sysdeps/powerpc/fpu/e_sqrt.c

   1 /* Double-precision floating point square root.
   2    Copyright (C) 1997, 2002-2004, 2008, 2011 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-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 = -(sg * sg - sx);
 103           fsgi = (xi0 + 0x40000000) >> 1 & 0x7ff00000;
 104           sy2 = sy + sy;
 105           sg = sy * sd + sg;    /* 16-bit approximation to sqrt(sx). */
 106
 107           /* schedule the INSERT_WORDS (fsg, fsgi, 0) to get separation
 108              between the store and the load.  */
 109           INSERT_WORDS (fsg, fsgi, 0);
 110           iw_u.parts.msw = fsgi;
 111           iw_u.parts.lsw = (0);
 112           e = -(sy * sg - almost_half);
 113           sd = -(sg * sg - sx);
 114           if ((xi0 & 0x7ff00000) == 0)
 115             goto denorm;
 116           sy = sy + e * sy2;
 117           sg = sg + sy * sd;    /* 32-bit approximation to sqrt(sx).  */
 118           sy2 = sy + sy;
 119           /* complete the INSERT_WORDS (fsg, fsgi, 0) operation.  */
 120           fsg = iw_u.value;
 121           e = -(sy * sg - almost_half);
 122           sd = -(sg * sg - sx);
 123           sy = sy + e * sy2;
 124           shx = sx * fsg;
 125           sg = sg + sy * sd;    /* 64-bit approximation to sqrt(sx),
 126                                    but perhaps rounded incorrectly.  */
 127           sy2 = sy + sy;
 128           g = sg * fsg;
 129           e = -(sy * sg - almost_half);
 130           d = -(g * sg - shx);
 131           sy = sy + e * sy2;
 132           fesetenv_register (fe);
 133           return g + sy * d;
 134         denorm:
 135           /* For denormalised numbers, we normalise, calculate the
 136              square root, and return an adjusted result.  */
 137           fesetenv_register (fe);
 138           return __slow_ieee754_sqrt (x * two108) * twom54;
 139         }
 140     }
 141   else if (x < 0)
 142     {
 143       /* For some reason, some PowerPC32 processors don't implement
 144          FE_INVALID_SQRT.  */
 145 #ifdef FE_INVALID_SQRT
 146       feraiseexcept (FE_INVALID_SQRT);
 147
 148       fenv_union_t u = { .fenv = fegetenv_register () };
 149       if ((u.l[1] & FE_INVALID) == 0)
 150 #endif
 151         feraiseexcept (FE_INVALID);
 152       x = a_nan.value;
 153     }
 154   return f_wash (x);
 155 }
 156
 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)