1 /* origin: FreeBSD /usr/src/lib/msun/src/s_cbrtl.c */
3 * ====================================================
4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5 * Copyright (c) 2009-2011, Bruce D. Evans, Steven G. Kargl, David Schultz.
7 * Developed at SunPro, a Sun Microsystems, Inc. business.
8 * Permission to use, copy, modify, and distribute this
9 * software is freely granted, provided that this notice
11 * ====================================================
13 * The argument reduction and testing for exceptional cases was
14 * written by Steven G. Kargl with input from Bruce D. Evans
15 * and David A. Schultz.
20 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
21 long double cbrtl(long double x)
25 #elif (LDBL_MANT_DIG == 64 || LDBL_MANT_DIG == 113) && LDBL_MAX_EXP == 16384
27 #define BIAS (LDBL_MAX_EXP - 1)
28 static const unsigned B1 = 709958130; /* B1 = (127-127.0/3-0.03306235651)*2**23 */
30 long double cbrtl(long double x)
32 union IEEEl2bits u, v;
33 long double r, s, t, w;
41 expsign = u.xbits.expsign;
45 * If x = +-Inf, then cbrt(x) = +-Inf.
46 * If x = NaN, then cbrt(x) = NaN.
48 if (k == BIAS + LDBL_MAX_EXP)
52 /* If x = +-0, then cbrt(x) = +-0. */
53 if ((u.bits.manh | u.bits.manl) == 0)
55 /* Adjust subnormal numbers. */
61 u.xbits.expsign = BIAS;
77 v.xbits.expsign = (expsign & 0x8000) | (BIAS + k / 3);
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.
85 /* ~5-bit estimate: */
87 GET_FLOAT_WORD(hx, fx);
88 SET_FLOAT_WORD(ft, ((hx & 0x7fffffff) / 3 + B1));
90 /* ~16-bit estimate: */
94 dt = dt * (dx + dx + dr) / (dx + dr + dr);
96 /* ~47-bit estimate: */
98 dt = dt * (dx + dx + dr) / (dx + dr + dr);
100 #if LDBL_MANT_DIG == 64
102 * dt is cbrtl(x) to ~47 bits (after x has been reduced to 1 <= x < 8).
103 * Round it away from zero to 32 bits (32 so that t*t is exact, and
104 * away from zero for technical reasons).
106 t = dt + (0x1.0p32L + 0x1.0p-31L) - 0x1.0p32;
107 #elif LDBL_MANT_DIG == 113
109 * Round dt away from zero to 47 bits. Since we don't trust the 47,
110 * add 2 47-bit ulps instead of 1 to round up. Rounding is slow and
111 * might be avoidable in this case, since on most machines dt will
112 * have been evaluated in 53-bit precision and the technical reasons
113 * for rounding up might not apply to either case in cbrtl() since
114 * dt is much more accurate than needed.
116 t = dt + 0x2.0p-46 + 0x1.0p60L - 0x1.0p60;
120 * Final step Newton iteration to 64 or 113 bits with
123 s = t*t; /* t*t is exact */
124 r = x/s; /* error <= 0.5 ulps; |r| < |t| */
125 w = t+t; /* t+t is exact */
126 r = (r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */
127 t = t+t*r; /* error <= 0.5 + 0.5/3 + epsilon */