLE_MSW64(x,hx);
82
83	/*
84	 * The following is the guts of s_cbrtf, with the handling of
85	 * special values removed and extra care for accuracy not taken,
86	 * but with most of the extra accuracy not discarded.
87	 */
88
89	/* ~5-bit estimate: */
90	fx = x;
91	GET_FLOAT_WORD(hx, fx);
92	SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1));
93
94	/* ~16-bit estimate: */
95	dx = x;
96	dt = ft;
97	dr = dt * dt * dt;
98	dt = dt * (dx + dx + dr) / (dx + dr + dr);
99
100	/* ~47-bit estimate: */
101	dr = dt * dt * dt;
102	dt = dt * (dx + dx + dr) / (dx + dr + dr);
103
104	/*
105	 * Round dt away from zero to 47 bits.  Since we don't trust the 47,
106	 * add 2 47-bit ulps instead of 1 to round up.  Rounding is slow and
107	 * might be avoidable in this case, since on most machines dt will
108	 * have been evaluated in 53-bit precision and the technical reasons
109	 * for rounding up might not apply to either case in cbrtl() since
110	 * dt is much more accurate than needed.
111	 */
112	t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60;
113
114	/*
115	 * Final step Newton iteration to 64 or 113 bits with
116	 * error < 0.667 ulps
117	 */
118	s=t*t;				/* t*t is exact */
119	r=x/s;				/* error <= 0.5 ulps; |r| < |t| */
120	w=t+t;				/* t+t is exact */
121	r=(r-t)/(w+r);			/* r-t is exact; w+r ~= 3*t */
122	t=t+t*r;			/* error <= 0.5 + 0.5/3 + epsilon */
123
124	t *= v;
125	return (t);
126}
127