1 /* Double-precision floating point square root.
2 Copyright (C) 1997-2023 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
4
5 The GNU C Library is free software; you can redistribute it and/or
6 modify it under the terms of the GNU Lesser General Public
7 License as published by the Free Software Foundation; either
8 version 2.1 of the License, or (at your option) any later version.
9
10 The GNU C Library is distributed in the hope that it will be useful,
11 but WITHOUT ANY WARRANTY; without even the implied warranty of
12 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
13 Lesser General Public License for more details.
14
15 You should have received a copy of the GNU Lesser General Public
16 License along with the GNU C Library; if not, see
17 <https://www.gnu.org/licenses/>. */
18
19 #include <math.h>
20 #include <math_private.h>
21 #include <fenv_libc.h>
22 #include <libm-alias-finite.h>
23 #include <math-use-builtins.h>
24
25 double
26 __ieee754_sqrt (double x)
27 {
28 #if USE_SQRT_BUILTIN
29 return __builtin_sqrt (x);
30 #else
31 /* The method is based on a description in
32 Computation of elementary functions on the IBM RISC System/6000 processor,
33 P. W. Markstein, IBM J. Res. Develop, 34(1) 1990.
34 Basically, it consists of two interleaved Newton-Raphson approximations,
35 one to find the actual square root, and one to find its reciprocal
36 without the expense of a division operation. The tricky bit here
37 is the use of the POWER/PowerPC multiply-add operation to get the
38 required accuracy with high speed.
39
40 The argument reduction works by a combination of table lookup to
41 obtain the initial guesses, and some careful modification of the
42 generated guesses (which mostly runs on the integer unit, while the
43 Newton-Raphson is running on the FPU). */
44
45 extern const float __t_sqrt[1024];
46
47 if (x > 0)
48 {
49 /* schedule the EXTRACT_WORDS to get separation between the store
50 and the load. */
51 ieee_double_shape_type ew_u;
52 ieee_double_shape_type iw_u;
53 ew_u.value = (x);
54 if (x != INFINITY)
55 {
56 /* Variables named starting with 's' exist in the
57 argument-reduced space, so that 2 > sx >= 0.5,
58 1.41... > sg >= 0.70.., 0.70.. >= sy > 0.35... .
59 Variables named ending with 'i' are integer versions of
60 floating-point values. */
61 double sx; /* The value of which we're trying to find the
62 square root. */
63 double sg, g; /* Guess of the square root of x. */
64 double sd, d; /* Difference between the square of the guess and x. */
65 double sy; /* Estimate of 1/2g (overestimated by 1ulp). */
66 double sy2; /* 2*sy */
67 double e; /* Difference between y*g and 1/2 (se = e * fsy). */
68 double shx; /* == sx * fsg */
69 double fsg; /* sg*fsg == g. */
70 fenv_t fe; /* Saved floating-point environment (stores rounding
71 mode and whether the inexact exception is
72 enabled). */
73 uint32_t xi0, xi1, sxi, fsgi;
74 const float *t_sqrt;
75
76 fe = fegetenv_register ();
77 /* complete the EXTRACT_WORDS (xi0,xi1,x) operation. */
78 xi0 = ew_u.parts.msw;
79 xi1 = ew_u.parts.lsw;
80 relax_fenv_state ();
81 sxi = (xi0 & 0x3fffffff) | 0x3fe00000;
82 /* schedule the INSERT_WORDS (sx, sxi, xi1) to get separation
83 between the store and the load. */
84 iw_u.parts.msw = sxi;
85 iw_u.parts.lsw = xi1;
86 t_sqrt = __t_sqrt + (xi0 >> (52 - 32 - 8 - 1) & 0x3fe);
87 sg = t_sqrt[0];
88 sy = t_sqrt[1];
89 /* complete the INSERT_WORDS (sx, sxi, xi1) operation. */
90 sx = iw_u.value;
91
92 /* Here we have three Newton-Raphson iterations each of a
93 division and a square root and the remainder of the
94 argument reduction, all interleaved. */
95 sd = -__builtin_fma (sg, sg, -sx);
96 fsgi = (xi0 + 0x40000000) >> 1 & 0x7ff00000;
97 sy2 = sy + sy;
98 sg = __builtin_fma (sy, sd, sg); /* 16-bit approximation to
99 sqrt(sx). */
100
101 /* schedule the INSERT_WORDS (fsg, fsgi, 0) to get separation
102 between the store and the load. */
103 INSERT_WORDS (fsg, fsgi, 0);
104 iw_u.parts.msw = fsgi;
105 iw_u.parts.lsw = (0);
106 e = -__builtin_fma (sy, sg, -0x1.0000000000001p-1);
107 sd = -__builtin_fma (sg, sg, -sx);
108 if ((xi0 & 0x7ff00000) == 0)
109 goto denorm;
110 sy = __builtin_fma (e, sy2, sy);
111 sg = __builtin_fma (sy, sd, sg); /* 32-bit approximation to
112 sqrt(sx). */
113 sy2 = sy + sy;
114 /* complete the INSERT_WORDS (fsg, fsgi, 0) operation. */
115 fsg = iw_u.value;
116 e = -__builtin_fma (sy, sg, -0x1.0000000000001p-1);
117 sd = -__builtin_fma (sg, sg, -sx);
118 sy = __builtin_fma (e, sy2, sy);
119 shx = sx * fsg;
120 sg = __builtin_fma (sy, sd, sg); /* 64-bit approximation to
121 sqrt(sx), but perhaps
122 rounded incorrectly. */
123 sy2 = sy + sy;
124 g = sg * fsg;
125 e = -__builtin_fma (sy, sg, -0x1.0000000000001p-1);
126 d = -__builtin_fma (g, sg, -shx);
127 sy = __builtin_fma (e, sy2, sy);
128 fesetenv_register (fe);
129 return __builtin_fma (sy, d, g);
130 denorm:
131 /* For denormalised numbers, we normalise, calculate the
132 square root, and return an adjusted result. */
133 fesetenv_register (fe);
134 return __ieee754_sqrt (x * 0x1p+108f) * 0x1p-54f;
135 }
136 }
137 else if (x < 0)
138 {
139 /* For some reason, some PowerPC32 processors don't implement
140 FE_INVALID_SQRT. */
141 # ifdef FE_INVALID_SQRT
142 __feraiseexcept (FE_INVALID_SQRT);
143
144 fenv_union_t u = { .fenv = fegetenv_register () };
145 if ((u.l & FE_INVALID) == 0)
146 # endif
147 __feraiseexcept (FE_INVALID);
148 x = NAN;
149 }
150 return f_wash (x);
151 #endif /* USE_SQRT_BUILTIN */
152 }
153
154 libm_alias_finite (__ieee754_sqrt, __sqrt)