s_cbrtl.c revision 238923 
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: head/lib/msun/src/s_cbrtl.c 238923 2012-07-30 21:55:49Z kargl $");
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	ENTERI();
55	if (k == 0) {
56		/* If x = +-0, then cbrt(x) = +-0. */
57		if ((u.bits.manh | u.bits.manl) == 0)
58			RETURNI(x);
59		/* Adjust subnormal numbers. */
60		u.e *= 0x1.0p514;
61		k = u.bits.exp;
62		k -= BIAS + 514;
63 	} else
64		k -= BIAS;
65	u.xbits.expsign = BIAS;
66	v.e = 1;
67
68	x = u.e;
69	switch (k % 3) {
70	case 1:
71	case -2:
72		x = 2*x;
73		k--;
74		break;
75	case 2:
76	case -1:
77		x = 4*x;
78		k -= 2;
79		break;
80	}
81	v.xbits.expsign = (expsign & 0x8000) | (BIAS + k / 3);
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#if LDBL_MANT_DIG == 64
105	/*
106	 * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8).
107	 * Round it away from zero to 32 bits (32 so that t*t is exact, and
108	 * away from zero for technical reasons).
109	 */
110	volatile double vd2 = 0x1.0p32;
111	volatile double vd1 = 0x1.0p-31;
112	#define vd ((long double)vd2 + vd1)
113
114	t = dt + vd - 0x1.0p32;
115#elif LDBL_MANT_DIG == 113
116	/*
117	 * Round dt away from zero to 47 bits.  Since we don't trust the 47,
118	 * add 2 47-bit ulps instead of 1 to round up.  Rounding is slow and
119	 * might be avoidable in this case, since on most machines dt will
120	 * have been evaluated in 53-bit precision and the technical reasons
121	 * for rounding up might not apply to either case in cbrtl() since
122	 * dt is much more accurate than needed.
123	 */
124	t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60;
125#else
126#error "Unsupported long double format"
127#endif
128
129	/*
130     	 * Final step Newton iteration to 64 or 113 bits with
131	 * error < 0.667 ulps
132	 */
133	s=t*t;				/* t*t is exact */
134	r=x/s;				/* error <= 0.5 ulps; |r| < |t| */
135	w=t+t;				/* t+t is exact */
136	r=(r-t)/(w+r);			/* r-t is exact; w+r ~= 3*t */
137	t=t+t*r;			/* error <= 0.5 + 0.5/3 + epsilon */
138
139	t *= v.e;
140	RETURNI(t);
141}
142