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#include <ieeefp.h>
22
23#include "fpmath.h"
24#include "math.h"
25#include "math_private.h"
26
27#define	BIAS	(LDBL_MAX_EXP - 1)
28
29static const unsigned
30    B1 = 709958130;	/* B1 = (127-127.0/3-0.03306235651)*2**23 */
31
32long double
33cbrtl(long double x)
34{
35	union IEEEl2bits u, v;
36	long double r, s, t, w;
37	double dr, dt, dx;
38	float ft, fx;
39	uint32_t hx;
40	uint16_t expsign;
41	int k;
42
43	u.e = x;
44	expsign = u.xbits.expsign;
45	k = expsign & 0x7fff;
46
47	/*
48	 * If x = +-Inf, then cbrt(x) = +-Inf.
49	 * If x = NaN, then cbrt(x) = NaN.
50	 */
51	if (k == BIAS + LDBL_MAX_EXP)
52		return (x + x);
53
54#ifdef __i386__
55	fp_prec_t oprec;
56
57	oprec = fpgetprec();
58	if (oprec != FP_PE)
59		fpsetprec(FP_PE);
60#endif
61
62	if (k == 0) {
63		/* If x = +-0, then cbrt(x) = +-0. */
64		if ((u.bits.manh | u.bits.manl) == 0) {
65#ifdef __i386__
66			if (oprec != FP_PE)
67				fpsetprec(oprec);
68#endif
69			return (x);
70	    	}
71		/* Adjust subnormal numbers. */
72		u.e *= 0x1.0p514;
73		k = u.bits.exp;
74		k -= BIAS + 514;
75 	} else
76		k -= BIAS;
77	u.xbits.expsign = BIAS;
78	v.e = 1;
79
80	x = u.e;
81	switch (k % 3) {
82	case 1:
83	case -2:
84		x = 2*x;
85		k--;
86		break;
87	case 2:
88	case -1:
89		x = 4*x;
90		k -= 2;
91		break;
92	}
93	v.xbits.expsign = (expsign & 0x8000) | (BIAS + k / 3);
94
95	/*
96	 * The following is the guts of s_cbrtf, with the handling of
97	 * special values removed and extra care for accuracy not taken,
98	 * but with most of the extra accuracy not discarded.
99	 */
100
101	/* ~5-bit estimate: */
102	fx = x;
103	GET_FLOAT_WORD(hx, fx);
104	SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1));
105
106	/* ~16-bit estimate: */
107	dx = x;
108	dt = ft;
109	dr = dt * dt * dt;
110	dt = dt * (dx + dx + dr) / (dx + dr + dr);
111
112	/* ~47-bit estimate: */
113	dr = dt * dt * dt;
114	dt = dt * (dx + dx + dr) / (dx + dr + dr);
115
116#if LDBL_MANT_DIG == 64
117	/*
118	 * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8).
119	 * Round it away from zero to 32 bits (32 so that t*t is exact, and
120	 * away from zero for technical reasons).
121	 */
122	volatile double vd2 = 0x1.0p32;
123	volatile double vd1 = 0x1.0p-31;
124	#define vd ((long double)vd2 + vd1)
125
126	t = dt + vd - 0x1.0p32;
127#elif LDBL_MANT_DIG == 113
128	/*
129	 * Round dt away from zero to 47 bits.  Since we don't trust the 47,
130	 * add 2 47-bit ulps instead of 1 to round up.  Rounding is slow and
131	 * might be avoidable in this case, since on most machines dt will
132	 * have been evaluated in 53-bit precision and the technical reasons
133	 * for rounding up might not apply to either case in cbrtl() since
134	 * dt is much more accurate than needed.
135	 */
136	t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60;
137#else
138#error "Unsupported long double format"
139#endif
140
141	/*
142     	 * Final step Newton iteration to 64 or 113 bits with
143	 * error < 0.667 ulps
144	 */
145	s=t*t;				/* t*t is exact */
146	r=x/s;				/* error <= 0.5 ulps; |r| < |t| */
147	w=t+t;				/* t+t is exact */
148	r=(r-t)/(w+r);			/* r-t is exact; w+r ~= 3*t */
149	t=t+t*r;			/* error <= 0.5 + 0.5/3 + epsilon */
150
151	t *= v.e;
152#ifdef __i386__
153	if (oprec != FP_PE)
154		fpsetprec(oprec);
155#endif
156	return (t);
157}
158