/* Double-precision floating point square root.

   Copyright (C) 1997-2018 Free Software Foundation, Inc.

   This file is part of the GNU C Library.

   The GNU C Library is free software; you can redistribute it and/or

   modify it under the terms of the GNU Lesser General Public

   License as published by the Free Software Foundation; either

   version 2.1 of the License, or (at your option) any later version.

   The GNU C Library is distributed in the hope that it will be useful,

   but WITHOUT ANY WARRANTY; without even the implied warranty of

   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU

   Lesser General Public License for more details.

   You should have received a copy of the GNU Lesser General Public

   License along with the GNU C Library; if not, see

   <http://www.gnu.org/licenses/>.  */

#include <math.h>

#include <math_private.h>

#include <fenv_libc.h>

#include <inttypes.h>

#include <stdint.h>

#include <sysdep.h>

#include <ldsodefs.h>

#ifndef _ARCH_PPCSQ

static const double almost_half = 0.5000000000000001;	/* 0.5 + 2^-53 */

static const ieee_float_shape_type a_nan = {.word = 0x7fc00000 };

static const ieee_float_shape_type a_inf = {.word = 0x7f800000 };

static const float two108 = 3.245185536584267269e+32;

static const float twom54 = 5.551115123125782702e-17;

extern const float __t_sqrt[1024];

/* The method is based on a description in

   Computation of elementary functions on the IBM RISC System/6000 processor,

   P. W. Markstein, IBM J. Res. Develop, 34(1) 1990.

   Basically, it consists of two interleaved Newton-Raphson approximations,

   one to find the actual square root, and one to find its reciprocal

   without the expense of a division operation.   The tricky bit here

   is the use of the POWER/PowerPC multiply-add operation to get the

   required accuracy with high speed.

   The argument reduction works by a combination of table lookup to

   obtain the initial guesses, and some careful modification of the

   generated guesses (which mostly runs on the integer unit, while the

   Newton-Raphson is running on the FPU).  */

double

__slow_ieee754_sqrt (double x)

{

  const float inf = a_inf.value;

  if (x > 0)

    {

      /* schedule the EXTRACT_WORDS to get separation between the store

	 and the load.  */

      ieee_double_shape_type ew_u;

      ieee_double_shape_type iw_u;

      ew_u.value = (x);

      if (x != inf)

	{

	  /* Variables named starting with 's' exist in the

	     argument-reduced space, so that 2 > sx >= 0.5,

	     1.41... > sg >= 0.70.., 0.70.. >= sy > 0.35... .

	     Variables named ending with 'i' are integer versions of

	     floating-point values.  */

	  double sx;	/* The value of which we're trying to find the

			   square root.  */

	  double sg, g;	/* Guess of the square root of x.  */

	  double sd, d;	/* Difference between the square of the guess and x.  */

	  double sy;	/* Estimate of 1/2g (overestimated by 1ulp).  */

	  double sy2;	/* 2*sy */

	  double e;	/* Difference between y*g and 1/2 (se = e * fsy).  */

	  double shx;	/* == sx * fsg */

	  double fsg;	/* sg*fsg == g.  */

	  fenv_t fe;	/* Saved floating-point environment (stores rounding

			   mode and whether the inexact exception is

			   enabled).  */

	  uint32_t xi0, xi1, sxi, fsgi;

	  const float *t_sqrt;

	  fe = fegetenv_register ();

	  /* complete the EXTRACT_WORDS (xi0,xi1,x) operation.  */

	  xi0 = ew_u.parts.msw;

	  xi1 = ew_u.parts.lsw;

	  relax_fenv_state ();

	  sxi = (xi0 & 0x3fffffff) | 0x3fe00000;

	  /* schedule the INSERT_WORDS (sx, sxi, xi1) to get separation

	     between the store and the load.  */

	  iw_u.parts.msw = sxi;

	  iw_u.parts.lsw = xi1;

	  t_sqrt = __t_sqrt + (xi0 >> (52 - 32 - 8 - 1) & 0x3fe);

	  sg = t_sqrt[0];

	  sy = t_sqrt[1];

	  /* complete the INSERT_WORDS (sx, sxi, xi1) operation.  */

	  sx = iw_u.value;

	  /* Here we have three Newton-Raphson iterations each of a

	     division and a square root and the remainder of the

	     argument reduction, all interleaved.   */

	  sd = -__builtin_fma (sg, sg, -sx);

	  fsgi = (xi0 + 0x40000000) >> 1 & 0x7ff00000;

	  sy2 = sy + sy;

	  sg = __builtin_fma (sy, sd, sg);	/* 16-bit approximation to

						   sqrt(sx). */

	  /* schedule the INSERT_WORDS (fsg, fsgi, 0) to get separation

	     between the store and the load.  */

	  INSERT_WORDS (fsg, fsgi, 0);

	  iw_u.parts.msw = fsgi;

	  iw_u.parts.lsw = (0);

	  e = -__builtin_fma (sy, sg, -almost_half);

	  sd = -__builtin_fma (sg, sg, -sx);

	  if ((xi0 & 0x7ff00000) == 0)

	    goto denorm;

	  sy = __builtin_fma (e, sy2, sy);

	  sg = __builtin_fma (sy, sd, sg);	/* 32-bit approximation to

						   sqrt(sx).  */

	  sy2 = sy + sy;

	  /* complete the INSERT_WORDS (fsg, fsgi, 0) operation.  */

	  fsg = iw_u.value;

	  e = -__builtin_fma (sy, sg, -almost_half);

	  sd = -__builtin_fma (sg, sg, -sx);

	  sy = __builtin_fma (e, sy2, sy);

	  shx = sx * fsg;

	  sg = __builtin_fma (sy, sd, sg);	/* 64-bit approximation to

						   sqrt(sx), but perhaps

						   rounded incorrectly.  */

	  sy2 = sy + sy;

	  g = sg * fsg;

	  e = -__builtin_fma (sy, sg, -almost_half);

	  d = -__builtin_fma (g, sg, -shx);

	  sy = __builtin_fma (e, sy2, sy);

	  fesetenv_register (fe);

	  return __builtin_fma (sy, d, g);

	denorm:

	  /* For denormalised numbers, we normalise, calculate the

	     square root, and return an adjusted result.  */

	  fesetenv_register (fe);

	  return __slow_ieee754_sqrt (x * two108) * twom54;

	}

    }

  else if (x < 0)

    {

      /* For some reason, some PowerPC32 processors don't implement

	 FE_INVALID_SQRT.  */

#ifdef FE_INVALID_SQRT

      __feraiseexcept (FE_INVALID_SQRT);

      fenv_union_t u = { .fenv = fegetenv_register () };

      if ((u.l & FE_INVALID) == 0)

#endif

	__feraiseexcept (FE_INVALID);

      x = a_nan.value;

    }

  return f_wash (x);

}

#endif /* _ARCH_PPCSQ  */

#undef __ieee754_sqrt

double

__ieee754_sqrt (double x)

{

  double z;

#ifdef _ARCH_PPCSQ

  asm ("fsqrt %0,%1\n" :"=f" (z):"f" (x));

#else

  z = __slow_ieee754_sqrt (x);

#endif

  return z;

}

strong_alias (__ieee754_sqrt, __sqrt_finite)