1 /* origin: FreeBSD /usr/src/lib/msun/src/s_cbrt.c */
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 /* cbrt(x)
15 * Return cube root of x
16 */
17
18 #include <math.h>
19 #include <stdint.h>
20
21 static const uint32_t
22 B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */
23 B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */
24
25 /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */
26 static const double
27 P0 = 1.87595182427177009643, /* 0x3ffe03e6, 0x0f61e692 */
28 P1 = -1.88497979543377169875, /* 0xbffe28e0, 0x92f02420 */
29 P2 = 1.621429720105354466140, /* 0x3ff9f160, 0x4a49d6c2 */
30 P3 = -0.758397934778766047437, /* 0xbfe844cb, 0xbee751d9 */
31 P4 = 0.145996192886612446982; /* 0x3fc2b000, 0xd4e4edd7 */
32
cbrt(double x)33 double cbrt(double x)
34 {
35 union {double f; uint64_t i;} u = {x};
36 double_t r,s,t,w;
37 uint32_t hx = u.i>>32 & 0x7fffffff;
38
39 if (hx >= 0x7ff00000) /* cbrt(NaN,INF) is itself */
40 return x+x;
41
42 /*
43 * Rough cbrt to 5 bits:
44 * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
45 * where e is integral and >= 0, m is real and in [0, 1), and "/" and
46 * "%" are integer division and modulus with rounding towards minus
47 * infinity. The RHS is always >= the LHS and has a maximum relative
48 * error of about 1 in 16. Adding a bias of -0.03306235651 to the
49 * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
50 * floating point representation, for finite positive normal values,
51 * ordinary integer divison of the value in bits magically gives
52 * almost exactly the RHS of the above provided we first subtract the
53 * exponent bias (1023 for doubles) and later add it back. We do the
54 * subtraction virtually to keep e >= 0 so that ordinary integer
55 * division rounds towards minus infinity; this is also efficient.
56 */
57 if (hx < 0x00100000) { /* zero or subnormal? */
58 u.f = x*0x1p54;
59 hx = u.i>>32 & 0x7fffffff;
60 if (hx == 0)
61 return x; /* cbrt(0) is itself */
62 hx = hx/3 + B2;
63 } else
64 hx = hx/3 + B1;
65 u.i &= 1ULL<<63;
66 u.i |= (uint64_t)hx << 32;
67 t = u.f;
68
69 /*
70 * New cbrt to 23 bits:
71 * cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
72 * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
73 * to within 2**-23.5 when |r - 1| < 1/10. The rough approximation
74 * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
75 * gives us bounds for r = t**3/x.
76 *
77 * Try to optimize for parallel evaluation as in __tanf.c.
78 */
79 r = (t*t)*(t/x);
80 t = t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4));
81
82 /*
83 * Round t away from zero to 23 bits (sloppily except for ensuring that
84 * the result is larger in magnitude than cbrt(x) but not much more than
85 * 2 23-bit ulps larger). With rounding towards zero, the error bound
86 * would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps
87 * in the rounded t, the infinite-precision error in the Newton
88 * approximation barely affects third digit in the final error
89 * 0.667; the error in the rounded t can be up to about 3 23-bit ulps
90 * before the final error is larger than 0.667 ulps.
91 */
92 u.f = t;
93 u.i = (u.i + 0x80000000) & 0xffffffffc0000000ULL;
94 t = u.f;
95
96 /* one step Newton iteration to 53 bits with error < 0.667 ulps */
97 s = t*t; /* t*t is exact */
98 r = x/s; /* error <= 0.5 ulps; |r| < |t| */
99 w = t+t; /* t+t is exact */
100 r = (r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */
101 t = t+t*r; /* error <= 0.5 + 0.5/3 + epsilon */
102 return t;
103 }
104