about summary refs log tree commit diff
path: root/sysdeps/powerpc/e_sqrt.c
blob: df80973f585de9401d5da10760bab95489458010 (plain) (blame)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
/* Single-precision floating point square root.
   Copyright (C) 1997 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 Library General Public License as
   published by the Free Software Foundation; either version 2 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
   Library General Public License for more details.

   You should have received a copy of the GNU Library General Public
   License along with the GNU C Library; see the file COPYING.LIB.  If not,
   write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330,
   Boston, MA 02111-1307, USA.  */

#include <math.h>
#include <math_private.h>
#include <fenv_libc.h>
#include <inttypes.h>

static const double almost_half = 0.5000000000000001;  /* 0.5 + 2^-53 */
static const uint32_t a_nan = 0x7fc00000;
static const uint32_t a_inf = 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-Rhapson 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-Rhapson is running on the FPU).  */
double
__sqrt(double x)
{
  const float inf = *(const float *)&a_inf;
  /* x = f_wash(x); *//* This ensures only one exception for SNaN. */
  if (x > 0)
    {
      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();
	  EXTRACT_WORDS (xi0,xi1,x);
	  relax_fenv_state();
	  sxi = xi0 & 0x3fffffff | 0x3fe00000;
	  INSERT_WORDS (sx, sxi, xi1);
	  t_sqrt = __t_sqrt + (xi0 >> 52-32-8-1  & 0x3fe);
	  sg = t_sqrt[0];
	  sy = t_sqrt[1];
	  
	  /* Here we have three Newton-Rhapson iterations each of a
	     division and a square root and the remainder of the
	     argument reduction, all interleaved.   */
	  sd  = -(sg*sg - sx);
	  fsgi = xi0 + 0x40000000 >> 1 & 0x7ff00000;
	  sy2 = sy + sy;
	  sg  = sy*sd + sg;  /* 16-bit approximation to sqrt(sx). */
	  INSERT_WORDS (fsg, fsgi, 0);
	  e   = -(sy*sg - almost_half);
	  sd  = -(sg*sg - sx);
	  if ((xi0 & 0x7ff00000) == 0)
	    goto denorm;
	  sy  = sy + e*sy2;
	  sg  = sg + sy*sd;  /* 32-bit approximation to sqrt(sx).  */
	  sy2 = sy + sy;
	  e   = -(sy*sg - almost_half);
	  sd  = -(sg*sg - sx);
	  sy  = sy + e*sy2;
	  shx = sx * fsg;
	  sg  = sg + sy*sd;  /* 64-bit approximation to sqrt(sx),
				but perhaps rounded incorrectly.  */
	  sy2 = sy + sy;
	  g   = sg * fsg;
	  e   = -(sy*sg - almost_half);
	  d   = -(g*sg - shx);
	  sy  = sy + e*sy2;
	  fesetenv_register (fe);
	  return g + sy*d;
	denorm:
	  /* For denormalised numbers, we normalise, calculate the
	     square root, and return an adjusted result.  */
	  fesetenv_register (fe);
	  return __sqrt(x * two108) * twom54;
	}
    }
  else if (x < 0)
    {
#ifdef FE_INVALID_SQRT
      feraiseexcept (FE_INVALID_SQRT);
      /* For some reason, some PowerPC processors don't implement
	 FE_INVALID_SQRT.  I guess no-one ever thought they'd be
	 used for square roots... :-) */
      if (!fetestexcept (FE_INVALID))
#endif
	feraiseexcept (FE_INVALID);
#ifndef _IEEE_LIBM
      if (_LIB_VERSION != _IEEE_)
	x = __kernel_standard(x,x,26);
      else
#endif
      x = *(const float*)&a_nan;
    }
  return f_wash(x);
}

weak_alias (__sqrt, sqrt)
/* Strictly, this is wrong, but the only places where _ieee754_sqrt is
   used will not pass in a negative result.  */
strong_alias(__sqrt,__ieee754_sqrt)