sysdeps/powerpc/fpu/e_sqrt.c

   1 /* Double-precision floating point square root.
   2    Copyright (C) 1997-2013 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 static const double almost_half = 0.5000000000000001;   /* 0.5 + 2^-53 */
  28 static const ieee_float_shape_type a_nan = {.word = 0x7fc00000 };
  29 static const ieee_float_shape_type a_inf = {.word = 0x7f800000 };
  30 static const float two108 = 3.245185536584267269e+32;
  31 static const float twom54 = 5.551115123125782702e-17;
  32 extern const float __t_sqrt[1024];
  33
  34 /* The method is based on a description in
  35    Computation of elementary functions on the IBM RISC System/6000 processor,
  36    P. W. Markstein, IBM J. Res. Develop, 34(1) 1990.
  37    Basically, it consists of two interleaved Newton-Raphson approximations,
  38    one to find the actual square root, and one to find its reciprocal
  39    without the expense of a division operation.   The tricky bit here
  40    is the use of the POWER/PowerPC multiply-add operation to get the
  41    required accuracy with high speed.
  42
  43    The argument reduction works by a combination of table lookup to
  44    obtain the initial guesses, and some careful modification of the
  45    generated guesses (which mostly runs on the integer unit, while the
  46    Newton-Raphson is running on the FPU).  */
  47
  48 double
  49 __slow_ieee754_sqrt (double x)
  50 {
  51   const float inf = a_inf.value;
  52
  53   if (x > 0)
  54     {
  55       /* schedule the EXTRACT_WORDS to get separation between the store
  56          and the load.  */
  57       ieee_double_shape_type ew_u;
  58       ieee_double_shape_type iw_u;
  59       ew_u.value = (x);
  60       if (x != inf)
  61         {
  62           /* Variables named starting with 's' exist in the
  63              argument-reduced space, so that 2 > sx >= 0.5,
  64              1.41... > sg >= 0.70.., 0.70.. >= sy > 0.35... .
  65              Variables named ending with 'i' are integer versions of
  66              floating-point values.  */
  67           double sx;    /* The value of which we're trying to find the
  68                            square root.  */
  69           double sg, g; /* Guess of the square root of x.  */
  70           double sd, d; /* Difference between the square of the guess and x.  */
  71           double sy;    /* Estimate of 1/2g (overestimated by 1ulp).  */
  72           double sy2;   /* 2*sy */
  73           double e;     /* Difference between y*g and 1/2 (se = e * fsy).  */
  74           double shx;   /* == sx * fsg */
  75           double fsg;   /* sg*fsg == g.  */
  76           fenv_t fe;    /* Saved floating-point environment (stores rounding
  77                            mode and whether the inexact exception is
  78                            enabled).  */
  79           uint32_t xi0, xi1, sxi, fsgi;
  80           const float *t_sqrt;
  81
  82           fe = fegetenv_register ();
  83           /* complete the EXTRACT_WORDS (xi0,xi1,x) operation.  */
  84           xi0 = ew_u.parts.msw;
  85           xi1 = ew_u.parts.lsw;
  86           relax_fenv_state ();
  87           sxi = (xi0 & 0x3fffffff) | 0x3fe00000;
  88           /* schedule the INSERT_WORDS (sx, sxi, xi1) to get separation
  89              between the store and the load.  */
  90           iw_u.parts.msw = sxi;
  91           iw_u.parts.lsw = xi1;
  92           t_sqrt = __t_sqrt + (xi0 >> (52 - 32 - 8 - 1) & 0x3fe);
  93           sg = t_sqrt[0];
  94           sy = t_sqrt[1];
  95           /* complete the INSERT_WORDS (sx, sxi, xi1) operation.  */
  96           sx = iw_u.value;
  97
  98           /* Here we have three Newton-Raphson iterations each of a
  99              division and a square root and the remainder of the
 100              argument reduction, all interleaved.   */
 101           sd = -(sg * sg - sx);
 102           fsgi = (xi0 + 0x40000000) >> 1 & 0x7ff00000;
 103           sy2 = sy + sy;
 104           sg = sy * sd + sg;    /* 16-bit approximation to sqrt(sx). */
 105
 106           /* schedule the INSERT_WORDS (fsg, fsgi, 0) to get separation
 107              between the store and the load.  */
 108           INSERT_WORDS (fsg, fsgi, 0);
 109           iw_u.parts.msw = fsgi;
 110           iw_u.parts.lsw = (0);
 111           e = -(sy * sg - almost_half);
 112           sd = -(sg * sg - sx);
 113           if ((xi0 & 0x7ff00000) == 0)
 114             goto denorm;
 115           sy = sy + e * sy2;
 116           sg = sg + sy * sd;    /* 32-bit approximation to sqrt(sx).  */
 117           sy2 = sy + sy;
 118           /* complete the INSERT_WORDS (fsg, fsgi, 0) operation.  */
 119           fsg = iw_u.value;
 120           e = -(sy * sg - almost_half);
 121           sd = -(sg * sg - sx);
 122           sy = sy + e * sy2;
 123           shx = sx * fsg;
 124           sg = sg + sy * sd;    /* 64-bit approximation to sqrt(sx),
 125                                    but perhaps rounded incorrectly.  */
 126           sy2 = sy + sy;
 127           g = sg * fsg;
 128           e = -(sy * sg - almost_half);
 129           d = -(g * sg - shx);
 130           sy = sy + e * sy2;
 131           fesetenv_register (fe);
 132           return g + sy * d;
 133         denorm:
 134           /* For denormalised numbers, we normalise, calculate the
 135              square root, and return an adjusted result.  */
 136           fesetenv_register (fe);
 137           return __slow_ieee754_sqrt (x * two108) * twom54;
 138         }
 139     }
 140   else if (x < 0)
 141     {
 142       /* For some reason, some PowerPC32 processors don't implement
 143          FE_INVALID_SQRT.  */
 144 #ifdef FE_INVALID_SQRT
 145       feraiseexcept (FE_INVALID_SQRT);
 146
 147       fenv_union_t u = { .fenv = fegetenv_register () };
 148       if ((u.l[1] & FE_INVALID) == 0)
 149 #endif
 150         feraiseexcept (FE_INVALID);
 151       x = a_nan.value;
 152     }
 153   return f_wash (x);
 154 }
 155
 156 #undef __ieee754_sqrt
 157 double
 158 __ieee754_sqrt (double x)
 159 {
 160   double z;
 161
 162   /* If the CPU is 64-bit we can use the optional FP instructions.  */
 163   if (__CPU_HAS_FSQRT)
 164     {
 165       /* Volatile is required to prevent the compiler from moving the
 166          fsqrt instruction above the branch.  */
 167       __asm __volatile ("       fsqrt   %0,%1\n"
 168                                 :"=f" (z):"f" (x));
 169     }
 170   else
 171     z = __slow_ieee754_sqrt (x);
 172
 173   return z;
 174 }
 175 strong_alias (__ieee754_sqrt, __sqrt_finite)