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/* mpfr_agm -- arithmetic-geometric mean of two floating-point numbers
Copyright 1999, 2000, 2001, 2002, 2003, 2004 Free Software Foundation.
This file is part of the MPFR Library.
The MPFR 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 MPFR 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 MPFR 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 "mpfr-impl.h"
int
mpfr_agm (mpfr_ptr r, mpfr_srcptr op2, mpfr_srcptr op1, mp_rnd_t rnd_mode)
{
int s, go_on, compare, inexact;
mp_prec_t p, q;
#if 0
double uo, vo;
#endif
mp_limb_t *up, *vp, *tmpp, *tmpup, *tmpvp, *ap, *bp;
mpfr_t u, v, tmp, tmpu, tmpv, a, b;
TMP_DECL(marker);
/* Deal with special values */
if (MPFR_ARE_SINGULAR(op1, op2))
{
/* If a or b is NaN, the result is NaN */
if (MPFR_IS_NAN(op1) || MPFR_IS_NAN(op2))
{
MPFR_SET_NAN(r);
MPFR_RET_NAN;
}
/* now one of a or b is Inf or 0 */
/* If a or b is -Inf or +Inf, the result is +Inf if both operands are
stricly positive, and NaN otherwise */
else if (MPFR_IS_INF(op1) || MPFR_IS_INF(op2))
{
if (!(MPFR_IS_STRICTPOS(op1) && MPFR_IS_STRICTPOS(op2)))
{
MPFR_SET_NAN(r);
MPFR_RET_NAN;
}
else /* one of a, b is +Inf, the other is a "normal" number */
{
MPFR_SET_INF(r);
MPFR_SET_SAME_SIGN(r, op1);
MPFR_RET(0); /* exact */
}
}
else /* a and b are neither NaN nor Inf, and one is zero */
{ /* If a or b is 0, the result is +0 since a sqrt is positive */
MPFR_ASSERTD(MPFR_IS_ZERO(op1) || MPFR_IS_ZERO(op2));
MPFR_SET_POS(r);
MPFR_SET_ZERO(r);
MPFR_RET(0); /* exact */
}
}
MPFR_CLEAR_FLAGS(r);
/* precision of the following calculus */
q = MPFR_PREC(r);
p = q + 15;
/* Initialisations */
go_on = 1;
TMP_MARK(marker);
s = (p - 1) / BITS_PER_MP_LIMB + 1;
MPFR_TMP_INIT(ap, a, p, s);
MPFR_TMP_INIT(bp, b, p, s);
MPFR_TMP_INIT(up, u, p, s);
MPFR_TMP_INIT(vp, v, p, s);
MPFR_TMP_INIT(tmpup, tmpu, p, s);
MPFR_TMP_INIT(tmpvp, tmpv, p, s);
MPFR_TMP_INIT(tmpp, tmp, p, s);
/* b and a are the 2 operands but we want b >= a */
if ((compare = mpfr_cmp (op1, op2)) > 0)
{
mpfr_set (b,op1,GMP_RNDN);
mpfr_set (a,op2,GMP_RNDN);
}
else if (compare == 0)
{
mpfr_set (r, op1, rnd_mode);
TMP_FREE(marker);
MPFR_RET(0); /* exact */
}
else
{
mpfr_set (b, op2, GMP_RNDN);
mpfr_set (a, op1, GMP_RNDN);
}
#if 0
vo = mpfr_get_d1 (b);
uo = mpfr_get_d1 (a);
#endif
mpfr_set (u, a, GMP_RNDN);
mpfr_set (v, b, GMP_RNDN);
/* Main loop */
while (go_on) {
int err, can_round;
mp_prec_t eq;
double erraux;
#if 0
erraux = (vo == uo) ? 0.0
: __gmpfr_ceil_exp2 (-2.0 * (double) p * uo / (vo - uo));
err = 1 + (int) ((3.0 / 2.0 * (double) __gmpfr_ceil_log2 ((double) p)
+ 1.0) * __gmpfr_ceil_exp2 (- (double) p)
+ 3.0 * erraux);
#else
/* since the argument of __gmpfr_ceil_exp2() is negative,
erraux is always bounded by 1.
We also have floor((3/2*ceil(log2(p))+1) * 2^(-p)) = 0,
since it the value f(p) inside floor() is 5/8 for p=2, 1/2 for p=3,
and f(2p) <= f(p)/2 + 1/2^p. So err=4.
*/
erraux = 1.0;
err = 4;
#endif
#if 0 /* this can not happen since p = q+15, and err=4 */
if (p - err - 3 <= q)
{
p = q + err + 4;
err = 1 + (int) ((3.0 / 2.0 * __gmpfr_ceil_log2 ((double) p) + 1.0)
* __gmpfr_ceil_exp2 (- (double) p) + 3.0 * erraux);
}
#endif
/* Calculus of un and vn */
do
{
mpfr_mul (tmp, u, v, GMP_RNDN);
mpfr_sqrt (tmpu, tmp, GMP_RNDN);
mpfr_add (tmp, u, v, GMP_RNDN);
mpfr_div_2ui (tmpv, tmp, 1, GMP_RNDN);
mpfr_set (u, tmpu, GMP_RNDN);
mpfr_set (v, tmpv, GMP_RNDN);
}
while (mpfr_cmp2 (u, v, &eq) != 0 && eq <= p - 2);
/* Roundability of the result */
can_round = mpfr_can_round (v, p - err - 3, GMP_RNDN, GMP_RNDZ,
q + (rnd_mode == GMP_RNDN));
if (can_round)
go_on = 0;
else {
go_on = 1;
p += 5;
s = (p - 1) / BITS_PER_MP_LIMB + 1;
MPFR_TMP_INIT(up, u, p, s);
MPFR_TMP_INIT(vp, v, p, s);
MPFR_TMP_INIT(tmpup, tmpu, p, s);
MPFR_TMP_INIT(tmpvp, tmpv, p, s);
MPFR_TMP_INIT(tmpp, tmp, p, s);
mpfr_set (u, a, GMP_RNDN);
mpfr_set (v, b, GMP_RNDN);
}
}
/* End of while */
/* Setting of the result */
inexact = mpfr_set (r, v, rnd_mode);
/* Let's clean */
TMP_FREE(marker);
return inexact; /* agm(u,v) can be exact for u, v rational only for u=v.
Proof (due to Nicolas Brisebarre): it suffices to consider
u=1 and v<1. Then 1/AGM(1,v) = 2F1(1/2,1/2,1;1-v^2),
and a theorem due to G.V. Chudnovsky states that for x a
non-zero algebraic number with |x|<1, then
2F1(1/2,1/2,1;x) and 2F1(-1/2,1/2,1;x) are algebraically
independent over Q. */
}
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