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