1 /* Single-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 float
26 __ieee754_sqrtf (float x)
27 {
28 #if USE_SQRTF_BUILTIN
29 return __builtin_sqrtf (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 if (x != INFINITY)
50 {
51 /* Variables named starting with 's' exist in the
52 argument-reduced space, so that 2 > sx >= 0.5,
53 1.41... > sg >= 0.70.., 0.70.. >= sy > 0.35... .
54 Variables named ending with 'i' are integer versions of
55 floating-point values. */
56 float sx; /* The value of which we're trying to find the square
57 root. */
58 float sg, g; /* Guess of the square root of x. */
59 float sd, d; /* Difference between the square of the guess and x. */
60 float sy; /* Estimate of 1/2g (overestimated by 1ulp). */
61 float sy2; /* 2*sy */
62 float e; /* Difference between y*g and 1/2 (note that e==se). */
63 float shx; /* == sx * fsg */
64 float fsg; /* sg*fsg == g. */
65 fenv_t fe; /* Saved floating-point environment (stores rounding
66 mode and whether the inexact exception is
67 enabled). */
68 uint32_t xi, sxi, fsgi;
69 const float *t_sqrt;
70
71 GET_FLOAT_WORD (xi, x);
72 fe = fegetenv_register ();
73 relax_fenv_state ();
74 sxi = (xi & 0x3fffffff) | 0x3f000000;
75 SET_FLOAT_WORD (sx, sxi);
76 t_sqrt = __t_sqrt + (xi >> (23 - 8 - 1) & 0x3fe);
77 sg = t_sqrt[0];
78 sy = t_sqrt[1];
79
80 /* Here we have three Newton-Raphson iterations each of a
81 division and a square root and the remainder of the
82 argument reduction, all interleaved. */
83 sd = -__builtin_fmaf (sg, sg, -sx);
84 fsgi = (xi + 0x40000000) >> 1 & 0x7f800000;
85 sy2 = sy + sy;
86 sg = __builtin_fmaf (sy, sd, sg); /* 16-bit approximation to
87 sqrt(sx). */
88 e = -__builtin_fmaf (sy, sg, -0x1.0000020365653p-1);
89 SET_FLOAT_WORD (fsg, fsgi);
90 sd = -__builtin_fmaf (sg, sg, -sx);
91 sy = __builtin_fmaf (e, sy2, sy);
92 if ((xi & 0x7f800000) == 0)
93 goto denorm;
94 shx = sx * fsg;
95 sg = __builtin_fmaf (sy, sd, sg); /* 32-bit approximation to
96 sqrt(sx), but perhaps
97 rounded incorrectly. */
98 sy2 = sy + sy;
99 g = sg * fsg;
100 e = -__builtin_fmaf (sy, sg, -0x1.0000020365653p-1);
101 d = -__builtin_fmaf (g, sg, -shx);
102 sy = __builtin_fmaf (e, sy2, sy);
103 fesetenv_register (fe);
104 return __builtin_fmaf (sy, d, g);
105 denorm:
106 /* For denormalised numbers, we normalise, calculate the
107 square root, and return an adjusted result. */
108 fesetenv_register (fe);
109 return __ieee754_sqrtf (x * 0x1p+48) * 0x1p-24;
110 }
111 }
112 else if (x < 0)
113 {
114 /* For some reason, some PowerPC32 processors don't implement
115 FE_INVALID_SQRT. */
116 # ifdef FE_INVALID_SQRT
117 feraiseexcept (FE_INVALID_SQRT);
118
119 fenv_union_t u = { .fenv = fegetenv_register () };
120 if ((u.l & FE_INVALID) == 0)
121 # endif
122 feraiseexcept (FE_INVALID);
123 x = NAN;
124 }
125 return f_washf (x);
126 #endif /* USE_SQRTF_BUILTIN */
127 }
128 libm_alias_finite (__ieee754_sqrtf, __sqrtf)