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 //__FBSDID("$FreeBSD: src/lib/msun/src/e_sqrtl.c,v 1.1 2008/03/02 01:47:58 das Exp $");
28
29 #ifndef _DOUBLE_DOUBLE_FLOAT
30 /* Return (x + ulp) for normal positive x. Assumes no overflow. */
31 static inline long double
inc(long double x)32 inc(long double x)
33 {
34 union IEEEl2bits u;
35
36 u.e = x;
37 if (++u.bits.manl == 0) {
38 if (++u.bits.manh == 0) {
39 u.bits.exp++;
40 u.bits.manh |= LDBL_NBIT;
41 }
42 }
43 return (u.e);
44 }
45
46 /* Return (x - ulp) for normal positive x. Assumes no underflow. */
47 static inline long double
dec(long double x)48 dec(long double x)
49 {
50 union IEEEl2bits u;
51
52 u.e = x;
53 if (u.bits.manl-- == 0) {
54 if (u.bits.manh-- == LDBL_NBIT) {
55 u.bits.exp--;
56 u.bits.manh |= LDBL_NBIT;
57 }
58 }
59 return (u.e);
60 }
61
62 #ifndef __GNUC__
63 #pragma STDC FENV_ACCESS ON
64 #endif
65
66 #endif
67
68 /*
69 * This is slow, but simple and portable. You should use hardware sqrt
70 * if possible.
71 */
72
73 #define BIAS (LDBL_MAX_EXP-1)
74
75 long double
sqrtl(long double x)76 sqrtl(long double x)
77 {
78 union IEEEl2bits u;
79 int k;
80 long double lo, xn;
81 fenv_t env;
82
83 u.e = x;
84
85 /* If x = NaN, then sqrt(x) = qNaN. */
86 /* If x = Inf, then sqrt(x) = Inf. */
87 /* If x = -Inf, then sqrt(x) = sNaN. */
88 if (u.bits.exp == LDBL_INF_NAN_EXP) {
89 if (u.bits.sign && u.bits.manh == LDBL_NBIT_INF && u.bits.manl == 0)
90 return __math_invalidl(x);
91 return x + x;
92 }
93
94 /* If x = +-0, then sqrt(x) = +-0. */
95 if ((u.bits.manh | u.bits.manl | u.bits.exp) == 0)
96 return (x);
97
98 /* If x < 0, then raise invalid and return NaN */
99 if (u.bits.sign)
100 return __math_invalidl(x);
101
102 feholdexcept(&env);
103
104 if (u.bits.exp == 0) {
105 /* Adjust subnormal numbers. */
106 u.e *= 0x1.0p514L;
107 k = -514;
108 } else {
109 k = 0;
110 }
111 /*
112 * u.e is a normal number, so break it into u.e = e*2^n where
113 * u.e = (2*e)*2^2k for odd n and u.e = (4*e)*2^2k for even n.
114 */
115 if ((u.bits.exp - (BIAS-1)) & 1) { /* n is odd. */
116 k += u.bits.exp - BIAS; /* 2k = n - 1. */
117 #ifdef _DOUBLE_DOUBLE_FLOAT
118 u.dbits.dl = scalbn(u.dbits.dl, BIAS - u.bits.exp);
119 #endif
120 u.bits.exp = BIAS; /* u.e in [1,2). */
121 } else {
122 k += u.bits.exp - (BIAS + 1); /* 2k = n - 2. */
123 #ifdef _DOUBLE_DOUBLE_FLOAT
124 u.dbits.dl = scalbn(u.dbits.dl, (BIAS + 1) - u.bits.exp);
125 #endif
126 u.bits.exp = (BIAS + 1); /* u.e in [2,4). */
127 }
128
129 /*
130 * Newton's iteration.
131 * Split u.e into a high and low part to achieve additional precision.
132 */
133 xn = (long double)sqrt((double)u.e); /* 53-bit estimate of sqrtl(x). */
134 xn = (xn + (u.e / xn)) * 0.5L; /* 106-bit estimate. */
135
136 lo = u.e;
137 #ifdef _DOUBLE_DOUBLE_FLOAT
138 u.dbits.dl = 0.0; /* Zero out lower double */
139 #else
140 u.bits.manl = 0; /* Zero out lower bits. */
141 #endif
142 lo = (lo - u.e) / xn; /* Low bits divided by xn. */
143 xn = xn + (u.e / xn); /* High portion of estimate. */
144 u.e = xn + lo; /* Combine everything. */
145
146 u.bits.exp += (k >> 1) - 1;
147 #ifdef _DOUBLE_DOUBLE_FLOAT
148 u.dbits.dl = scalbn(u.dbits.dl, (k>>1) -1);
149 #endif
150
151 #if defined(FE_INEXACT) && defined(FE_TOWARDZERO) && defined(FE_TONEAREST) && defined(FE_UPWARD) && !defined(_DOUBLE_DOUBLE_FLOAT)
152 {
153 int r;
154 feclearexcept(FE_INEXACT);
155 r = fegetround();
156 fesetround(FE_TOWARDZERO); /* Set to round-toward-zero. */
157 xn = x / u.e; /* Chopped quotient (inexact?). */
158
159 if (!fetestexcept(FE_INEXACT)) { /* Quotient is exact. */
160 if (xn == u.e) {
161 fesetenv(&env);
162 return (u.e);
163 }
164 /* Round correctly for inputs like x = y**2 - ulp. */
165 xn = dec(xn); /* xn = xn - ulp. */
166 }
167
168 if (r == FE_TONEAREST) {
169 xn = inc(xn); /* xn = xn + ulp. */
170 } else if (r == FE_UPWARD) {
171 u.e = inc(u.e); /* u.e = u.e + ulp. */
172 xn = inc(xn); /* xn = xn + ulp. */
173 }
174 u.e = u.e + xn; /* Chopped sum. */
175 feupdateenv(&env); /* Restore env and raise inexact */
176 u.bits.exp--;
177 }
178 #endif
179 return (u.e);
180 }
181
