/* mpfr_agm -- arithmetic-geometric mean of two floating-point numbers
Copyright 1999-2017 Free Software Foundation, Inc.
Contributed by the AriC and Caramba projects, INRIA.
This file is part of the GNU MPFR Library.
The GNU 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 3 of the License, or (at your
option) any later version.
The GNU 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 GNU MPFR Library; see the file COPYING.LESSER. If not, see
http://www.gnu.org/licenses/ or write to the Free Software Foundation, Inc.,
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA. */
#define MPFR_NEED_LONGLONG_H
#include "mpfr-impl.h"
/* agm(x,y) is between x and y, so we don't need to save exponent range */
int
mpfr_agm (mpfr_ptr r, mpfr_srcptr op2, mpfr_srcptr op1, mpfr_rnd_t rnd_mode)
{
int compare, inexact;
mp_size_t s;
mpfr_prec_t p, q;
mp_limb_t *up, *vp, *ufp, *vfp;
mpfr_t u, v, uf, vf, sc1, sc2;
mpfr_exp_t scaleop = 0, scaleit;
unsigned long n; /* number of iterations */
MPFR_ZIV_DECL (loop);
MPFR_TMP_DECL(marker);
MPFR_SAVE_EXPO_DECL (expo);
MPFR_LOG_FUNC
(("op2[%Pu]=%.*Rg op1[%Pu]=%.*Rg rnd=%d",
mpfr_get_prec (op2), mpfr_log_prec, op2,
mpfr_get_prec (op1), mpfr_log_prec, op1, rnd_mode),
("r[%Pu]=%.*Rg inexact=%d",
mpfr_get_prec (r), mpfr_log_prec, r, inexact));
/* 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 and b is +Inf, the result is +Inf.
Otherwise if a or b is -Inf or 0, the result is NaN */
else if (MPFR_IS_INF(op1) || MPFR_IS_INF(op2))
{
if (MPFR_IS_STRICTPOS(op1) && MPFR_IS_STRICTPOS(op2))
{
MPFR_SET_INF(r);
MPFR_SET_SAME_SIGN(r, op1);
MPFR_RET(0); /* exact */
}
else
{
MPFR_SET_NAN(r);
MPFR_RET_NAN;
}
}
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 */
}
}
/* If a or b is negative (excluding -Infinity), the result is NaN */
if (MPFR_UNLIKELY(MPFR_IS_NEG(op1) || MPFR_IS_NEG(op2)))
{
MPFR_SET_NAN(r);
MPFR_RET_NAN;
}
/* Precision of the following calculus */
q = MPFR_PREC(r);
p = q + MPFR_INT_CEIL_LOG2(q) + 15;
MPFR_ASSERTD (p >= 7); /* see algorithms.tex */
s = MPFR_PREC2LIMBS (p);
/* b (op2) and a (op1) are the 2 operands but we want b >= a */
compare = mpfr_cmp (op1, op2);
if (MPFR_UNLIKELY( compare == 0 ))
return mpfr_set (r, op1, rnd_mode);
else if (compare > 0)
{
mpfr_srcptr t = op1;
op1 = op2;
op2 = t;
}
/* Now b (=op2) > a (=op1) */
MPFR_SAVE_EXPO_MARK (expo);
MPFR_TMP_MARK(marker);
/* Main loop */
MPFR_ZIV_INIT (loop, p);
for (;;)
{
mpfr_prec_t eq;
unsigned long err = 0; /* must be set to 0 at each Ziv iteration */
MPFR_BLOCK_DECL (flags);
/* Init temporary vars */
MPFR_TMP_INIT (up, u, p, s);
MPFR_TMP_INIT (vp, v, p, s);
MPFR_TMP_INIT (ufp, uf, p, s);
MPFR_TMP_INIT (vfp, vf, p, s);
/* Calculus of un and vn */
retry:
MPFR_BLOCK (flags,
mpfr_mul (u, op1, op2, MPFR_RNDN);
/* mpfr_mul(...): faster since PREC(op) < PREC(u) */
mpfr_add (v, op1, op2, MPFR_RNDN);
/* mpfr_add with !=prec is still good */);
if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags) || MPFR_UNDERFLOW (flags)))
{
mpfr_exp_t e1 , e2;
MPFR_ASSERTN (scaleop == 0);
e1 = MPFR_GET_EXP (op1);
e2 = MPFR_GET_EXP (op2);
/* Let's determine scaleop to avoid an overflow/underflow. */
if (MPFR_OVERFLOW (flags))
{
/* Let's recall that emin <= e1 <= e2 <= emax.
