sysdeps/powerpc/fpu/e_sqrt.c

   1 /* Double-precision floating point square root.
   2    Copyright (C) 1997-2015 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    <http://www.gnu.org/licenses/>.  */
  18
  19 #include <math.h>
  20 #include <math_private.h>
  21 #include <fenv_libc.h>
  22 #include <inttypes.h>
  23 #include <stdint.h>
  24 #include <sysdep.h>
  25 #include <ldsodefs.h>
  26
  27 #ifndef _ARCH_PPCSQ
  28 static const double almost_half = 0.5000000000000001;   /* 0.5 + 2^-53 */
  29 static const ieee_float_shape_type a_nan = {.word = 0x7fc00000 };
  30 static const ieee_float_shape_type a_inf = {.word = 0x7f800000 };
  31 static const float two108 = 3.245185536584267269e+32;
  32 static const float twom54 = 5.551115123125782702e-17;
  33 extern const float __t_sqrt[1024];
  34
  35 /* The method is based on a description in
  36    Computation of elementary functions on the IBM RISC System/6000 processor,
  37    P. W. Markstein, IBM J. Res. Develop, 34(1) 1990.
  38    Basically, it consists of two interleaved Newton-Raphson approximations,
  39    one to find the actual square root, and one to find its reciprocal
  40    without the expense of a division operation.   The tricky bit here
  41    is the use of the POWER/PowerPC multiply-add operation to get the
  42    required accuracy with high speed.
  43
  44    The argument reduction works by a combination of table lookup to
  45    obtain the initial guesses, and some careful modification of the
  46    generated guesses (which mostly runs on the integer unit, while the
  47    Newton-Raphson is running on the FPU).  */
  48
  49 double
  50 __slow_ieee754_sqrt (double x)
  51 {
  52   const float inf = a_inf.value;
  53
  54   if (x > 0)
  55     {
  56       /* schedule the EXTRACT_WORDS to get separation between the store
  57          and the load.  */
  58       ieee_double_shape_type ew_u;
  59       ieee_double_shape_type iw_u;
  60       ew_u.value = (x);
  61       if (x != inf)
  62         {
  63           /* Variables named starting with 's' exist in the
  64              argument-reduced space, so that 2 > sx >= 0.5,
  65              1.41... > sg >= 0.70.., 0.70.. >= sy > 0.35... .
  66              Variables named ending with 'i' are integer versions of
  67              floating-point values.  */
  68           double sx;    /* The value of which we're trying to find the
  69                            square root.  */
  70           double sg, g; /* Guess of the square root of x.  */
  71           double sd, d; /* Difference between the square of the guess and x.  */
  72           double sy;    /* Estimate of 1/2g (overestimated by 1ulp).  */
  73           double sy2;   /* 2*sy */
  74           double e;     /* Difference between y*g and 1/2 (se = e * fsy).  */
  75           double shx;   /* == sx * fsg */
  76           double fsg;   /* sg*fsg == g.  */
  77           fenv_t fe;    /* Saved floating-point environment (stores rounding
  78                            mode and whether the inexact exception is
  79                            enabled).  */
  80           uint32_t xi0, xi1, sxi, fsgi;
  81           const float *t_sqrt;
  82
  83           fe = fegetenv_register ();
  84           /* complete the EXTRACT_WORDS (xi0,xi1,x) operation.  */
  85           xi0 = ew_u.parts.msw;
  86           xi1 = ew_u.parts.lsw;
  87           relax_fenv_state ();
  88           sxi = (xi0 & 0x3fffffff) | 0x3fe00000;
  89           /* schedule the INSERT_WORDS (sx, sxi, xi1) to get separation
  90              between the store and the load.  */
  91           iw_u.parts.msw = sxi;
  92           iw_u.parts.lsw = xi1;
  93           t_sqrt = __t_sqrt + (xi0 >> (52 - 32 - 8 - 1) & 0x3fe);
  94           sg = t_sqrt[0];
  95           sy = t_sqrt[1];
  96           /* complete the INSERT_WORDS (sx, sxi, xi1) operation.  */
  97           sx = iw_u.value;
  98
  99           /* Here we have three Newton-Raphson iterations each of a
 100              division and a square root and the remainder of the
 101              argument reduction, all interleaved.   */
 102           sd = -__builtin_fma (sg, sg, -sx);
 103           fsgi = (xi0 + 0x40000000) >> 1 & 0x7ff00000;
 104           sy2 = sy + sy;
 105           sg = __builtin_fma (sy, sd, sg);      /* 16-bit approximation to
 106                                                    sqrt(sx). */
 107
 108           /* schedule the INSERT_WORDS (fsg, fsgi, 0) to get separation
 109              between the store and the load.  */
 110           INSERT_WORDS (fsg, fsgi, 0);
 111           iw_u.parts.msw = fsgi;
 112           iw_u.parts.lsw = (0);
 113           e = -__builtin_fma (sy, sg, -almost_half);
 114           sd = -__builtin_fma (sg, sg, -sx);
 115           if ((xi0 & 0x7ff00000) == 0)
 116             goto denorm;
 117           sy = __builtin_fma (e, sy2, sy);
 118           sg = __builtin_fma (sy, sd, sg);      /* 32-bit approximation to
 119                                                    sqrt(sx).  */
 120           sy2 = sy + sy;
 121           /* complete the INSERT_WORDS (fsg, fsgi, 0) operation.  */
 122           fsg = iw_u.value;
 123           e = -__builtin_fma (sy, sg, -almost_half);
 124           sd = -__builtin_fma (sg, sg, -sx);
 125           sy = __builtin_fma (e, sy2, sy);
 126           shx = sx * fsg;
 127           sg = __builtin_fma (sy, sd, sg);      /* 64-bit approximation to
 128                                                    sqrt(sx), but perhaps
 129                                                    rounded incorrectly.  */
 130           sy2 = sy + sy;
 131           g = sg * fsg;
 132           e = -__builtin_fma (sy, sg, -almost_half);
 133           d = -__builtin_fma (g, sg, -shx);
 134           sy = __builtin_fma (e, sy2, sy);
 135           fesetenv_register (fe);
 136           return __builtin_fma (sy, d, g);
 137         denorm:
 138           /* For denormalised numbers, we normalise, calculate the
 139              square root, and return an adjusted result.  */
 140           fesetenv_register (fe);
 141           return __slow_ieee754_sqrt (x * two108) * twom54;
 142         }
 143     }
 144   else if (x < 0)
 145     {
 146       /* For some reason, some PowerPC32 processors don't implement
 147          FE_INVALID_SQRT.  */
 148 #ifdef FE_INVALID_SQRT
 149       __feraiseexcept (FE_INVALID_SQRT);
 150
 151       fenv_union_t u = { .fenv = fegetenv_register () };
 152       if ((u.l & FE_INVALID) == 0)
 153 #endif
 154         __feraiseexcept (FE_INVALID);
 155       x = a_nan.value;
 156     }
 157   return f_wash (x);
 158 }
 159 #endif /* _ARCH_PPCSQ  */
 160
 161 #undef __ieee754_sqrt
 162 double
 163 __ieee754_sqrt (double x)
 164 {
 165   double z;
 166
 167 #ifdef _ARCH_PPCSQ
 168   asm ("fsqrt %0,%1\n" :"=f" (z):"f" (x));
 169 #else
 170   z = __slow_ieee754_sqrt (x);
 171 #endif
 172
 173   return z;
 174 }
 175 strong_alias (__ieee754_sqrt, __sqrt_finite)