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Diffstat (limited to 'mpc/src/sin_cos.c')
| -rw-r--r-- | mpc/src/sin_cos.c | 402 |
1 files changed, 402 insertions, 0 deletions
diff --git a/mpc/src/sin_cos.c b/mpc/src/sin_cos.c new file mode 100644 index 0000000000..0cff45acd5 --- /dev/null +++ b/mpc/src/sin_cos.c @@ -0,0 +1,402 @@ +/* mpc_sin_cos -- combined sine and cosine of a complex number. + +Copyright (C) 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" + +static int +mpc_sin_cos_nonfinite (mpc_ptr rop_sin, mpc_ptr rop_cos, mpc_srcptr op, + mpc_rnd_t rnd_sin, mpc_rnd_t rnd_cos) + /* assumes that op (that is, its real or imaginary part) is not finite */ +{ + int overlap; + mpc_t op_loc; + + overlap = (rop_sin == op || rop_cos == op); + if (overlap) { + mpc_init3 (op_loc, MPC_PREC_RE (op), MPC_PREC_IM (op)); + mpc_set (op_loc, op, MPC_RNDNN); + } + else + op_loc [0] = op [0]; + + if (rop_sin != NULL) { + if (mpfr_nan_p (mpc_realref (op_loc)) || mpfr_nan_p (mpc_imagref (op_loc))) { + mpc_set (rop_sin, op_loc, rnd_sin); + if (mpfr_nan_p (mpc_imagref (op_loc))) { + /* sin(x +i*NaN) = NaN +i*NaN, except for x=0 */ + /* sin(-0 +i*NaN) = -0 +i*NaN */ + /* sin(+0 +i*NaN) = +0 +i*NaN */ + if (!mpfr_zero_p (mpc_realref (op_loc))) + mpfr_set_nan (mpc_realref (rop_sin)); + } + else /* op = NaN + i*y */ + if (!mpfr_inf_p (mpc_imagref (op_loc)) && !mpfr_zero_p (mpc_imagref (op_loc))) + /* sin(NaN -i*Inf) = NaN -i*Inf */ + /* sin(NaN -i*0) = NaN -i*0 */ + /* sin(NaN +i*0) = NaN +i*0 */ + /* sin(NaN +i*Inf) = NaN +i*Inf */ + /* sin(NaN +i*y) = NaN +i*NaN, when 0<|y|<Inf */ + mpfr_set_nan (mpc_imagref (rop_sin)); + } + else if (mpfr_inf_p (mpc_realref (op_loc))) { + mpfr_set_nan (mpc_realref (rop_sin)); + + if (!mpfr_inf_p (mpc_imagref (op_loc)) && !mpfr_zero_p (mpc_imagref (op_loc))) + /* sin(+/-Inf +i*y) = NaN +i*NaN, when 0<|y|<Inf */ + mpfr_set_nan (mpc_imagref (rop_sin)); + else + /* sin(+/-Inf -i*Inf) = NaN -i*Inf */ + /* sin(+/-Inf +i*Inf) = NaN +i*Inf */ + /* sin(+/-Inf -i*0) = NaN -i*0 */ + /* sin(+/-Inf +i*0) = NaN +i*0 */ + mpfr_set (mpc_imagref (rop_sin), mpc_imagref (op_loc), MPC_RND_IM (rnd_sin)); + } + else if (mpfr_zero_p (mpc_realref (op_loc))) { + /* sin(-0 -i*Inf) = -0 -i*Inf */ + /* sin(+0 -i*Inf) = +0 -i*Inf */ + /* sin(-0 +i*Inf) = -0 +i*Inf */ + /* sin(+0 +i*Inf) = +0 +i*Inf */ + mpc_set (rop_sin, op_loc, rnd_sin); + } + else { + /* sin(x -i*Inf) = +Inf*(sin(x) -i*cos(x)) */ + /* sin(x +i*Inf) = +Inf*(sin(x) +i*cos(x)) */ + mpfr_t