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186 lines
6.0 KiB
C
186 lines
6.0 KiB
C
/* Double-precision floating point square root.
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Copyright (C) 1997, 2002, 2003, 2004 Free Software Foundation, Inc.
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This file is part of the GNU C Library.
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The GNU C Library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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The GNU C Library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with the GNU C Library; if not, write to the Free
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Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA
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02111-1307 USA. */
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#include <math.h>
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#include <math_private.h>
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#include <fenv_libc.h>
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#include <inttypes.h>
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#include <sysdep.h>
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#include <ldsodefs.h>
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#include <dl-procinfo.h>
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static const double almost_half = 0.5000000000000001; /* 0.5 + 2^-53 */
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static const ieee_float_shape_type a_nan = {.word = 0x7fc00000 };
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static const ieee_float_shape_type a_inf = {.word = 0x7f800000 };
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static const float two108 = 3.245185536584267269e+32;
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static const float twom54 = 5.551115123125782702e-17;
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extern const float __t_sqrt[1024];
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/* The method is based on a description in
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Computation of elementary functions on the IBM RISC System/6000 processor,
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P. W. Markstein, IBM J. Res. Develop, 34(1) 1990.
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Basically, it consists of two interleaved Newton-Rhapson approximations,
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one to find the actual square root, and one to find its reciprocal
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without the expense of a division operation. The tricky bit here
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is the use of the POWER/PowerPC multiply-add operation to get the
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required accuracy with high speed.
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The argument reduction works by a combination of table lookup to
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obtain the initial guesses, and some careful modification of the
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generated guesses (which mostly runs on the integer unit, while the
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Newton-Rhapson is running on the FPU). */
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#ifdef __STDC__
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double
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__slow_ieee754_sqrt (double x)
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#else
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double
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__slow_ieee754_sqrt (x)
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double x;
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#endif
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{
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const float inf = a_inf.value;
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if (x > 0)
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{
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/* schedule the EXTRACT_WORDS to get separation between the store
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and the load. */
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ieee_double_shape_type ew_u;
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ieee_double_shape_type iw_u;
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ew_u.value = (x);
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if (x != inf)
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{
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/* Variables named starting with 's' exist in the
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argument-reduced space, so that 2 > sx >= 0.5,
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1.41... > sg >= 0.70.., 0.70.. >= sy > 0.35... .
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Variables named ending with 'i' are integer versions of
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floating-point values. */
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double sx; /* The value of which we're trying to find the
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square root. */
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double sg, g; /* Guess of the square root of x. */
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double sd, d; /* Difference between the square of the guess and x. */
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double sy; /* Estimate of 1/2g (overestimated by 1ulp). */
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double sy2; /* 2*sy */
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double e; /* Difference between y*g and 1/2 (se = e * fsy). */
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double shx; /* == sx * fsg */
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double fsg; /* sg*fsg == g. */
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fenv_t fe; /* Saved floating-point environment (stores rounding
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mode and whether the inexact exception is
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enabled). */
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uint32_t xi0, xi1, sxi, fsgi;
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const float *t_sqrt;
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fe = fegetenv_register ();
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/* complete the EXTRACT_WORDS (xi0,xi1,x) operation. */
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xi0 = ew_u.parts.msw;
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xi1 = ew_u.parts.lsw;
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relax_fenv_state ();
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sxi = (xi0 & 0x3fffffff) | 0x3fe00000;
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/* schedule the INSERT_WORDS (sx, sxi, xi1) to get separation
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between the store and the load. */
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iw_u.parts.msw = sxi;
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iw_u.parts.lsw = xi1;
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t_sqrt = __t_sqrt + (xi0 >> (52 - 32 - 8 - 1) & 0x3fe);
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sg = t_sqrt[0];
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sy = t_sqrt[1];
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/* complete the INSERT_WORDS (sx, sxi, xi1) operation. */
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sx = iw_u.value;
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/* Here we have three Newton-Rhapson iterations each of a
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division and a square root and the remainder of the
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argument reduction, all interleaved. */
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sd = -(sg * sg - sx);
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fsgi = (xi0 + 0x40000000) >> 1 & 0x7ff00000;
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sy2 = sy + sy;
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sg = sy * sd + sg; /* 16-bit approximation to sqrt(sx). */
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/* schedule the INSERT_WORDS (fsg, fsgi, 0) to get separation
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between the store and the load. */
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INSERT_WORDS (fsg, fsgi, 0);
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iw_u.parts.msw = fsgi;
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iw_u.parts.lsw = (0);
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e = -(sy * sg - almost_half);
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sd = -(sg * sg - sx);
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if ((xi0 & 0x7ff00000) == 0)
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goto denorm;
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sy = sy + e * sy2;
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sg = sg + sy * sd; /* 32-bit approximation to sqrt(sx). */
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sy2 = sy + sy;
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/* complete the INSERT_WORDS (fsg, fsgi, 0) operation. */
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fsg = iw_u.value;
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e = -(sy * sg - almost_half);
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sd = -(sg * sg - sx);
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sy = sy + e * sy2;
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shx = sx * fsg;
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sg = sg + sy * sd; /* 64-bit approximation to sqrt(sx),
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but perhaps rounded incorrectly. */
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sy2 = sy + sy;
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g = sg * fsg;
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e = -(sy * sg - almost_half);
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d = -(g * sg - shx);
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sy = sy + e * sy2;
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fesetenv_register (fe);
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return g + sy * d;
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denorm:
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/* For denormalised numbers, we normalise, calculate the
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square root, and return an adjusted result. */
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fesetenv_register (fe);
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return __slow_ieee754_sqrt (x * two108) * twom54;
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}
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}
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else if (x < 0)
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{
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/* For some reason, some PowerPC32 processors don't implement
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FE_INVALID_SQRT. */
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#ifdef FE_INVALID_SQRT
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feraiseexcept (FE_INVALID_SQRT);
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if (!fetestexcept (FE_INVALID))
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#endif
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feraiseexcept (FE_INVALID);
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x = a_nan.value;
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}
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return f_wash (x);
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}
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#ifdef __STDC__
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double
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__ieee754_sqrt (double x)
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#else
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double
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__ieee754_sqrt (x)
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double x;
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#endif
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{
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double z;
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/* If the CPU is 64-bit we can use the optional FP instructions we. */
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if ((GLRO (dl_hwcap) & PPC_FEATURE_64) != 0)
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{
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/* Volatile is required to prevent the compiler from moving the
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fsqrt instruction above the branch. */
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__asm __volatile (" fsqrt %0,%1\n"
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:"=f" (z):"f" (x));
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}
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else
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z = __slow_ieee754_sqrt (x);
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return z;
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}
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