1/* @(#)s_cbrt.c 5.1 93/09/24 */
2/*
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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 * Optimized by Bruce D. Evans.
13 */
14
15#include <sys/cdefs.h>
16__FBSDID("$FreeBSD$");
17
18#include <float.h>
19#include "math.h"
20#include "math_private.h"
21
22/* cbrt(x)
23 * Return cube root of x
24 */
25static const u_int32_t
26	B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */
27	B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */
28
29/* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */
30static const double
31P0 =  1.87595182427177009643,		/* 0x3ffe03e6, 0x0f61e692 */
32P1 = -1.88497979543377169875,		/* 0xbffe28e0, 0x92f02420 */
33P2 =  1.621429720105354466140,		/* 0x3ff9f160, 0x4a49d6c2 */
34P3 = -0.758397934778766047437,		/* 0xbfe844cb, 0xbee751d9 */
35P4 =  0.145996192886612446982;		/* 0x3fc2b000, 0xd4e4edd7 */
36
37double
38cbrt(double x)
39{
40	int32_t	hx;
41	union {
42	    double value;
43	    uint64_t bits;
44	} u;
45	double r,s,t=0.0,w;
46	u_int32_t sign;
47	u_int32_t high,low;
48
49	EXTRACT_WORDS(hx,low,x);
50	sign=hx&0x80000000; 		/* sign= sign(x) */
51	hx  ^=sign;
52	if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */
53
54    /*
55     * Rough cbrt to 5 bits:
56     *    cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
57     * where e is integral and >= 0, m is real and in [0, 1), and "/" and
58     * "%" are integer division and modulus with rounding towards minus
59     * infinity.  The RHS is always >= the LHS and has a maximum relative
60     * error of about 1 in 16.  Adding a bias of -0.03306235651 to the
61     * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
62     * floating point representation, for finite positive normal values,
63     * ordinary integer division of the value in bits magically gives
64     * almost exactly the RHS of the above provided we first subtract the
65     * exponent bias (1023 for doubles) and later add it back.  We do the
66     * subtraction virtually to keep e >= 0 so that ordinary integer
67     * division rounds towards minus infinity; this is also efficient.
68     */
69	if(hx<0x00100000) { 		/* zero or subnormal? */
70	    if((hx|low)==0)
71		return(x);		/* cbrt(0) is itself */
72	    SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */
73	    t*=x;
74	    GET_HIGH_WORD(high,t);
75	    INSERT_WORDS(t,sign|((high&0x7fffffff)/3+B2),0);
76	} else
77	    INSERT_WORDS(t,sign|(hx/3+B1),0);
78
79    /*
80     * New cbrt to 23 bits:
81     *    cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
82     * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
83     * to within 2**-23.5 when |r - 1| < 1/10.  The rough approximation
84     * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
85     * gives us bounds for r = t**3/x.
86     *
87     * Try to optimize for parallel evaluation as in k_tanf.c.
88     */
89	r=(t*t)*(t/x);
90	t=t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4));
91
92    /*
93     * Round t away from zero to 23 bits (sloppily except for ensuring that
94     * the result is larger in magnitude than cbrt(x) but not much more than
95     * 2 23-bit ulps larger).  With rounding towards zero, the error bound
96     * would be ~5/6 instead of ~4/6.  With a maximum error of 2 23-bit ulps
97     * in the rounded t, the infinite-precision error in the Newton
98     * approximation barely affects third digit in the final error
99     * 0.667; the error in the rounded t can be up to about 3 23-bit ulps
100     * before the final error is larger than 0.667 ulps.
101     */
102	u.value=t;
103	u.bits=(u.bits+0x80000000)&0xffffffffc0000000ULL;
104	t=u.value;
105
106    /* one step Newton iteration to 53 bits with error < 0.667 ulps */
107	s=t*t;				/* t*t is exact */
108	r=x/s;				/* error <= 0.5 ulps; |r| < |t| */
109	w=t+t;				/* t+t is exact */
110	r=(r-t)/(w+r);			/* r-t is exact; w+r ~= 3*t */
111	t=t+t*r;			/* error <= 0.5 + 0.5/3 + epsilon */
112
113	return(t);
114}
115
116#if (LDBL_MANT_DIG == 53)
117__weak_reference(cbrt, cbrtl);
118#endif
119