sysdeps/powerpc/e_sqrt.c

   1 /* Single-precision floating point square root.
   2    Copyright (C) 1997 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 Library General Public License as
   7    published by the Free Software Foundation; either version 2 of the
   8    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    Library General Public License for more details.
  14
  15    You should have received a copy of the GNU Library General Public
  16    License along with the GNU C Library; see the file COPYING.LIB.  If not,
  17    write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330,
  18    Boston, MA 02111-1307, USA.  */
  19
  20 #include <math.h>
  21 #include <math_private.h>
  22 #include <fenv_libc.h>
  23 #include <inttypes.h>
  24
  25 static const double almost_half = 0.5000000000000001;  /* 0.5 + 2^-53 */
  26 static const uint32_t a_nan = 0x7fc00000;
  27 static const uint32_t a_inf = 0x7f800000;
  28 static const float two108 = 3.245185536584267269e+32;
  29 static const float twom54 = 5.551115123125782702e-17;
  30 extern const float __t_sqrt[1024];
  31
  32 /* The method is based on a description in
  33    Computation of elementary functions on the IBM RISC System/6000 processor,
  34    P. W. Markstein, IBM J. Res. Develop, 34(1) 1990.
  35    Basically, it consists of two interleaved Newton-Rhapson approximations,
  36    one to find the actual square root, and one to find its reciprocal
  37    without the expense of a division operation.   The tricky bit here
  38    is the use of the POWER/PowerPC multiply-add operation to get the
  39    required accuracy with high speed.
  40
  41    The argument reduction works by a combination of table lookup to
  42    obtain the initial guesses, and some careful modification of the
  43    generated guesses (which mostly runs on the integer unit, while the
  44    Newton-Rhapson is running on the FPU).  */
  45 double
  46 __sqrt(double x)
  47 {
  48   const float inf = *(const float *)&a_inf;
  49   /* x = f_wash(x); *//* This ensures only one exception for SNaN. */
  50   if (x > 0)
  51     {
  52       if (x != inf)
  53         {
  54           /* Variables named starting with 's' exist in the
  55              argument-reduced space, so that 2 > sx >= 0.5,
  56              1.41... > sg >= 0.70.., 0.70.. >= sy > 0.35... .
  57              Variables named ending with 'i' are integer versions of
  58              floating-point values.  */
  59           double sx;   /* The value of which we're trying to find the
  60                           square root.  */
  61           double sg,g; /* Guess of the square root of x.  */
  62           double sd,d; /* Difference between the square of the guess and x.  */
  63           double sy;   /* Estimate of 1/2g (overestimated by 1ulp).  */
  64           double sy2;  /* 2*sy */
  65           double e;    /* Difference between y*g and 1/2 (se = e * fsy).  */
  66           double shx;  /* == sx * fsg */
  67           double fsg;  /* sg*fsg == g.  */
  68           fenv_t fe;  /* Saved floating-point environment (stores rounding
  69                          mode and whether the inexact exception is
  70                          enabled).  */
  71           uint32_t xi0, xi1, sxi, fsgi;
  72           const float *t_sqrt;
  73
  74           fe = fegetenv_register();
  75           EXTRACT_WORDS (xi0,xi1,x);
  76           relax_fenv_state();
  77           sxi = xi0 & 0x3fffffff | 0x3fe00000;
  78           INSERT_WORDS (sx, sxi, xi1);
  79           t_sqrt = __t_sqrt + (xi0 >> 52-32-8-1  & 0x3fe);
  80           sg = t_sqrt[0];
  81           sy = t_sqrt[1];
  82
  83           /* Here we have three Newton-Rhapson iterations each of a
  84              division and a square root and the remainder of the
  85              argument reduction, all interleaved.   */
  86           sd  = -(sg*sg - sx);
  87           fsgi = xi0 + 0x40000000 >> 1 & 0x7ff00000;
  88           sy2 = sy + sy;
  89           sg  = sy*sd + sg;  /* 16-bit approximation to sqrt(sx). */
  90           INSERT_WORDS (fsg, fsgi, 0);
  91           e   = -(sy*sg - almost_half);
  92           sd  = -(sg*sg - sx);
  93           if ((xi0 & 0x7ff00000) == 0)
  94             goto denorm;
  95           sy  = sy + e*sy2;
  96           sg  = sg + sy*sd;  /* 32-bit approximation to sqrt(sx).  */
  97           sy2 = sy + sy;
  98           e   = -(sy*sg - almost_half);
  99           sd  = -(sg*sg - sx);
 100           sy  = sy + e*sy2;
 101           shx = sx * fsg;
 102           sg  = sg + sy*sd;  /* 64-bit approximation to sqrt(sx),
 103                                 but perhaps rounded incorrectly.  */
 104           sy2 = sy + sy;
 105           g   = sg * fsg;
 106           e   = -(sy*sg - almost_half);
 107           d   = -(g*sg - shx);
 108           sy  = sy + e*sy2;
 109           fesetenv_register (fe);
 110           return g + sy*d;
 111         denorm:
 112           /* For denormalised numbers, we normalise, calculate the
 113              square root, and return an adjusted result.  */
 114           fesetenv_register (fe);
 115           return __sqrt(x * two108) * twom54;
 116         }
 117     }
 118   else if (x < 0)
 119     {
 120 #ifdef FE_INVALID_SQRT
 121       feraiseexcept (FE_INVALID_SQRT);
 122       /* For some reason, some PowerPC processors don't implement
 123          FE_INVALID_SQRT.  I guess no-one ever thought they'd be
 124          used for square roots... :-) */
 125       if (!fetestexcept (FE_INVALID))
 126 #endif
 127         feraiseexcept (FE_INVALID);
 128 #ifndef _IEEE_LIBM
 129       if (_LIB_VERSION != _IEEE_)
 130         x = __kernel_standard(x,x,26);
 131       else
 132 #endif
 133       x = *(const float*)&a_nan;
 134     }
 135   return f_wash(x);
 136 }
 137
 138 weak_alias (__sqrt, sqrt)
 139 /* Strictly, this is wrong, but the only places where _ieee754_sqrt is
 140    used will not pass in a negative result.  */
 141 strong_alias(__sqrt,__ieee754_sqrt)