1 /*
2 * ====================================================
3 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
4 *
5 * Developed at SunPro, a Sun Microsystems, Inc. business.
6 * Permission to use, copy, modify, and distribute this
7 * software is freely granted, provided that this notice
8 * is preserved.
9 * ====================================================
10 *
11 * Optimized by Bruce D. Evans.
12 */
13
14 #include <float.h>
15 #include "math.h"
16 #include "math_private.h"
17
18 /* cbrt(x)
19 * Return cube root of x
20 */
21 static const u_int32_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
33 double
cbrt(double x)34 cbrt(double x)
35 {
36 int32_t hx;
37 union {
38 double value;
39 uint64_t bits;
40 } u;
41 double r,s,t=0.0,w;
42 u_int32_t sign;
43 u_int32_t high,low;
44
45 EXTRACT_WORDS(hx,low,x);
46 sign=hx&0x80000000; /* sign= sign(x) */
47 hx ^=sign;
48 if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */
49
50 /*
51 * Rough cbrt to 5 bits:
52 * cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
53 * where e is integral and >= 0, m is real and in [0, 1), and "/" and
54 * "%" are integer division and modulus with rounding towards minus
55 * infinity. The RHS is always >= the LHS and has a maximum relative
56 * error of about 1 in 16. Adding a bias of -0.03306235651 to the
57 * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
58 * floating point representation, for finite positive normal values,
59 * ordinary integer division of the value in bits magically gives
60 * almost exactly the RHS of the above provided we first subtract the
61 * exponent bias (1023 for doubles) and later add it back. We do the
62 * subtraction virtually to keep e >= 0 so that ordinary integer
63 * division rounds towards minus infinity; this is also efficient.
64 */
65 if(hx<0x00100000) { /* zero or subnormal? */
66 if((hx|low)==0)
67 return(x); /* cbrt(0) is itself */
68 SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */
69 t*=x;
70 GET_HIGH_WORD(high,t);
71 INSERT_WORDS(t,sign|((high&0x7fffffff)/3+B2),0);
72 } else
73 INSERT_WORDS(t,sign|(hx/3+B1),0);
74
75 /*
76 * New cbrt to 23 bits:
77 * cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
78 * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
79 * to within 2**-23.5 when |r - 1| < 1/10. The rough approximation
80 * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
81 * gives us bounds for r = t**3/x.
82 *
83 * Try to optimize for parallel evaluation as in k_tanf.c.
84 */
85 r=(t*t)*(t/x);
86 t=t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4));
87
88 /*
89 * Round t away from zero to 23 bits (sloppily except for ensuring that
90 * the result is larger in magnitude than cbrt(x) but not much more than
91 * 2 23-bit ulps larger). With rounding towards zero, the error bound
92 * would be ~5/6 instead of ~4/6. With a maximum error of 2 23-bit ulps
93 * in the rounded t, the infinite-precision error in the Newton
94 * approximation barely affects third digit in the final error
95 * 0.667; the error in the rounded t can be up to about 3 23-bit ulps
96 * before the final error is larger than 0.667 ulps.
97 */
98 u.value=t;
99 u.bits=(u.bits+0x80000000)&0xffffffffc0000000ULL;
100 t=u.value;
101
102 /* one step Newton iteration to 53 bits with error < 0.667 ulps */
103 s=t*t; /* t*t is exact */
104 r=x/s; /* error <= 0.5 ulps; |r| < |t| */
105 w=t+t; /* t+t is exact */
106 r=(r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */
107 t=t+t*r; /* error <= (0.5 + 0.5/3) * ulp */
108
109 return(t);
110 }
111
112 #if (LDBL_MANT_DIG == 53)
113 __weak_reference(cbrt, cbrtl);
114 #endif
115