sysdeps/powerpc/fpu/e_sqrt.c

   1 /* Double-precision floating point square root.
   2    Copyright (C) 1997, 2002, 2003, 2004 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, write to the Free
  17    Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA
  18    02111-1307 USA.  */
  19
  20 #include <math.h>
  21 #include <math_private.h>
  22 #include <fenv_libc.h>
  23 #include <inttypes.h>
  24
  25 #include <sysdep.h>
  26 #include <ldsodefs.h>
  27
  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-Rhapson 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-Rhapson is running on the FPU).  */
  48
  49 #ifdef __STDC__
  50 double
  51 __slow_ieee754_sqrt (double x)
  52 #else
  53 double
  54 __slow_ieee754_sqrt (x)
  55      double x;
  56 #endif
  57 {
  58   const float inf = a_inf.value;
  59
  60   if (x > 0)
  61     {
  62       /* schedule the EXTRACT_WORDS to get separation between the store
  63          and the load.  */
  64       ieee_double_shape_type ew_u;
  65       ieee_double_shape_type iw_u;
  66       ew_u.value = (x);
  67       if (x != inf)
  68         {
  69           /* Variables named starting with 's' exist in the
  70              argument-reduced space, so that 2 > sx >= 0.5,
  71              1.41... > sg >= 0.70.., 0.70.. >= sy > 0.35... .
  72              Variables named ending with 'i' are integer versions of
  73              floating-point values.  */
  74           double sx;    /* The value of which we're trying to find the
  75                            square root.  */
  76           double sg, g; /* Guess of the square root of x.  */
  77           double sd, d; /* Difference between the square of the guess and x.  */
  78           double sy;    /* Estimate of 1/2g (overestimated by 1ulp).  */
  79           double sy2;   /* 2*sy */
  80           double e;     /* Difference between y*g and 1/2 (se = e * fsy).  */
  81           double shx;   /* == sx * fsg */
  82           double fsg;   /* sg*fsg == g.  */
  83           fenv_t fe;    /* Saved floating-point environment (stores rounding
  84                            mode and whether the inexact exception is
  85                            enabled).  */
  86           uint32_t xi0, xi1, sxi, fsgi;
  87           const float *t_sqrt;
  88
  89           fe = fegetenv_register ();
  90           /* complete the EXTRACT_WORDS (xi0,xi1,x) operation.  */
  91           xi0 = ew_u.parts.msw;
  92           xi1 = ew_u.parts.lsw;
  93           relax_fenv_state ();
  94           sxi = (xi0 & 0x3fffffff) | 0x3fe00000;
  95           /* schedule the INSERT_WORDS (sx, sxi, xi1) to get separation
  96              between the store and the load.  */
  97           iw_u.parts.msw = sxi;
  98           iw_u.parts.lsw = xi1;
  99           t_sqrt = __t_sqrt + (xi0 >> (52 - 32 - 8 - 1) & 0x3fe);
 100           sg = t_sqrt[0];
 101           sy = t_sqrt[1];
 102           /* complete the INSERT_WORDS (sx, sxi, xi1) operation.  */
 103           sx = iw_u.value;
 104
 105           /* Here we have three Newton-Rhapson iterations each of a
 106              division and a square root and the remainder of the
 107              argument reduction, all interleaved.   */
 108           sd = -(sg * sg - sx);
 109           fsgi = (xi0 + 0x40000000) >> 1 & 0x7ff00000;
 110           sy2 = sy + sy;
 111           sg = sy * sd + sg;    /* 16-bit approximation to sqrt(sx). */
 112
 113           /* schedule the INSERT_WORDS (fsg, fsgi, 0) to get separation
 114              between the store and the load.  */
 115           INSERT_WORDS (fsg, fsgi, 0);
 116           iw_u.parts.msw = fsgi;
 117           iw_u.parts.lsw = (0);
 118           e = -(sy * sg - almost_half);
 119           sd = -(sg * sg - sx);
 120           if ((xi0 & 0x7ff00000) == 0)
 121             goto denorm;
 122           sy = sy + e * sy2;
 123           sg = sg + sy * sd;    /* 32-bit approximation to sqrt(sx).  */
 124           sy2 = sy + sy;
 125           /* complete the INSERT_WORDS (fsg, fsgi, 0) operation.  */
 126           fsg = iw_u.value;
 127           e = -(sy * sg - almost_half);
 128           sd = -(sg * sg - sx);
 129           sy = sy + e * sy2;
 130           shx = sx * fsg;
 131           sg = sg + sy * sd;    /* 64-bit approximation to sqrt(sx),
 132                                    but perhaps rounded incorrectly.  */
 133           sy2 = sy + sy;
 134           g = sg * fsg;
 135           e = -(sy * sg - almost_half);
 136           d = -(g * sg - shx);
 137           sy = sy + e * sy2;
 138           fesetenv_register (fe);
 139           return g + sy * d;
 140         denorm:
 141           /* For denormalised numbers, we normalise, calculate the
 142              square root, and return an adjusted result.  */
 143           fesetenv_register (fe);
 144           return __slow_ieee754_sqrt (x * two108) * twom54;
 145         }
 146     }
 147   else if (x < 0)
 148     {
 149       /* For some reason, some PowerPC32 processors don't implement
 150          FE_INVALID_SQRT.  */
 151 #ifdef FE_INVALID_SQRT
 152       feraiseexcept (FE_INVALID_SQRT);
 153       if (!fetestexcept (FE_INVALID))
 154 #endif
 155         feraiseexcept (FE_INVALID);
 156       x = a_nan.value;
 157     }
 158   return f_wash (x);
 159 }
 160
 161 #ifdef __STDC__
 162 double
 163 __ieee754_sqrt (double x)
 164 #else
 165 double
 166 __ieee754_sqrt (x)
 167      double x;
 168 #endif
 169 {
 170   double z;
 171
 172   /* If the CPU is 64-bit we can use the optional FP instructions.  */
 173   if (__CPU_HAS_FSQRT)
 174     {
 175       /* Volatile is required to prevent the compiler from moving the
 176          fsqrt instruction above the branch.  */
 177       __asm __volatile ("       fsqrt   %0,%1\n"
 178                                 :"=f" (z):"f" (x));
 179     }
 180   else
 181     z = __slow_ieee754_sqrt (x);
 182
 183   return z;
 184 }