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Diffstat (limited to 'mpc/src/asin.c')
-rw-r--r-- | mpc/src/asin.c | 226 |
1 files changed, 226 insertions, 0 deletions
diff --git a/mpc/src/asin.c b/mpc/src/asin.c new file mode 100644 index 0000000000..bd4e3132de --- /dev/null +++ b/mpc/src/asin.c @@ -0,0 +1,226 @@ +/* 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; +} |