There has been an overflow. Thus e2 >= emax/2.
If the mpfr_mul overflowed, then e1 + e2 > emax.
If the mpfr_add overflowed, then e2 = emax.
We want: (e1 + scale) + (e2 + scale) <= emax,
i.e. scale <= (emax - e1 - e2) / 2. Let's take
scale = min(floor((emax - e1 - e2) / 2), -1).
This is OK, as:
1. emin <= scale <= -1.
2. e1 + scale >= emin. Indeed:
* If e1 + e2 > emax, then
e1 + scale >= e1 + (emax - e1 - e2) / 2 - 1
>= (emax + e1 - emax) / 2 - 1
>= e1 / 2 - 1 >= emin.
* Otherwise, mpfr_mul didn't overflow, therefore
mpfr_add overflowed and e2 = emax, so that
e1 > emin (see restriction below).
e1 + scale > emin - 1, thus e1 + scale >= emin.
3. e2 + scale <= emax, since scale < 0. */
if (e1 + e2 > MPFR_EXT_EMAX)
{
scaleop = - (((e1 + e2) - MPFR_EXT_EMAX + 1) / 2);
MPFR_ASSERTN (scaleop < 0);
}
else
{
/* The addition necessarily overflowed. */
MPFR_ASSERTN (e2 == MPFR_EXT_EMAX);
/* The case where e1 = emin and e2 = emax is not supported
here. This would mean that the precision of e2 would be
huge (and possibly not supported in practice anyway). */
MPFR_ASSERTN (e1 > MPFR_EXT_EMIN);
scaleop = -1;
}
}
else /* underflow only (in the multiplication) */
{
/* We have e1 + e2 <= emin (so, e1 <= e2 <= 0).
We want: (e1 + scale) + (e2 + scale) >= emin + 1,
i.e. scale >= (emin + 1 - e1 - e2) / 2. let's take
scale = ceil((emin + 1 - e1 - e2) / 2). This is OK, as:
1. 1 <= scale <= emax.
2. e1 + scale >= emin + 1 >= emin.
3. e2 + scale <= scale <= emax. */
MPFR_ASSERTN (e1 <= e2 && e2 <= 0);
scaleop = (MPFR_EXT_EMIN + 2 - e1 - e2) / 2;
MPFR_ASSERTN (scaleop > 0);
}
MPFR_ALIAS (sc1, op1, MPFR_SIGN (op1), e1 + scaleop);
MPFR_ALIAS (sc2, op2, MPFR_SIGN (op2), e2 + scaleop);
op1 = sc1;
op2 = sc2;
MPFR_LOG_MSG (("Exception in pre-iteration, scale = %"
MPFR_EXP_FSPEC "d\n", scaleop));
goto retry;
}
mpfr_clear_flags ();
mpfr_sqrt (u, u, MPFR_RNDN);
mpfr_div_2ui (v, v, 1, MPFR_RNDN);
scaleit = 0;
n = 1;
while (mpfr_cmp2 (u, v, &eq) != 0 && eq <= p - 2)
{
MPFR_BLOCK_DECL (flags2);
MPFR_LOG_MSG (("Iteration n = %lu\n", n));
retry2:
mpfr_add (vf, u, v, MPFR_RNDN); /* No overflow? */
mpfr_div_2ui (vf, vf, 1, MPFR_RNDN);
/* See proof in algorithms.tex */
if (4*eq > p)
{
mpfr_t w;
MPFR_BLOCK_DECL (flags3);
MPFR_LOG_MSG (("4*eq > p\n", 0));
/* vf = V(k) */
mpfr_init2 (w, (p + 1) / 2);
MPFR_BLOCK
(flags3,
mpfr_sub (w, v, u, MPFR_RNDN); /* e = V(k-1)-U(k-1) */
