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
28 #ifndef _DOUBLE_DOUBLE_FLOAT
29 /* Return (x + ulp) for normal positive x. Assumes no overflow. */
30 static inline long double
inc(long double x)31 inc(long double x)
32 {
33 union IEEEl2bits u;
34
35 u.e = x;
36 if (++u.bits.manl == 0) {
37 if (++u.bits.manh == 0) {
38 u.bits.exp++;
39 u.bits.manh |= LDBL_NBIT;
40 }
41 }
42 return (u.e);
43 }
44
45 /* Return (x - ulp) for normal positive x. Assumes no underflow. */
46 static inline long double
dec(long double x)47 dec(long double x)
48 {
49 union IEEEl2bits u;
50
51 u.e = x;
52 if (u.bits.manl-- == 0) {
53 if (u.bits.manh-- == LDBL_NBIT) {
54 u.bits.exp--;
55 u.bits.manh |= LDBL_NBIT;
56 }
57 }
58 return (u.e);
59 }
60
61 #ifndef __GNUC__
62 #pragma STDC FENV_ACCESS ON
63 #endif
64
65 #endif
66
67 /*
68 * This is slow, but simple and portable. You should use hardware sqrt
69 * if possible.
70 */
71
72 #define BIAS (LDBL_MAX_EXP-1)
73
74 long double
sqrtl(long double x)75 sqrtl(long double x)
76 {
77 union IEEEl2bits u;
78 int k;
79 long double lo, xn;
80 fenv_t env;
81
82 u.e = x;
83
84 /* If x = NaN, then sqrt(x) = qNaN. */
85 /* If x = Inf, then sqrt(x) = Inf. */
86 /* If x = -Inf, then sqrt(x) = sNaN. */
87 if (u.bits.exp == LDBL_INF_NAN_EXP) {
88 if (u.bits.sign && u.bits.manh == LDBL_NBIT_INF && u.bits.manl == 0)
89 return __math_invalidl(x);
90 return x + x;
91 }
92
93 /* If x = +-0, then sqrt(x) = +-0. */
94 if ((u.bits.manh | u.bits.manl | u.bits.exp) == 0)
95 return (x);
96
97 /* If x < 0, then raise invalid and return NaN */
98 if (u.bits.sign)
99 return __math_invalidl(x);
100
101 feholdexcept(&env);
102
103 if (u.bits.exp == 0) {
104 /* Adjust subnormal numbers. */
105 u.e *= 0x1.0p514L;
106 k = -514;
107 } else {
108 k = 0;
109 }
110 /*
111 * u.e is a normal number, so break it into u.e = e*2^n where
112 * u.e = (2*e)*2^2k for odd n and u.e = (4*e)*2^2k for even n.
113 */
114 if ((u.bits.exp - (BIAS-1)) & 1) { /* n is odd. */
115 k += u.bits.exp - BIAS; /* 2k = n - 1. */
116 #ifdef _DOUBLE_DOUBLE_FLOAT
117 u.dbits.dl = scalbn(u.dbits.dl, BIAS - u.bits.exp);
118 #endif
119 u.bits.exp = BIAS; /* u.e in [1,2). */
120 } else {
121 k += u.bits.exp - (BIAS + 1); /* 2k = n - 2. */
122 #ifdef _DOUBLE_DOUBLE_FLOAT
123 u.dbits.dl = scalbn(u.dbits.dl, (BIAS + 1) - u.bits.exp);
124 #endif
125 u.bits.exp = (BIAS + 1); /* u.e in [2,4). */
126 }
127
128 /*
129 * Newton's iteration.
130 * Split u.e into a high and low part to achieve additional precision.
131 */
132 xn = (long double)sqrt((double)u.e); /* 53-bit estimate of sqrtl(x). */
133 xn = (xn + (u.e / xn)) * 0.5L; /* 106-bit estimate. */
134
135 lo = u.e;
136 #ifdef _DOUBLE_DOUBLE_FLOAT
137 u.dbits.dl = 0.0; /* Zero out lower double */
138 #else
139 u.bits.manl = 0; /* Zero out lower bits. */
140 #endif
141 lo = (lo - u.e) / xn; /* Low bits divided by xn. */
142 xn = xn + (u.e / xn); /* High portion of estimate. */
143 u.e = xn + lo; /* Combine everything. */
144
145 u.bits.exp += (k >> 1) - 1;
146 #ifdef _DOUBLE_DOUBLE_FLOAT
147 u.dbits.dl = scalbn(u.dbits.dl, (k>>1) -1);
148 #endif
149
150 #if defined(FE_INEXACT) && defined(FE_TOWARDZERO) && defined(FE_TONEAREST) && defined(FE_UPWARD) && !defined(_DOUBLE_DOUBLE_FLOAT)
151 {
152 int r;
153 feclearexcept(FE_INEXACT);
154 r = fegetround();
155 fesetround(FE_TOWARDZERO); /* Set to round-toward-zero. */
156 xn = x / u.e; /* Chopped quotient (inexact?). */
157
158 if (!fetestexcept(FE_INEXACT)) { /* Quotient is exact. */
159 if (xn == u.e) {
160 fesetenv(&env);
161 return (u.e);
162 }
163 /* Round correctly for inputs like x = y**2 - ulp. */
164 xn = dec(xn); /* xn = xn - ulp. */
165 }
166
167 if (r == FE_TONEAREST) {
168 xn = inc(xn); /* xn = xn + ulp. */
169 } else if (r == FE_UPWARD) {
170 u.e = inc(u.e); /* u.e = u.e + ulp. */
171 xn = inc(xn); /* xn = xn + ulp. */
172 }
173 u.e = u.e + xn; /* Chopped sum. */
174 feupdateenv(&env); /* Restore env and raise inexact */
175 u.bits.exp--;
176 }
177 #endif
178 return (u.e);
179 }
180
