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-rw-r--r--mpc/src/asin.c226
1 files changed, 226 insertions, 0 deletions
diff --git a/mpc/src/asin.c b/mpc/src/asin.c
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+/* mpc_asin -- arcsine of a complex number.
+
+Copyright (C) 2009, 2010, 2011 INRIA
+
+This file is part of GNU MPC.
+
+GNU MPC 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.
+
+GNU MPC 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 this program. If not, see http://www.gnu.org/licenses/ .
+*/
+
+#include "mpc-impl.h"
+
+int
+mpc_asin (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd)
+{
+ mpfr_prec_t p, p_re, p_im, incr_p = 0;
+ mpfr_rnd_t rnd_re, rnd_im;
+ mpc_t z1;
+ int inex;
+
+ /* special values */
+ if (mpfr_nan_p (mpc_realref (op)) || mpfr_nan_p (mpc_imagref (op)))
+ {
+ if (mpfr_inf_p (mpc_realref (op)) || mpfr_inf_p (mpc_imagref (op)))
+ {
+ mpfr_set_nan (mpc_realref (rop));
+ mpfr_set_inf (mpc_imagref (rop), mpfr_signbit (mpc_imagref (op)) ? -1 : +1);
+ }
+ else if (mpfr_zero_p (mpc_realref (op)))
+ {
+ mpfr_set (mpc_realref (rop), mpc_realref (op), GMP_RNDN);
+ mpfr_set_nan (mpc_imagref (rop));
+ }
+ else
+ {
+ mpfr_set_nan (mpc_realref (rop));
+ mpfr_set_nan (mpc_imagref (rop));
+ }
+
+ return 0;
+ }
+
+ if (mpfr_inf_p (mpc_realref (op)) || mpfr_inf_p (mpc_imagref (op)))
+ {
+ int inex_re;
+ if (mpfr_inf_p (mpc_realref (op)))
+ {
+ int inf_im = mpfr_inf_p (mpc_imagref (op));
+
+ inex_re = set_pi_over_2 (mpc_realref (rop),
+ (mpfr_signbit (mpc_realref (op)) ? -1 : 1), MPC_RND_RE (rnd));
+ mpfr_set_inf (mpc_imagref (rop), (mpfr_signbit (mpc_imagref (op)) ? -1 : 1));
+
+ if (inf_im)
+ mpfr_div_2ui (mpc_realref (rop), mpc_realref (rop), 1, GMP_RNDN);
+ }
+ else
+ {
+ mpfr_set_zero (mpc_realref (rop), (mpfr_signbit (mpc_realref (op)) ? -1 : 1));
+ inex_re = 0;
+ mpfr_set_inf (mpc_imagref (rop), (mpfr_signbit (mpc_imagref (op)) ? -1 : 1));
+ }
+
+ return MPC_INEX (inex_re, 0);
+ }
+
+ /* pure real argument */
+ if (mpfr_zero_p (mpc_imagref (op)))
+ {
+ int inex_re;
+ int inex_im;
+ int s_im;
+ s_im = mpfr_signbit (mpc_imagref (op));
+
+ if (mpfr_cmp_ui (mpc_realref (op), 1) > 0)
+ {
+ if (s_im)
+ inex_im = -mpfr_acosh (mpc_imagref (rop), mpc_realref (op),
+ INV_RND (MPC_RND_IM (rnd)));
+ else
+ inex_im = mpfr_acosh (mpc_imagref (rop), mpc_realref (op),
+ MPC_RND_IM (rnd));
+ inex_re = set_pi_over_2 (mpc_realref (rop),
