1/* origin: FreeBSD /usr/src/lib/msun/src/s_cbrtl.c */
2/*-
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 * Copyright (c) 2009-2011, Bruce D. Evans, Steven G. Kargl, David Schultz.
6 *
7 * Developed at SunPro, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
10 * is preserved.
11 * ====================================================
12 *
13 * The argument reduction and testing for exceptional cases was
14 * written by Steven G. Kargl with input from Bruce D. Evans
15 * and David A. Schultz.
16 */
17
18#include "libm.h"
19
20#if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
21long double cbrtl(long double x)
22{
23	return cbrt(x);
24}
25#elif (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
26static const unsigned B1 = 709958130; /* B1 = (127-127.0/3-0.03306235651)*2**23 */
27
28long double cbrtl(long double x)
29{
30	union ldshape u = {x}, v;
31	union {float f; uint32_t i;} uft;
32	long double r, s, t, w;
33	double_t dr, dt, dx;
34	float_t ft;
35	int e = u.i.se & 0x7fff;
36	int sign = u.i.se & 0x8000;
37
38	/*
39	 * If x = +-Inf, then cbrt(x) = +-Inf.
40	 * If x = NaN, then cbrt(x) = NaN.
41	 */
42	if (e == 0x7fff)
43		return x + x;
44	if (e == 0) {
45		/* Adjust subnormal numbers. */
46		u.f *= 0x1p120;
47		e = u.i.se & 0x7fff;
48		/* If x = +-0, then cbrt(x) = +-0. */
49		if (e == 0)
50			return x;
51		e -= 120;
52	}
53	e -= 0x3fff;
54	u.i.se = 0x3fff;
55	x = u.f;
56	switch (e % 3) {
57	case 1:
58	case -2:
59		x *= 2;
60		e--;
61		break;
62	case 2:
63	case -1:
64		x *= 4;
65		e -= 2;
66		break;
67	}
68	v.f = 1.0;
69	v.i.se = sign | (0x3fff + e/3);
70
71	/*
72	 * The following is the guts of s_cbrtf, with the handling of
73	 * special values removed and extra care for accuracy not taken,
74	 * but with most of the extra accuracy not discarded.
75	 */
76
77	/* ~5-bit estimate: */
78	uft.f = x;
79	uft.i = (uft.i & 0x7fffffff)/3 + B1;
80	ft = uft.f;
81
82	/* ~16-bit estimate: */
83	dx = x;
84	dt = ft;
85	dr = dt * dt * dt;
86	dt = dt * (dx + dx + dr) / (dx + dr + dr);
87
88	/* ~47-bit estimate: */
89	dr = dt * dt * dt;
90	dt = dt * (dx + dx + dr) / (dx + dr + dr);
91
92#if LDBL_MANT_DIG == 64
93	/*
94	 * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8).
95	 * Round it away from zero to 32 bits (32 so that t*t is exact, and
96	 * away from zero for technical reasons).
97	 */
98	t = dt + (0x1.0p32L + 0x1.0p-31L) - 0x1.0p32;
99#elif LDBL_MANT_DIG == 113
100	/*
101	 * Round dt away from zero to 47 bits.  Since we don't trust the 47,
102	 * add 2 47-bit ulps instead of 1 to round up.  Rounding is slow and
103	 * might be avoidable in this case, since on most machines dt will
104	 * have been evaluated in 53-bit precision and the technical reasons
105	 * for rounding up might not apply to either case in cbrtl() since
106	 * dt is much more accurate than needed.
107	 */
108	t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60;
109#endif
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.f;
122	return t;
123}
124#endif
125