mpfr_sqr (w, w, MPFR_RNDN); /* e = e^2 */
mpfr_div_2ui (w, w, 4, MPFR_RNDN); /* e*= (1/2)^2*1/4 */
mpfr_div (w, w, vf, MPFR_RNDN); /* 1/4*e^2/V(k) */
);
if (MPFR_LIKELY (! MPFR_UNDERFLOW (flags3)))
{
mpfr_sub (v, vf, w, MPFR_RNDN);
err = MPFR_GET_EXP (vf) - MPFR_GET_EXP (v); /* 0 or 1 */
mpfr_clear (w);
break;
}
/* There has been an underflow because of the cancellation
between V(k-1) and U(k-1). Let's use the conventional
method. */
MPFR_LOG_MSG (("4*eq > p -> underflow\n", 0));
mpfr_clear (w);
mpfr_clear_underflow ();
}
/* U(k) increases, so that U.V can overflow (but not underflow). */
MPFR_BLOCK (flags2, mpfr_mul (uf, u, v, MPFR_RNDN););
if (MPFR_UNLIKELY (MPFR_OVERFLOW (flags2)))
{
mpfr_exp_t scale2;
scale2 = - (((MPFR_GET_EXP (u) + MPFR_GET_EXP (v))
- MPFR_EXT_EMAX + 1) / 2);
MPFR_EXP (u) += scale2;
MPFR_EXP (v) += scale2;
scaleit += scale2;
MPFR_LOG_MSG (("Overflow in iteration n = %lu, scaleit = %"
MPFR_EXP_FSPEC "d (%" MPFR_EXP_FSPEC "d)\n",
n, scaleit, scale2));
mpfr_clear_overflow ();
goto retry2;
}
mpfr_sqrt (u, uf, MPFR_RNDN);
mpfr_swap (v, vf);
n ++;
}
MPFR_LOG_MSG (("End of iterations (n = %lu)\n", n));
/* the error on v is bounded by (18n+51) ulps, or twice if there
was an exponent loss in the final subtraction */
err += MPFR_INT_CEIL_LOG2(18 * n + 51); /* 18n+51 should not overflow
since n is about log(p) */
/* we should have n+2 <= 2^(p/4) [see algorithms.tex] */
if (MPFR_LIKELY (MPFR_INT_CEIL_LOG2(n + 2) <= p / 4 &&
MPFR_CAN_ROUND (v, p - err, q, rnd_mode)))
break; /* Stop the loop */
/* Next iteration */
MPFR_ZIV_NEXT (loop, p);
s = MPFR_PREC2LIMBS (p);
}
MPFR_ZIV_FREE (loop);
if (MPFR_UNLIKELY ((__gmpfr_flags & (MPFR_FLAGS_ALL ^ MPFR_FLAGS_INEXACT))
!= 0))
{
MPFR_ASSERTN (! mpfr_overflow_p ()); /* since mpfr_clear_flags */
MPFR_ASSERTN (! mpfr_underflow_p ()); /* since mpfr_clear_flags */
MPFR_ASSERTN (! mpfr_divby0_p ()); /* since mpfr_clear_flags */
MPFR_ASSERTN (! mpfr_nanflag_p ()); /* since mpfr_clear_flags */
}
/* Setting of the result */
inexact = mpfr_set (r, v, rnd_mode);
MPFR_EXP (r) -= scaleop + scaleit;
/* Let's clean */
MPFR_TMP_FREE(marker);
MPFR_SAVE_EXPO_FREE (expo);
/* From the definition of the AGM, underflow and overflow
are not possible. */
return mpfr_check_range (r, inexact, rnd_mode);
/* 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. */
}