+ (mpfr_signbit (mpc_realref (op)) ? -1 : 1), MPC_RND_RE (rnd));
+ if (s_im)
+ mpc_conj (rop, rop, MPC_RNDNN);
+ }
+ else if (mpfr_cmp_si (mpc_realref (op), -1) < 0)
+ {
+ mpfr_t minus_op_re;
+ minus_op_re[0] = mpc_realref (op)[0];
+ MPFR_CHANGE_SIGN (minus_op_re);
+
+ if (s_im)
+ inex_im = -mpfr_acosh (mpc_imagref (rop), minus_op_re,
+ INV_RND (MPC_RND_IM (rnd)));
+ else
+ inex_im = mpfr_acosh (mpc_imagref (rop), minus_op_re,
+ MPC_RND_IM (rnd));
+ inex_re = set_pi_over_2 (mpc_realref (rop),
+ (mpfr_signbit (mpc_realref (op)) ? -1 : 1), MPC_RND_RE (rnd));
+ if (s_im)
+ mpc_conj (rop, rop, MPC_RNDNN);
+ }
+ else
+ {
+ inex_im = mpfr_set_ui (mpc_imagref (rop), 0, MPC_RND_IM (rnd));
+ if (s_im)
+ mpfr_neg (mpc_imagref (rop), mpc_imagref (rop), GMP_RNDN);
+ inex_re = mpfr_asin (mpc_realref (rop), mpc_realref (op), MPC_RND_RE (rnd));
+ }
+
+ return MPC_INEX (inex_re, inex_im);
+ }
+
+ /* pure imaginary argument */
+ if (mpfr_zero_p (mpc_realref (op)))
+ {
+ int inex_im;
+ int s;
+ s = mpfr_signbit (mpc_realref (op));
+ mpfr_set_ui (mpc_realref (rop), 0, GMP_RNDN);
+ if (s)
+ mpfr_neg (mpc_realref (rop), mpc_realref (rop), GMP_RNDN);
+ inex_im = mpfr_asinh (mpc_imagref (rop), mpc_imagref (op), MPC_RND_IM (rnd));
+
+ return MPC_INEX (0, inex_im);
+ }
+
+ /* regular complex: asin(z) = -i*log(i*z+sqrt(1-z^2)) */
+ p_re = mpfr_get_prec (mpc_realref(rop));
+ p_im = mpfr_get_prec (mpc_imagref(rop));
+ rnd_re = MPC_RND_RE(rnd);
+ rnd_im = MPC_RND_IM(rnd);
+ p = p_re >= p_im ? p_re : p_im;
+ mpc_init2 (z1, p);
+ while (1)
+ {
+ mpfr_exp_t ex, ey, err;
+
+ p += mpc_ceil_log2 (p) + 3 + incr_p; /* incr_p is zero initially */
+ incr_p = p / 2;
+ mpfr_set_prec (mpc_realref(z1), p);
+ mpfr_set_prec (mpc_imagref(z1), p);
+
+ /* z1 <- z^2 */
+ mpc_sqr (z1, op, MPC_RNDNN);
+ /* err(x) <= 1/2 ulp(x), err(y) <= 1/2 ulp(y) */
+ /* z1 <- 1-z1 */
+ ex = mpfr_get_exp (mpc_realref(z1));
+ mpfr_ui_sub (mpc_realref(z1), 1, mpc_realref(z1), GMP_RNDN);
+ mpfr_neg (mpc_imagref(z1), mpc_imagref(z1), GMP_RNDN);
+ ex = ex - mpfr_get_exp (mpc_realref(z1));
+ ex = (ex <= 0) ? 0 : ex;
+ /* err(x) <= 2^ex * ulp(x) */
+ ex = ex + mpfr_get_exp (mpc_realref(z1)) - p;
+ /* err(x) <= 2^ex */
+ ey = mpfr_get_exp (mpc_imagref(z1)) - p - 1;
+ /* err(y) <= 2^ey */
+ ex = (ex >= ey) ? ex : ey; /* err(x), err(y) <= 2^ex, i.e., the norm
+ of the error is bounded by |h|<=2^(ex+1/2) */
+ /* z1 <- sqrt(z1): if z1 = z + h, then sqrt(z1) = sqrt(z) + h/2/sqrt(t) */
