/* mpfr_atanu -- atanu(x) = atan(x)*u/(2*pi) mpfr_atanpi -- atanpi(x) = atan(x)/pi Copyright 2021-2023 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 https://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" /* put in y the correctly rounded value of atan(x)*u/(2*pi) */ int mpfr_atanu (mpfr_ptr y, mpfr_srcptr x, unsigned long u, mpfr_rnd_t rnd_mode) { mpfr_t tmp, pi; mpfr_prec_t prec; mpfr_exp_t expx; int inex; MPFR_SAVE_EXPO_DECL (expo); MPFR_ZIV_DECL (loop); MPFR_LOG_FUNC (("x[%Pd]=%.*Rg u=%lu rnd=%d", mpfr_get_prec(x), mpfr_log_prec, x, u, rnd_mode), ("y[%Pd]=%.*Rg inexact=%d", mpfr_get_prec (y), mpfr_log_prec, y, inex)); /* Singular cases */ if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) { if (MPFR_IS_NAN (x)) { MPFR_SET_NAN (y); MPFR_RET_NAN; } else if (MPFR_IS_INF (x)) { /* atanu(+Inf,u) = u/4, atanu(-Inf,u) = -u/4 */ if (MPFR_IS_POS (x)) return mpfr_set_ui_2exp (y, u, -2, rnd_mode); else { inex = mpfr_set_ui_2exp (y, u, -2, MPFR_INVERT_RND (rnd_mode)); MPFR_CHANGE_SIGN (y); return -inex; } } else /* necessarily x=0 */ { MPFR_ASSERTD(MPFR_IS_ZERO(x)); /* atan(0)=0 with same sign, even when u=0 to ensure atanu(-x,u) = -atanu(x,u) */ MPFR_SET_ZERO (y); MPFR_SET_SAME_SIGN (y, x); MPFR_RET (0); /* exact result */ } } if (u == 0) /* return 0 with sign of x, which is coherent with case x=0 */ { MPFR_SET_ZERO (y); MPFR_SET_SAME_SIGN (y, x); MPFR_RET (0); } if (mpfr_cmpabs_ui (x, 1) == 0) { /* |x| = 1: atanu(1,u) = u/8, atanu(-1,u)=-u/8 */ /* we can't use mpfr_set_si_2exp with -u since -u might not be representable as long */ if (MPFR_SIGN(x) > 0) return mpfr_set_ui_2exp (y, u, -3, rnd_mode); else { inex = mpfr_set_ui_2exp (y, u, -3, MPFR_INVERT_RND(rnd_mode)); MPFR_CHANGE_SIGN(y); return -inex; } } /* For x>=1, we have pi/2-1/x < atan(x) < pi/2, thus u/4-u/(2*pi*x) < atanu(x,u) < u/4, and the relative difference between atanu(x,u) and u/4 is less than 2/(pi*x) < 1/x <= 2^(1-EXP(x)). If the relative difference is <= 2^(-prec-2), then the difference between atanu(x,u) and u/4 is <= 1/4*ulp(u/4) <= 1/2*ulp(RN(u/4)). We also require x >= 2^64, which implies x > 2*u/pi, so that (u-1)/4 < u/4-u/(2*pi*x) < u/4. */ expx = MPFR_GET_EXP(x); if (expx >= 65 && expx - 1 >= MPFR_PREC(y) + 2) { prec = (MPFR_PREC(y) <= 63) ? 65 : MPFR_PREC(y) + 2; /* now prec > 64 and prec > MPFR_PREC(y)+1 */ mpfr_init2 (tmp, prec); /* since expx >= 65, we have emax >= 65, thus u is representable here, and we don't need to work in an extended exponent range */ inex = mpfr_set_ui (tmp, u, MPFR_RNDN); /* exact since prec >= 64 */ MPFR_ASSERTD(inex == 0); mpfr_nextbelow (tmp); /* Since prec >= 65, the last significant bit of tmp is 1, and since prec > PREC(y), tmp is not representable in the target precision, which ensures we will get a correct ternary value below. */ MPFR_ASSERTD(mpfr_min_prec(tmp) > MPFR_PREC(y)); if (MPFR_SIGN(x) < 0) MPFR_CHANGE_SIGN(tmp); /* since prec >= PREC(y)+2, the rounding of tmp is correct */ inex = mpfr_div_2ui (y, tmp, 2, rnd_mode); mpfr_clear (tmp); return inex; } prec = MPFR_PREC (y); MPFR_SAVE_EXPO_MARK (expo); prec += MPFR_INT_CEIL_LOG2(prec) + 10; mpfr_init2 (tmp, prec); mpfr_init2 (pi, prec); MPFR_ZIV_INIT (loop, prec); for (;;) { /* In the error analysis below, each thetax denotes a variable such that |thetax| <= 2^(1-prec) */ mpfr_atan (tmp, x, MPFR_RNDA); /* tmp = atan(x) * (1 + theta1), and tmp cannot be zero since we rounded away from zero, and the case x=0 was treated before */ /* first multiply by u to avoid underflow issues */ mpfr_mul_ui (tmp, tmp, u, MPFR_RNDA); /* tmp = atan(x)*u * (1 + theta2)^2, and |tmp| >= 0.5*2^emin */ mpfr_const_pi (pi, MPFR_RNDZ); /* round toward zero since we we will divide by pi, to round tmp away */ /* pi = Pi * (1 + theta3) */ mpfr_div (tmp, tmp, pi, MPFR_RNDA); /* tmp = atan(x)*u/Pi * (1 + theta4)^4, with |tmp| > 0 */ /* since we rounded away from 0, if we get 0.5*2^emin here, it means |atanu(x,u)| < 0.25*2^emin (pi is not exact) thus we have underflow */ if (MPFR_EXP(tmp) == __gmpfr_emin) { /* mpfr_underflow rounds away for RNDN */ mpfr_clear (tmp); mpfr_clear (pi); MPFR_SAVE_EXPO_FREE (expo); return mpfr_underflow (y, (rnd_mode == MPFR_RNDN) ? MPFR_RNDZ : rnd_mode, 1); } mpfr_div_2ui (tmp, tmp, 1, MPFR_RNDA); /* exact */ /* tmp = atan(x)*u/(2*Pi) * (1 + theta4)^4 */ /* since |(1 + theta4)^4 - 1| <= 8*|theta4| for prec >= 3, the relative error is less than 2^(4-prec) */ MPFR_ASSERTD(!MPFR_IS_ZERO(tmp)); if (MPFR_LIKELY (MPFR_CAN_ROUND (tmp, prec - 4, MPFR_PREC (y), rnd_mode))) break; MPFR_ZIV_NEXT (loop, prec); mpfr_set_prec (tmp, prec); mpfr_set_prec (pi, prec); } MPFR_ZIV_FREE (loop); inex = mpfr_set (y, tmp, rnd_mode); mpfr_clear (tmp); mpfr_clear (pi); MPFR_SAVE_EXPO_FREE (expo); return mpfr_check_range (y, inex, rnd_mode); } int mpfr_atanpi (mpfr_ptr y, mpfr_srcptr x, mpfr_rnd_t rnd_mode) { return mpfr_atanu (y, x, 2, rnd_mode); }