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 <float.h>
18 #include <ieeefp.h>
19 #include <math.h>
20 
21 #include "math_private.h"
22 
23 #define	BIAS	(LDBL_MAX_EXP - 1)
24 
25 static const unsigned
26     B1 = 709958130;	/* B1 = (127-127.0/3-0.03306235651)*2**23 */
27 
28 long double
29 cbrtl(long double x)
30 {
31 	long double v, r, s, t, w;
32 	double dr, dt, dx;
33 	float ft, fx;
34 	uint32_t hx, lx;
35 	uint16_t expsign, es;
36 	int k;
37 	volatile double vd1, vd2;
38 
39 	GET_LDOUBLE_EXP(expsign,x);
40 	k = expsign & 0x7fff;
41 
42 	/*
43 	 * If x = +-Inf, then cbrt(x) = +-Inf.
44 	 * If x = NaN, then cbrt(x) = NaN.
45 	 */
46 	if (k == BIAS + LDBL_MAX_EXP)
47 		return (x + x);
48 
49 	if (k == 0) {
50 		/* If x = +-0, then cbrt(x) = +-0. */
51 		GET_LDOUBLE_WORDS(es,hx,lx,x);
52 		if ((hx|lx) == 0) {
53 			return (x);
54 		}
55 		/* Adjust subnormal numbers. */
56 		x *= 0x1.0p514;
57 		GET_LDOUBLE_EXP(k,x);
58 		k &= 0x7fff;
59 		k -= BIAS + 514;
60 	} else
61 		k -= BIAS;
62 	SET_LDOUBLE_EXP(x,BIAS);
63 	v = 1;
64 
65 	switch (k % 3) {
66 	case 1:
67 	case -2:
68 		x = 2*x;
69 		k--;
70 		break;
71 	case 2:
72 	case -1:
73 		x = 4*x;
74 		k -= 2;
75 		break;
76 	}
77 	SET_LDOUBLE_EXP(v, (expsign & 0x8000) | (BIAS + k / 3));
78 
79 	/*
80 	 * The following is the guts of s_cbrtf, with the handling of
81 	 * special values removed and extra care for accuracy not taken,
82 	 * but with most of the extra accuracy not discarded.
83 	 */
84 
85 	/* ~5-bit estimate: */
86 	fx = x;
87 	GET_FLOAT_WORD(hx, fx);
88 	SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1));
89 
90 	/* ~16-bit estimate: */
91 	dx = x;
92 	dt = ft;
93 	dr = dt * dt * dt;
94 	dt = dt * (dx + dx + dr) / (dx + dr + dr);
95 
96 	/* ~47-bit estimate: */
97 	dr = dt * dt * dt;
98 	dt = dt * (dx + dx + dr) / (dx + dr + dr);
99 
100 	/*
101 	 * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8).
102 	 * Round it away from zero to 32 bits (32 so that t*t is exact, and
103 	 * away from zero for technical reasons).
104 	 */
105 	vd2 = 0x1.0p32;
106 	vd1 = 0x1.0p-31;
107 	#define vd ((long double)vd2 + vd1)
108 
109 	t = dt + vd - 0x1.0p32;
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;
122 	return (t);
123 }
124