s, c; + mpfr_init2 (s, 2); + mpfr_init2 (c, 2); + mpfr_sin_cos (s, c, mpc_realref (op_loc), GMP_RNDZ); + mpfr_set_inf (mpc_realref (rop_sin), MPFR_SIGN (s)); + mpfr_set_inf (mpc_imagref (rop_sin), MPFR_SIGN (c)*MPFR_SIGN (mpc_imagref (op_loc))); + mpfr_clear (s); + mpfr_clear (c); + } + } + + if (rop_cos != NULL) { + if (mpfr_nan_p (mpc_realref (op_loc))) { + /* cos(NaN + i * NaN) = NaN + i * NaN */ + /* cos(NaN - i * Inf) = +Inf + i * NaN */ + /* cos(NaN + i * Inf) = +Inf + i * NaN */ + /* cos(NaN - i * 0) = NaN - i * 0 */ + /* cos(NaN + i * 0) = NaN + i * 0 */ + /* cos(NaN + i * y) = NaN + i * NaN, when y != 0 */ + if (mpfr_inf_p (mpc_imagref (op_loc))) + mpfr_set_inf (mpc_realref (rop_cos), +1); + else + mpfr_set_nan (mpc_realref (rop_cos)); + + if (mpfr_zero_p (mpc_imagref (op_loc))) + mpfr_set (mpc_imagref (rop_cos), mpc_imagref (op_loc), MPC_RND_IM (rnd_cos)); + else + mpfr_set_nan (mpc_imagref (rop_cos)); + } + else if (mpfr_nan_p (mpc_imagref (op_loc))) { + /* cos(-Inf + i * NaN) = NaN + i * NaN */ + /* cos(+Inf + i * NaN) = NaN + i * NaN */ + /* cos(-0 + i * NaN) = NaN - i * 0 */ + /* cos(+0 + i * NaN) = NaN + i * 0 */ + /* cos(x + i * NaN) = NaN + i * NaN, when x != 0 */ + if (mpfr_zero_p (mpc_realref (op_loc))) + mpfr_set (mpc_imagref (rop_cos), mpc_realref (op_loc), MPC_RND_IM (rnd_cos)); + else + mpfr_set_nan (mpc_imagref (rop_cos)); + + mpfr_set_nan (mpc_realref (rop_cos)); + } + else if (mpfr_inf_p (mpc_realref (op_loc))) { + /* cos(-Inf -i*Inf) = cos(+Inf +i*Inf) = -Inf +i*NaN */ + /* cos(-Inf +i*Inf) = cos(+Inf -i*Inf) = +Inf +i*NaN */ + /* cos(-Inf -i*0) = cos(+Inf +i*0) = NaN -i*0 */ + /* cos(-Inf +i*0) = cos(+Inf -i*0) = NaN +i*0 */ + /* cos(-Inf +i*y) = cos(+Inf +i*y) = NaN +i*NaN, when y != 0 */ + + const int same_sign = + mpfr_signbit (mpc_realref (op_loc)) == mpfr_signbit (mpc_imagref (op_loc)); + + if (mpfr_inf_p (mpc_imagref (op_loc))) + mpfr_set_inf (mpc_realref (rop_cos), (same_sign ? -1 : +1)); + else + mpfr_set_nan (mpc_realref (rop_cos)); + + if (mpfr_zero_p (mpc_imagref (op_loc))) + mpfr_setsign (mpc_imagref (rop_cos), mpc_imagref (op_loc), same_sign, + MPC_RND_IM(rnd_cos)); + else + mpfr_set_nan (mpc_imagref (rop_cos)); + } + else if (mpfr_zero_p (mpc_realref (op_loc))) { + /* cos(-0 -i*Inf) = cos(+0 +i*Inf) = +Inf -i*0 */ + /* cos(-0 +i*Inf) = cos(+0 -i*Inf) = +Inf +i*0 */ + const int same_sign = + mpfr_signbit (mpc_realref (op_loc)) == mpfr_signbit (mpc_imagref (op_loc)); + + mpfr_setsign (mpc_imagref (rop_cos), mpc_realref (op_loc), same_sign, + MPC_RND_IM (rnd_cos)); + mpfr_set_inf (mpc_realref (rop_cos), +1); + } + else { + /* cos(x -i*Inf) = +Inf*cos(x) +i*Inf*sin(x), when x != 0 */ + /* cos(x +i*Inf) = +Inf*cos(x) -i*Inf*sin(x), when x != 0 */ + mpfr_t s, c; + mpfr_init2 (c, 2); + mpfr_init2 (s, 2); + mpfr_sin_cos (s, c, mpc_realref (op_loc), GMP_RNDN); + mpfr_set_inf (mpc_realref (rop_cos), mpfr_sgn (c)); + mpfr_set_inf (mpc_imagref (rop_cos), + (mpfr_sgn (mpc_imagref (op_loc)) == mpfr_sgn (s) ? -1 : +1)); + mpfr_clear (s); + mpfr_clear (c); + } + } + + if (overlap) + mpc_clear (op_loc); + + return MPC_INEX12 (MPC_INEX (0,0), MPC_INEX (0,0)); + /* everything is exact */ +} + + +static int +mpc_sin_cos_real (mpc_ptr rop_sin, mpc_ptr rop_cos, mpc_srcptr op, + mpc_rnd_t rnd_sin, mpc_rnd_t rnd_cos) + /* assumes that op is real */ +{ + int inex_sin_re = 0, inex_cos_re = 0; + /* Until further notice, assume computations exact; in particular, + by definition, for not computed values. */ + mpfr_t s, c; + int inex_s, inex_c; + int sign_im = mpfr_signbit (mpc_imagref (op)); + + /* sin(x +-0*i) = sin(x) +-0*i*sign(cos(x)) */ + /* cos(x +-i*0) = cos(x) -+i*0*sign(sin(x)) */ + if (rop_sin != 0) + mpfr_init2 (s, MPC_PREC_RE (rop_sin)); + else + mpfr_init2 (s, 2); /* We need only the sign. */ + if (rop_cos != NULL) + mpfr_init2 (c, MPC_PREC_RE (rop_cos)); + else + mpfr_init2 (c, 2); + inex_s = mpfr_sin (s, mpc_realref (op), MPC_RND_RE (rnd_sin)); + inex_c = mpfr_cos (c, mpc_realref (op), MPC_RND_RE (rnd_cos)); + /* We cannot use mpfr_sin_cos since we may need two distinct rounding + modes and the exact return values. If we need only the sign, an + arbitrary rounding mode will work. */ + + if (rop_sin != NULL) { + mpfr_set (mpc_realref (rop_sin), s, GMP_RNDN); /* exact */ + inex_sin_re = inex_s; + mpfr_set_zero (mpc_imagref (rop_sin), + ( ( sign_im && !mpfr_signbit(c)) + || (!sign_im && mpfr_signbit(c)) ? -1 : 1)); + } + + if (rop_cos != NULL) { + mpfr_set (mpc_realref (rop_cos), c, GMP_RNDN); /* exact */ + inex_cos_re = inex_c; + mpfr_set_zero (mpc_imagref (rop_cos), + ( ( sign_im && mpfr_signbit(s)) + || (!sign_im && !mpfr_signbit(s)) ? -1 : 1)); + } + + mpfr_clear (s); + mpfr_clear (c); + + return MPC_INEX12 (MPC_INEX (inex_sin_re, 0), MPC_INEX (inex_cos_re, 0)); +} + + +static int +mpc_sin_cos_imag (mpc_ptr rop_sin, mpc_ptr rop_cos, mpc_srcptr op, + mpc_rnd_t rnd_sin, mpc_rnd_t rnd_cos) + /* assumes that op is purely imaginary, but not zero */ +{ + int inex_sin_im = 0, inex_cos_re = 0; + /* assume exact if not computed */ + int overlap; + mpc_t op_loc; + + overlap = (rop_sin == op || rop_cos == op); + if (overlap) { + mpc_init3 (op_loc, MPC_PREC_RE (op), MPC_PREC_IM (op)); + mpc_set (op_loc, op, MPC_RNDNN); + } + else + op_loc [0] = op [0]; + + if (rop_sin != NULL) { + /* sin(+-O +i*y) = +-0 +i*sinh(y) */ + mpfr_set (mpc_realref(rop_sin), mpc_realref(op_loc), GMP_RNDN); + inex_sin_im = mpfr_sinh (mpc_imagref(rop_sin), mpc_imagref(op_loc), MPC_RND_IM(rnd_sin)); + } + + if (rop_cos != NULL) { + /* cos(-0 - i * y) = cos(+0 + i * y) = cosh(y) - i * 0, + cos(-0 + i * y) = cos(+0 - i * y) = cosh(y) + i * 0, + where y > 0 */ + inex_cos_re = mpfr_cosh (mpc_realref (rop_cos), mpc_imagref (op_loc), MPC_RND_RE (rnd_cos)); + + mpfr_set_ui (mpc_imagref (rop_cos), 0ul, MPC_RND_IM (rnd_cos)); + if (mpfr_signbit (mpc_realref (op_loc)) == mpfr_signbit (mpc_imagref (op_loc))) + MPFR_CHANGE_SIGN (mpc_imagref (rop_cos)); + } + + if (overlap) + mpc_clear (op_loc); + + return MPC_INEX12 (MPC_INEX (0, inex_sin_im), MPC_INEX (inex_cos_re, 0)); +} + + +int +mpc_sin_cos (mpc_ptr rop_sin, mpc_ptr rop_cos, mpc_srcptr op, + mpc_rnd_t rnd_sin, mpc_rnd_t rnd_cos) + /* Feature not documented in the texinfo file: One of rop_sin or + rop_cos may be NULL, in which case it is not computed, and the + corresponding ternary inexact value is set to 0 (exact). */ +{ + if (!mpc_fin_p (op)) + return mpc_sin_cos_nonfinite (rop_sin, rop_cos, op, rnd_sin, rnd_cos); + else if (mpfr_zero_p (mpc_imagref (op))) + return mpc_sin_cos_real (rop_sin, rop_cos, op, rnd_sin, rnd_cos); + else if (mpfr_zero_p (mpc_realref (op))) + return mpc_sin_cos_imag (rop_sin, rop_cos, op, rnd_sin, rnd_cos); + else { + /* let op = a + i*b, then sin(op) = sin(a)*cosh(b) + i*cos(a)*sinh(b) + and cos(op) = cos(a)*cosh(b) - i*sin(a)*sinh(b). + + For Re(sin(op)) (and analogously, the other parts), we use the + following algorithm, with rounding to nearest for all operations + and working precision w: + + (1) x = o(sin(a)) + (2) y = o(cosh(b)) + (3) r = o(x*y) + then the error on r is at most 4 ulps, since we can write + r = sin(a)*cosh(b)*(1+t)^3 with |t| <= 2^(-w), + thus for w >= 2, r = sin(a)*cosh(b)*(1+4*t) with |t| <= 2^(-w), + thus the relative error is bounded by 4*2^(-w) <= 4*ulp(r). + */ + mpfr_t s, c, sh, ch, sch, csh; + mpfr_prec_t prec; + int ok; + int inex_re, inex_im, inex_sin, inex_cos; + + prec = 2; + if (rop_sin != NULL) + prec = MPC_MAX (prec, MPC_MAX_PREC (rop_sin)); + if (rop_cos != NULL) + prec = MPC_MAX (prec, MPC_MAX_PREC (rop_cos)); + + mpfr_init2 (s, 2); + mpfr_init2 (c, 2); + mpfr_init2 (sh, 2); + mpfr_init2 (ch, 2); + mpfr_init2 (sch, 2); + mpfr_init2 (csh, 2); + + do { + ok = 1; + prec += mpc_ceil_log2 (prec) + 5; + + mpfr_set_prec (s, prec); + mpfr_set_prec (c, prec); + mpfr_set_prec (sh, prec); + mpfr_set_prec (ch, prec); + mpfr_set_prec (sch, prec); + mpfr_set_prec (csh, prec); + + mpfr_sin_cos (s, c, mpc_realref(op), GMP_RNDN); + mpfr_sinh_cosh (sh, ch, mpc_imagref(op), GMP_RNDN); + + if (rop_sin != NULL) { + /* real part of sine */ + mpfr_mul (sch, s, ch, GMP_RNDN); + ok = (!mpfr_number_p (sch)) + || mpfr_can_round (sch, prec - 2, GMP_RNDN, GMP_RNDZ, + MPC_PREC_RE (rop_sin) + + (MPC_RND_RE (rnd_sin) == GMP_RNDN)); + + if (ok) { + /* imaginary part of sine */ + mpfr_mul (csh, c, sh, GMP_RNDN); + ok = (!mpfr_number_p (csh)) + || mpfr_can_round (csh, prec - 2, GMP_RNDN, GMP_RNDZ, + MPC_PREC_IM (rop_sin) + + (MPC_RND_IM (rnd_sin) == GMP_RNDN)); + } + } + + if (rop_cos != NULL && ok) { + /* real part of cosine */ + mpfr_mul (c, c, ch, GMP_RNDN); + ok = (!mpfr_number_p (c)) + || mpfr_can_round (c, prec - 2, GMP_RNDN, GMP_RNDZ, + MPC_PREC_RE (rop_cos) + + (MPC_RND_RE (rnd_cos) == GMP_RNDN)); + + if (ok) { + /* imaginary part of cosine */ + mpfr_mul (s, s, sh, GMP_RNDN); + mpfr_neg (s, s, GMP_RNDN); + ok = (!mpfr_number_p (s)) + || mpfr_can_round (s, prec - 2, GMP_RNDN, GMP_RNDZ, + MPC_PREC_IM (rop_cos) + + (MPC_RND_IM (rnd_cos) == GMP_RNDN)); + } + } + } while (ok == 0); + + if (rop_sin != NULL) { + inex_re = mpfr_set (mpc_realref (rop_sin), sch, MPC_RND_RE (rnd_sin)); + if (mpfr_inf_p (sch)) + inex_re = mpfr_sgn (sch); + inex_im = mpfr_set (mpc_imagref (rop_sin), csh, MPC_RND_IM (rnd_sin)); + if (mpfr_inf_p (csh)) + inex_im = mpfr_sgn (csh); + inex_sin = MPC_INEX (inex_re, inex_im); + } + else + inex_sin = MPC_INEX (0,0); /* return exact if not computed */ + + if (rop_cos != NULL) { + inex_re = mpfr_set (mpc_realref (rop_cos), c, MPC_RND_RE (rnd_cos)); + if (mpfr_inf_p (c)) + inex_re = mpfr_sgn (c); + inex_im = mpfr_set (mpc_imagref (rop_cos), s, MPC_RND_IM (rnd_cos)); + if (mpfr_inf_p (s)) + inex_im = mpfr_sgn (s); + inex_cos = MPC_INEX (inex_re, inex_im); + } + else + inex_cos = MPC_INEX (0,0); /* return exact if not computed */ + + mpfr_clear (s); + mpfr_clear (c); + mpfr_clear (sh); + mpfr_clear (ch); + mpfr_clear (sch); + mpfr_clear (csh); + + return (MPC_INEX12 (inex_sin, inex_cos)); + } +} |