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 <sys/cdefs.h>
18__FBSDID("$FreeBSD$");
19
20#include <float.h>
21#ifdef __i386__
22#include <ieeefp.h>
23#endif
24
25#include "fpmath.h"
26#include "math.h"
27#include "math_private.h"
28
29#define	BIAS	(LDBL_MAX_EXP - 1)
30
31static const unsigned
32    B1 = 709958130;	/* B1 = (127-127.0/3-0.03306235651)*2**23 */
33
34long double
35cbrtl(long double x)
36{
37	union IEEEl2bits u, v;
38	long double r, s, t, w;
39	double dr, dt, dx;
40	float ft, fx;
41	uint32_t hx;
42	uint16_t expsign;
43	int k;
44
45	u.e = x;
46	expsign = u.xbits.expsign;
47	k = expsign & 0x7fff;
48
49	/*
50	 * If x = +-Inf, then cbrt(x) = +-Inf.
51	 * If x = NaN, then cbrt(x) = NaN.
52	 */
53	if (k == BIAS + LDBL_MAX_EXP)
54		return (x + x);
55
56	ENTERI();
57	if (k == 0) {
58		/* If x = +-0, then cbrt(x) = +-0. */
59		if ((u.bits.manh | u.bits.manl) == 0)
60			RETURNI(x);
61		/* Adjust subnormal numbers. */
62		u.e *= 0x1.0p514;
63		k = u.bits.exp;
64		k -= BIAS + 514;
65 	} else
66		k -= BIAS;
67	u.xbits.expsign = BIAS;
68	v.e = 1;
69
70	x = u.e;
71	switch (k % 3) {
72	case 1:
73	case -2:
74		x = 2*x;
75		k--;
76		break;
77	case 2:
78	case -1:
79		x = 4*x;
80		k -= 2;
81		break;
82	}
83	v.xbits.expsign = (expsign & 0x8000) | (BIAS + k / 3);
84
85	/*
86	 * The following is the guts of s_cbrtf, with the handling of
87	 * special values removed and extra care for accuracy not taken,
88	 * but with most of the extra accuracy not discarded.
89	 */
90
91	/* ~5-bit estimate: */
92	fx = x;
93	GET_FLOAT_WORD(hx, fx);
94	SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1));
95
96	/* ~16-bit estimate: */
97	dx = x;
98	dt = ft;
99	dr = dt * dt * dt;
100	dt = dt * (dx + dx + dr) / (dx + dr + dr);
101
102	/* ~47-bit estimate: */
103	dr = dt * dt * dt;
104	dt = dt * (dx + dx + dr) / (dx + dr + dr);
105
106#if LDBL_MANT_DIG == 64
107	/*
108	 * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8).
109	 * Round it away from zero to 32 bits (32 so that t*t is exact, and
110	 * away from zero for technical reasons).
111	 */
112	volatile double vd2 = 0x1.0p32;
113	volatile double vd1 = 0x1.0p-31;
114	#define vd ((long double)vd2 + vd1)
115
116	t = dt + vd - 0x1.0p32;
117#elif LDBL_MANT_DIG == 113
118	/*
119	 * Round dt away from zero to 47 bits.  Since we don't trust the 47,
120	 * add 2 47-bit ulps instead of 1 to round up.  Rounding is slow and
121	 * might be avoidable in this case, since on most machines dt will
122	 * have been evaluated in 53-bit precision and the technical reasons
123	 * for rounding up might not apply to either case in cbrtl() since
124	 * dt is much more accurate than needed.
125	 */
126	t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60;
127#else
128#error "Unsupported long double format"
129#endif
130
131	/*
132     	 * Final step Newton iteration to 64 or 113 bits with
133	 * error < 0.667 ulps
134	 */
135	s=t*t;				/* t*t is exact */
136	r=x/s;				/* error <= 0.5 ulps; |r| < |t| */
137	w=t+t;				/* t+t is exact */
138	r=(r-t)/(w+r);			/* r-t is exact; w+r ~= 3*t */
139	t=t+t*r;			/* error <= 0.5 + 0.5/3 + epsilon */
140
141	t *= v.e;
142	RETURNI(t);
143}
144