sysdeps/powerpc/fpu/e_sqrt.c

   1 /* Double-precision floating point square root.
   2    Copyright (C) 1997-2024 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)