1 /* Double-precision floating point square root.
2 Copyright (C) 1997-2020 Free Software Foundation, Inc.
3 This file is part of the GNU C Library.
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.
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.
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/>. */
20 #include <math_private.h>
22 #include <fenv_libc.h>
29 static const double almost_half
= 0.5000000000000001; /* 0.5 + 2^-53 */
30 static const ieee_float_shape_type a_nan
= {.word
= 0x7fc00000 };
31 static const ieee_float_shape_type a_inf
= {.word
= 0x7f800000 };
32 static const float two108
= 3.245185536584267269e+32;
33 static const float twom54
= 5.551115123125782702e-17;
34 extern const float __t_sqrt
[1024];
36 /* The method is based on a description in
37 Computation of elementary functions on the IBM RISC System/6000 processor,
38 P. W. Markstein, IBM J. Res. Develop, 34(1) 1990.
39 Basically, it consists of two interleaved Newton-Raphson approximations,
40 one to find the actual square root, and one to find its reciprocal
41 without the expense of a division operation. The tricky bit here
42 is the use of the POWER/PowerPC multiply-add operation to get the
43 required accuracy with high speed.
45 The argument reduction works by a combination of table lookup to
46 obtain the initial guesses, and some careful modification of the
47 generated guesses (which mostly runs on the integer unit, while the
48 Newton-Raphson is running on the FPU). */
51 __slow_ieee754_sqrt (double x
)
53 const float inf
= a_inf
.value
;
57 /* schedule the EXTRACT_WORDS to get separation between the store
59 ieee_double_shape_type ew_u
;
60 ieee_double_shape_type iw_u
;
64 /* Variables named starting with 's' exist in the
65 argument-reduced space, so that 2 > sx >= 0.5,
66 1.41... > sg >= 0.70.., 0.70.. >= sy > 0.35... .
67 Variables named ending with 'i' are integer versions of
68 floating-point values. */
69 double sx
; /* The value of which we're trying to find the
71 double sg
, g
; /* Guess of the square root of x. */
72 double sd
, d
; /* Difference between the square of the guess and x. */
73 double sy
; /* Estimate of 1/2g (overestimated by 1ulp). */
74 double sy2
; /* 2*sy */
75 double e
; /* Difference between y*g and 1/2 (se = e * fsy). */
76 double shx
; /* == sx * fsg */
77 double fsg
; /* sg*fsg == g. */
78 fenv_t fe
; /* Saved floating-point environment (stores rounding
79 mode and whether the inexact exception is
81 uint32_t xi0
, xi1
, sxi
, fsgi
;
84 fe
= fegetenv_register ();
85 /* complete the EXTRACT_WORDS (xi0,xi1,x) operation. */
89 sxi
= (xi0
& 0x3fffffff) | 0x3fe00000;
90 /* schedule the INSERT_WORDS (sx, sxi, xi1) to get separation
91 between the store and the load. */
94 t_sqrt
= __t_sqrt
+ (xi0
>> (52 - 32 - 8 - 1) & 0x3fe);
97 /* complete the INSERT_WORDS (sx, sxi, xi1) operation. */
100 /* Here we have three Newton-Raphson iterations each of a
101 division and a square root and the remainder of the
102 argument reduction, all interleaved. */
103 sd
= -__builtin_fma (sg
, sg
, -sx
);
104 fsgi
= (xi0
+ 0x40000000) >> 1 & 0x7ff00000;
106 sg
= __builtin_fma (sy
, sd
, sg
); /* 16-bit approximation to
109 /* schedule the INSERT_WORDS (fsg, fsgi, 0) to get separation
110 between the store and the load. */
111 INSERT_WORDS (fsg
, fsgi
, 0);
112 iw_u
.parts
.msw
= fsgi
;
113 iw_u
.parts
.lsw
= (0);
114 e
= -__builtin_fma (sy
, sg
, -almost_half
);
115 sd
= -__builtin_fma (sg
, sg
, -sx
);
116 if ((xi0
& 0x7ff00000) == 0)
118 sy
= __builtin_fma (e
, sy2
, sy
);
119 sg
= __builtin_fma (sy
, sd
, sg
); /* 32-bit approximation to
122 /* complete the INSERT_WORDS (fsg, fsgi, 0) operation. */
124 e
= -__builtin_fma (sy
, sg
, -almost_half
);
125 sd
= -__builtin_fma (sg
, sg
, -sx
);
126 sy
= __builtin_fma (e
, sy2
, sy
);
128 sg
= __builtin_fma (sy
, sd
, sg
); /* 64-bit approximation to
129 sqrt(sx), but perhaps
130 rounded incorrectly. */
133 e
= -__builtin_fma (sy
, sg
, -almost_half
);
134 d
= -__builtin_fma (g
, sg
, -shx
);
135 sy
= __builtin_fma (e
, sy2
, sy
);
136 fesetenv_register (fe
);
137 return __builtin_fma (sy
, d
, g
);
139 /* For denormalised numbers, we normalise, calculate the
140 square root, and return an adjusted result. */
141 fesetenv_register (fe
);
142 return __slow_ieee754_sqrt (x
* two108
) * twom54
;
147 /* For some reason, some PowerPC32 processors don't implement
149 #ifdef FE_INVALID_SQRT
150 __feraiseexcept (FE_INVALID_SQRT
);
152 fenv_union_t u
= { .fenv
= fegetenv_register () };
153 if ((u
.l
& FE_INVALID
) == 0)
155 __feraiseexcept (FE_INVALID
);
160 #endif /* _ARCH_PPCSQ */
162 #undef __ieee754_sqrt
164 __ieee754_sqrt (double x
)
169 asm ("fsqrt %0,%1\n" :"=f" (z
):"f" (x
));
171 z
= __slow_ieee754_sqrt (x
);
176 strong_alias (__ieee754_sqrt
, __sqrt_finite
)