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	uint32_t hx, lx;
35	uint16_t expsign, es;
36	int k;
37	volatile double vd1, vd2;
38
39	GET_LDOUBLE_EXP(expsign,x);
40	k = expsign & 0x7fff;
41
42	/*
43	 * If x = +-Inf, then cbrt(x) = +-Inf.
44	 * If x = NaN, then cbrt(x) = NaN.
45	 */
46	if (k == BIAS + LDBL_MAX_EXP)
47		return (x + x);
48
49	if (k == 0) {
50		/* If x = +-0, then cbrt(x) = +-0. */
51		GET_LDOUBLE_WORDS(es,hx,lx,x);
52		if ((hx|lx) == 0) {
53			return (x);
54		}
55		/* Adjust subnormal numbers. */
56		x *= 0x1.0p514;
57		GET_LDOUBLE_EXP(k,x);
58		k &= 0x7fff;
59		k -= BIAS + 514;
60	} else
61		k -= BIAS;
62	SET_LDOUBLE_EXP(x,BIAS);
63	v = 1;
64
65	switch (k % 3) {
66	case 1:
67	case -2:
68		x = 2*x;
69		k--;
70		break;
71	case 2:
72	case -1:
73		x = 4*x;
74		k -= 2;
75		break;
76	}
77	SET_LDOUBLE_EXP(v, (expsign & 0x8000) | (BIAS + k / 3));
78
79	/*
80	 * The following is the guts of s_cbrtf, with the handling of
81	 * special values removed and extra care for accuracy not taken,
82	 * but with most of the extra accuracy not discarded.
83	 */
84
85	/* ~5-bit estimate: */
86	fx = x;
87	GET_FLOAT_WORD(hx, fx);
88	SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1));
89
90	/* ~16-bit estimate: */
91	dx = x;
92	dt = ft;
93	dr = dt * dt * dt;
94	dt = dt * (dx + dx + dr) / (dx + dr + dr);
95
96	/* ~47-bit estimate: */
97	dr = dt * dt * dt;
98	dt = dt * (dx + dx + dr) / (dx + dr + dr);
99
100	/*
101	 * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8).
102	 * Round it away from zero to 32 bits (32 so that t*t is exact, and
103	 * away from zero for technical reasons).
104	 */
105	vd2 = 0x1.0p32;
106	vd1 = 0x1.0p-31;
107	#define vd ((long double)vd2 + vd1)
108
109	t = dt + vd - 0x1.0p32;
110
111	/*
112	 * Final step Newton iteration to 64 or 113 bits with
113	 * error < 0.667 ulps
114	 */
115	s=t*t;				/* t*t is exact */
116	r=x/s;				/* error <= 0.5 ulps; |r| < |t| */
117	w=t+t;				/* t+t is exact */
118	r=(r-t)/(w+r);			/* r-t is exact; w+r ~= 3*t */
119	t=t+t*r;			/* error <= 0.5 + 0.5/3 + epsilon */
120
121	t *= v;
122	return (t);
123}
124