1 /*-
2 * Copyright (c) 2007 Steven G. Kargl
3 * All rights reserved.
4 *
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
7 * are met:
8 * 1. Redistributions of source code must retain the above copyright
9 * notice unmodified, this list of conditions, and the following
10 * disclaimer.
11 * 2. Redistributions in binary form must reproduce the above copyright
12 * notice, this list of conditions and the following disclaimer in the
13 * documentation and/or other materials provided with the distribution.
14 *
15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
16 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
17 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
18 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
19 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
20 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
21 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
22 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
23 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
24 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
25 */
26
27 #include <sys/cdefs.h>
28 #if 0
29 __FBSDID("$FreeBSD: head/lib/msun/src/e_sqrtl.c 176720 2008-03-02 01:47:58Z das $");
30 #endif
31 __RCSID("$NetBSD: e_sqrtl.c,v 1.4 2013/11/22 20:15:06 martin Exp $");
32
33 #include <machine/ieee.h>
34 #include <float.h>
35
36 #include "math.h"
37 #include "math_private.h"
38
39 #ifdef __HAVE_LONG_DOUBLE
40
41 #ifdef HAVE_FENV_H
42 #include <fenv.h>
43 #endif
44
45 #ifdef LDBL_IMPLICIT_NBIT
46 #define LDBL_NBIT 0
47 #endif
48
49 #ifdef HAVE_FENV_H
50
51 /* Return (x + ulp) for normal positive x. Assumes no overflow. */
52 static inline long double
inc(long double x)53 inc(long double x)
54 {
55 union ieee_ext_u ux = { .extu_ld = x, };
56
57 if (++ux.extu_fracl == 0) {
58 if (++ux.extu_frach == 0) {
59 ux.extu_exp++;
60 ux.extu_frach |= LDBL_NBIT;
61 }
62 }
63 return (ux.extu_ld);
64 }
65
66 /* Return (x - ulp) for normal positive x. Assumes no underflow. */
67 static inline long double
dec(long double x)68 dec(long double x)
69 {
70 union ieee_ext_u ux = { .extu_ld = x, };
71
72 if (ux.extu_fracl-- == 0) {
73 if (ux.extu_frach-- == LDBL_NBIT) {
74 ux.extu_exp--;
75 ux.extu_frach |= LDBL_NBIT;
76 }
77 }
78 return (ux.extu_ld);
79 }
80
81 /*
82 * This is slow, but simple and portable. You should use hardware sqrt
83 * if possible.
84 */
85
86 long double
__ieee754_sqrtl(long double x)87 __ieee754_sqrtl(long double x)
88 {
89 union ieee_ext_u ux = { .extu_ld = x, };
90 int k, r;
91 long double lo, xn;
92 fenv_t env;
93
94 /* If x = NaN, then sqrt(x) = NaN. */
95 /* If x = Inf, then sqrt(x) = Inf. */
96 /* If x = -Inf, then sqrt(x) = NaN. */
97 if (ux.extu_exp == LDBL_MAX_EXP * 2 - 1)
98 return (x * x + x);
99
100 /* If x = +-0, then sqrt(x) = +-0. */
101 if ((ux.extu_frach | ux.extu_fracl | ux.extu_exp) == 0)
102 return (x);
103
104 /* If x < 0, then raise invalid and return NaN */
105 if (ux.extu_sign)
106 return ((x - x) / (x - x));
107
108 feholdexcept(&env);
109
110 if (ux.extu_exp == 0) {
111 /* Adjust subnormal numbers. */
112 ux.extu_ld *= 0x1.0p514;
113 k = -514;
114 } else {
115 k = 0;
116 }
117 /*
118 * ux.extu_ld is a normal number, so break it into ux.extu_ld = e*2^n where
119 * ux.extu_ld = (2*e)*2^2k for odd n and ux.extu_ld = (4*e)*2^2k for even n.
120 */
121 if ((ux.extu_exp - EXT_EXP_BIAS) & 1) { /* n is even. */
122 k += ux.extu_exp - EXT_EXP_BIAS - 1; /* 2k = n - 2. */
123 ux.extu_exp = EXT_EXP_BIAS + 1; /* ux.extu_ld in [2,4). */
124 } else {
125 k += ux.extu_exp - EXT_EXP_BIAS; /* 2k = n - 1. */
126 ux.extu_exp = EXT_EXP_BIAS; /* ux.extu_ld in [1,2). */
127 }
128
129 /*
130 * Newton's iteration.
131 * Split ux.extu_ld into a high and low part to achieve additional precision.
132 */
133 xn = sqrt(ux.extu_ld); /* 53-bit estimate of sqrtl(x). */
134 #if LDBL_MANT_DIG > 100
135 xn = (xn + (ux.extu_ld / xn)) * 0.5; /* 106-bit estimate. */
136 #endif
137 lo = ux.extu_ld;
138 ux.extu_fracl = 0; /* Zero out lower bits. */
139 lo = (lo - ux.extu_ld) / xn; /* Low bits divided by xn. */
140 xn = xn + (ux.extu_ld / xn); /* High portion of estimate. */
141 ux.extu_ld = xn + lo; /* Combine everything. */
142 ux.extu_exp += (k >> 1) - 1;
143
144 feclearexcept(FE_INEXACT);
145 r = fegetround();
146 fesetround(FE_TOWARDZERO); /* Set to round-toward-zero. */
147 xn = x / ux.extu_ld; /* Chopped quotient (inexact?). */
148
149 if (!fetestexcept(FE_INEXACT)) { /* Quotient is exact. */
150 if (xn == ux.extu_ld) {
151 fesetenv(&env);
152 return (ux.extu_ld);
153 }
154 /* Round correctly for inputs like x = y**2 - ulp. */
155 xn = dec(xn); /* xn = xn - ulp. */
156 }
157
158 if (r == FE_TONEAREST) {
159 xn = inc(xn); /* xn = xn + ulp. */
160 } else if (r == FE_UPWARD) {
161 ux.extu_ld = inc(ux.extu_ld); /* ux.extu_ld = ux.extu_ld + ulp. */
162 xn = inc(xn); /* xn = xn + ulp. */
163 }
164 ux.extu_ld = ux.extu_ld + xn; /* Chopped sum. */
165 feupdateenv(&env); /* Restore env and raise inexact */
166 ux.extu_exp--;
167 return (ux.extu_ld);
168 }
169
170 #else
171
172 /*
173 * No fenv support:
174 * poor man's version: just use double
175 */
176 long double
__ieee754_sqrtl(long double x)177 __ieee754_sqrtl(long double x)
178 {
179 return __ieee754_sqrt((double)x);
180 }
181
182 #endif
183
184 #endif
185