1 /*-
2 * SPDX-License-Identifier: BSD-2-Clause
3 *
4 * Copyright (c) 2007 Steven G. Kargl
5 * All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 * 1. Redistributions of source code must retain the above copyright
11 * notice unmodified, this list of conditions, and the following
12 * disclaimer.
13 * 2. Redistributions in binary form must reproduce the above copyright
14 * notice, this list of conditions and the following disclaimer in the
15 * documentation and/or other materials provided with the distribution.
16 *
17 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
18 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
19 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
20 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
21 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
22 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
23 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
24 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
25 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
26 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
27 */
28
29 #include <fenv.h>
30 #include <float.h>
31
32 #include "fpmath.h"
33 #include "math.h"
34
35 /* Return (x + ulp) for normal positive x. Assumes no overflow. */
36 static inline long double
inc(long double x)37 inc(long double x)
38 {
39 union IEEEl2bits u;
40
41 u.e = x;
42 if (++u.bits.manl == 0) {
43 if (++u.bits.manh == 0) {
44 u.bits.exp++;
45 u.bits.manh |= LDBL_NBIT;
46 }
47 }
48 return (u.e);
49 }
50
51 /* Return (x - ulp) for normal positive x. Assumes no underflow. */
52 static inline long double
dec(long double x)53 dec(long double x)
54 {
55 union IEEEl2bits u;
56
57 u.e = x;
58 if (u.bits.manl-- == 0) {
59 if (u.bits.manh-- == LDBL_NBIT) {
60 u.bits.exp--;
61 u.bits.manh |= LDBL_NBIT;
62 }
63 }
64 return (u.e);
65 }
66
67 #pragma STDC FENV_ACCESS ON
68
69 /*
70 * This is slow, but simple and portable. You should use hardware sqrt
71 * if possible.
72 */
73
74 long double
sqrtl(long double x)75 sqrtl(long double x)
76 {
77 union IEEEl2bits u;
78 int k, r;
79 long double lo, xn;
80 fenv_t env;
81
82 u.e = x;
83
84 /* If x = NaN, then sqrt(x) = NaN. */
85 /* If x = Inf, then sqrt(x) = Inf. */
86 /* If x = -Inf, then sqrt(x) = NaN. */
87 if (u.bits.exp == LDBL_MAX_EXP * 2 - 1)
88 return (x * x + x);
89
90 /* If x = +-0, then sqrt(x) = +-0. */
91 if ((u.bits.manh | u.bits.manl | u.bits.exp) == 0)
92 return (x);
93
94 /* If x < 0, then raise invalid and return NaN */
95 if (u.bits.sign)
96 return ((x - x) / (x - x));
97
98 feholdexcept(&env);
99
100 if (u.bits.exp == 0) {
101 /* Adjust subnormal numbers. */
102 u.e *= 0x1.0p514;
103 k = -514;
104 } else {
105 k = 0;
106 }
107 /*
108 * u.e is a normal number, so break it into u.e = e*2^n where
109 * u.e = (2*e)*2^2k for odd n and u.e = (4*e)*2^2k for even n.
110 */
111 if ((u.bits.exp - 0x3ffe) & 1) { /* n is odd. */
112 k += u.bits.exp - 0x3fff; /* 2k = n - 1. */
113 u.bits.exp = 0x3fff; /* u.e in [1,2). */
114 } else {
115 k += u.bits.exp - 0x4000; /* 2k = n - 2. */
116 u.bits.exp = 0x4000; /* u.e in [2,4). */
117 }
118
119 /*
120 * Newton's iteration.
121 * Split u.e into a high and low part to achieve additional precision.
122 */
123 xn = sqrt(u.e); /* 53-bit estimate of sqrtl(x). */
124 #if LDBL_MANT_DIG > 100
125 xn = (xn + (u.e / xn)) * 0.5; /* 106-bit estimate. */
126 #endif
127 lo = u.e;
128 u.bits.manl = 0; /* Zero out lower bits. */
129 lo = (lo - u.e) / xn; /* Low bits divided by xn. */
130 xn = xn + (u.e / xn); /* High portion of estimate. */
131 u.e = xn + lo; /* Combine everything. */
132 u.bits.exp += (k >> 1) - 1;
133
134 feclearexcept(FE_INEXACT);
135 r = fegetround();
136 fesetround(FE_TOWARDZERO); /* Set to round-toward-zero. */
137 xn = x / u.e; /* Chopped quotient (inexact?). */
138
139 if (!fetestexcept(FE_INEXACT)) { /* Quotient is exact. */
140 if (xn == u.e) {
141 fesetenv(&env);
142 return (u.e);
143 }
144 /* Round correctly for inputs like x = y**2 - ulp. */
145 xn = dec(xn); /* xn = xn - ulp. */
146 }
147
148 if (r == FE_TONEAREST) {
149 xn = inc(xn); /* xn = xn + ulp. */
150 } else if (r == FE_UPWARD) {
151 u.e = inc(u.e); /* u.e = u.e + ulp. */
152 xn = inc(xn); /* xn = xn + ulp. */
153 }
154 u.e = u.e + xn; /* Chopped sum. */
155 feupdateenv(&env); /* Restore env and raise inexact */
156 u.bits.exp--;
157 return (u.e);
158 }
159