+ ey = mpfr_get_exp (mpc_realref(z1)) >= mpfr_get_exp (mpc_imagref(z1))
+ ? mpfr_get_exp (mpc_realref(z1)) : mpfr_get_exp (mpc_imagref(z1));
+ /* we have |z1| >= 2^(ey-1) thus 1/|z1| <= 2^(1-ey) */
+ mpc_sqrt (z1, z1, MPC_RNDNN);
+ ex = (2 * ex + 1) - 2 - (ey - 1); /* |h^2/4/|t| <= 2^ex */
+ ex = (ex + 1) / 2; /* ceil(ex/2) */
+ /* express ex in terms of ulp(z1) */
+ ey = mpfr_get_exp (mpc_realref(z1)) <= mpfr_get_exp (mpc_imagref(z1))
+ ? mpfr_get_exp (mpc_realref(z1)) : mpfr_get_exp (mpc_imagref(z1));
+ ex = ex - ey + p;
+ /* take into account the rounding error in the mpc_sqrt call */
+ err = (ex <= 0) ? 1 : ex + 1;
+ /* err(x) <= 2^err * ulp(x), err(y) <= 2^err * ulp(y) */
+ /* z1 <- i*z + z1 */
+ ex = mpfr_get_exp (mpc_realref(z1));
+ ey = mpfr_get_exp (mpc_imagref(z1));
+ mpfr_sub (mpc_realref(z1), mpc_realref(z1), mpc_imagref(op), GMP_RNDN);
+ mpfr_add (mpc_imagref(z1), mpc_imagref(z1), mpc_realref(op), GMP_RNDN);
+ if (mpfr_cmp_ui (mpc_realref(z1), 0) == 0 || mpfr_cmp_ui (mpc_imagref(z1), 0) == 0)
+ continue;
+ ex -= mpfr_get_exp (mpc_realref(z1)); /* cancellation in x */
+ ey -= mpfr_get_exp (mpc_imagref(z1)); /* cancellation in y */
+ ex = (ex >= ey) ? ex : ey; /* maximum cancellation */
+ err += ex;
+ err = (err <= 0) ? 1 : err + 1; /* rounding error in sub/add */
+ /* z1 <- log(z1): if z1 = z + h, then log(z1) = log(z) + h/t with
+ |t| >= min(|z1|,|z|) */
+ ex = mpfr_get_exp (mpc_realref(z1));
+ ey = mpfr_get_exp (mpc_imagref(z1));
+ ex = (ex >= ey) ? ex : ey;
+ err += ex - p; /* revert to absolute error <= 2^err */
+ mpc_log (z1, z1, GMP_RNDN);
+ err -= ex - 1; /* 1/|t| <= 1/|z| <= 2^(1-ex) */
+ /* express err in terms of ulp(z1) */
+ ey = mpfr_get_exp (mpc_realref(z1)) <= mpfr_get_exp (mpc_imagref(z1))
+ ? mpfr_get_exp (mpc_realref(z1)) : mpfr_get_exp (mpc_imagref(z1));
+ err = err - ey + p;
+ /* take into account the rounding error in the mpc_log call */
+ err = (err <= 0) ? 1 : err + 1;
+ /* z1 <- -i*z1 */
+ mpfr_swap (mpc_realref(z1), mpc_imagref(z1));
+ mpfr_neg (mpc_imagref(z1), mpc_imagref(z1), GMP_RNDN);
+ if (mpfr_can_round (mpc_realref(z1), p - err, GMP_RNDN, GMP_RNDZ,
+ p_re + (rnd_re == GMP_RNDN)) &&
+ mpfr_can_round (mpc_imagref(z1), p - err, GMP_RNDN, GMP_RNDZ,
+ p_im + (rnd_im == GMP_RNDN)))
+ break;
+ }
+
+ inex = mpc_set (rop, z1, rnd);
+ mpc_clear (z1);
+
+ return inex;
+}
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