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#include <sys/cdefs.h>
17__RCSID("$NetBSD: s_cbrtl.c,v 1.4 2024/04/03 18:53:42 christos Exp $");
18
19
20#include "namespace.h"
21#include <machine/ieee.h>
22#include <float.h>
23
24#include "math.h"
25#include "math_private.h"
26
27#ifdef __HAVE_LONG_DOUBLE
28__weak_alias(cbrtl, _cbrtl)
29
30#define	BIAS	(LDBL_MAX_EXP - 1)
31
32static const unsigned
33    B1 = 709958130;	/* B1 = (127-127.0/3-0.03306235651)*2**23 */
34
35long double
36cbrtl(long double x)
37{
38	union ieee_ext_u u, v;
39	long double r, s, t, w;
40	double dr, dt, dx;
41	float ft, fx;
42	uint32_t hx;
43	uint16_t expsign;
44	int k;
45
46	u.extu_ld = x;
47	expsign = GET_EXPSIGN(&u);
48	k = expsign & 0x7fff;
49
50	/*
51	 * If x = +-Inf, then cbrt(x) = +-Inf.
52	 * If x = NaN, then cbrt(x) = NaN.
53	 */
54	if (k == BIAS + LDBL_MAX_EXP)
55		return (x + x);
56
57	ENTERI();
58	if (k == 0) {
59		/* If x = +-0, then cbrt(x) = +-0. */
60		if ((u.extu_frach | u.extu_fracl) == 0)
61			RETURNI(x);
62		/* Adjust subnormal numbers. */
63		u.extu_ld *= 0x1.0p514;
64		k = u.extu_exp;
65		k -= BIAS + 514;
66 	} else
67		k -= BIAS;
68	SET_EXPSIGN(&u, BIAS);
69	v.extu_ld = 1;
70
71	x = u.extu_ld;
72	switch (k % 3) {
73	case 1:
74	case -2:
75		x = 2*x;
76		k--;
77		break;
78	case 2:
79	case -1:
80		x = 4*x;
81		k -= 2;
82		break;
83	}
84	SET_EXPSIGN(&v, (expsign & 0x8000) | (BIAS + k / 3));
85
86	/*
87	 * The following is the guts of s_cbrtf, with the handling of
88	 * special values removed and extra care for accuracy not taken,
89	 * but with most of the extra accuracy not discarded.
90	 */
91
92	/* ~5-bit estimate: */
93	fx = x;
94	GET_FLOAT_WORD(hx, fx);
95	SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1));
96
97	/* ~16-bit estimate: */
98	dx = x;
99	dt = ft;
100	dr = dt * dt * dt;
101	dt = dt * (dx + dx + dr) / (dx + dr + dr);
102
103	/* ~47-bit estimate: */
104	dr = dt * dt * dt;
105	dt = dt * (dx + dx + dr) / (dx + dr + dr);
106
107#if LDBL_MANT_DIG == 64
108	/*
109	 * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8).
110	 * Round it away from zero to 32 bits (32 so that t*t is exact, and
111	 * away from zero for technical reasons).
112	 */
113	volatile double vd2 = 0x1.0p32;
114	volatile double vd1 = 0x1.0p-31;
115	#define vd ((long double)vd2 + vd1)
116
117	t = dt + vd - 0x1.0p32;
118#elif LDBL_MANT_DIG == 113
119	/*
120	 * Round dt away from zero to 47 bits.  Since we don't trust the 47,
121	 * add 2 47-bit ulps instead of 1 to round up.  Rounding is slow and
122	 * might be avoidable in this case, since on most machines dt will
123	 * have been evaluated in 53-bit precision and the technical reasons
124	 * for rounding up might not apply to either case in cbrtl() since
125	 * dt is much more accurate than needed.
126	 */
127	t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60;
128#else
129#error "Unsupported long double format"
130#endif
131
132	/*
133     	 * Final step Newton iteration to 64 or 113 bits with
134	 * error < 0.667 ulps
135	 */
136	s=t*t;				/* t*t is exact */
137	r=x/s;				/* error <= 0.5 ulps; |r| < |t| */
138	w=t+t;				/* t+t is exact */
139	r=(r-t)/(w+r);			/* r-t is exact; w+r ~= 3*t */
140	t=t+t*r;			/* error <= (0.5 + 0.5/3) * ulp */
141
142	t *= v.extu_ld;
143	RETURNI(t);
144}
145#endif /* __HAVE_LONG_DOUBLE */
146