/* mpc_norm -- Square of the norm of a complex number. Copyright (C) 2002, 2005, 2008, 2009, 2010, 2011, 2012 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 /* for MPC_ASSERT */ #include "mpc-impl.h" /* a <- norm(b) = b * conj(b) (the rounding mode is mpfr_rnd_t here since we return an mpfr number) */ int mpc_norm (mpfr_ptr a, mpc_srcptr b, mpfr_rnd_t rnd) { int inexact; int saved_underflow, saved_overflow; /* handling of special values; consistent with abs in that norm = abs^2; so norm (+-inf, xxx) = norm (xxx, +-inf) = +inf */ if (!mpc_fin_p (b)) return mpc_abs (a, b, rnd); else if (mpfr_zero_p (mpc_realref (b))) { if (mpfr_zero_p (mpc_imagref (b))) return mpfr_set_ui (a, 0, rnd); /* +0 */ else return mpfr_sqr (a, mpc_imagref (b), rnd); } else if (mpfr_zero_p (mpc_imagref (b))) return mpfr_sqr (a, mpc_realref (b), rnd); /* Re(b) <> 0 */ else /* everything finite and non-zero */ { mpfr_t u, v, res; mpfr_prec_t prec, prec_u, prec_v; int loops; const int max_loops = 2; /* switch to exact squarings when loops==max_loops */ prec = mpfr_get_prec (a); mpfr_init (u); mpfr_init (v); mpfr_init (res); /* save the underflow or overflow flags from MPFR */ saved_underflow = mpfr_underflow_p (); saved_overflow = mpfr_overflow_p (); loops = 0; mpfr_clear_underflow (); mpfr_clear_overflow (); do { loops++; prec += mpc_ceil_log2 (prec) + 3; if (loops >= max_loops) { prec_u = 2 * MPC_PREC_RE (b); prec_v = 2 * MPC_PREC_IM (b); } else { prec_u = MPC_MIN (prec, 2 * MPC_PREC_RE (b)); prec_v = MPC_MIN (prec, 2 * MPC_PREC_IM (b)); } mpfr_set_prec (u, prec_u); mpfr_set_prec (v, prec_v); inexact = mpfr_sqr (u, mpc_realref(b), MPFR_RNDD); /* err <= 1 ulp in prec */ inexact |= mpfr_sqr (v, mpc_imagref(b), MPFR_RNDD); /* err <= 1 ulp in prec */ /* If loops = max_loops, inexact should be 0 here, except in case of underflow or overflow. If loops < max_loops and inexact is zero, we can exit the while-loop since it only remains to add u and v into a. */ if (inexact) { mpfr_set_prec (res, prec); mpfr_add (res, u, v, MPFR_RNDD); /* err <= 3 ulp in prec */ } } while (loops < max_loops && inexact != 0 && !mpfr_can_round (res, prec - 2, MPFR_RNDD, MPFR_RNDU, mpfr_get_prec (a) + (rnd == MPFR_RNDN))); if (!inexact) /* squarings were exact, neither underflow nor overflow */ inexact = mpfr_add (a, u, v, rnd); /* if there was an overflow in Re(b)^2 or Im(b)^2 or their sum, since the norm is larger, there is an overflow for the norm */ else if (mpfr_overflow_p ()) { /* replace by "correctly rounded overflow" */ mpfr_set_ui (a, 1ul, MPFR_RNDN); inexact = mpfr_mul_2ui (a, a, mpfr_get_emax (), rnd); } else if (mpfr_underflow_p ()) { /* necessarily one of the squarings did underflow (otherwise their sum could not underflow), thus one of u, v is zero. */ mpfr_exp_t emin = mpfr_get_emin (); /* Now either both u and v are zero, or u is zero and v exact, or v is zero and u exact. In the latter case, Im(b)^2 < 2^(emin-1). If ulp(u) >= 2^(emin+1) and norm(b) is not exactly representable at the target precision, then rounding u+Im(b)^2 is equivalent to rounding u+2^(emin-1). For instance, if exp(u)>0 and the target precision is smaller than about |emin|, the norm is not representable. To make the scaling in the "else" case work without underflow, we test whether exp(u) is larger than a small negative number instead. The second case is handled analogously. */ if (!mpfr_zero_p (u) && mpfr_get_exp (u) - 2 * (mpfr_exp_t) prec_u > emin && mpfr_get_exp (u) > -10) { mpfr_set_prec (v, MPFR_PREC_MIN); mpfr_set_ui_2exp (v, 1, emin - 1, MPFR_RNDZ); inexact = mpfr_add (a, u, v, rnd); } else if (!mpfr_zero_p (v) && mpfr_get_exp (v) - 2 * (mpfr_exp_t) prec_v > emin && mpfr_get_exp (v) > -10) { mpfr_set_prec (u, MPFR_PREC_MIN); mpfr_set_ui_2exp (u, 1, emin - 1, MPFR_RNDZ); inexact = mpfr_add (a, u, v, rnd); } else { unsigned long int scale, exp_re, exp_im; int inex_underflow; /* scale the input to an average exponent close to 0 */ exp_re = (unsigned long int) (-mpfr_get_exp (mpc_realref (b))); exp_im = (unsigned long int) (-mpfr_get_exp (mpc_imagref (b))); scale = exp_re / 2 + exp_im / 2 + (exp_re % 2 + exp_im % 2) / 2; /* (exp_re + exp_im) / 2, computed in a way avoiding integer overflow */ if (mpfr_zero_p (u)) { /* recompute the scaled value exactly */ mpfr_mul_2ui (u, mpc_realref (b), scale, MPFR_RNDN); mpfr_sqr (u, u, MPFR_RNDN); } else /* just scale */ mpfr_mul_2ui (u, u, 2*scale, MPFR_RNDN); if (mpfr_zero_p (v)) { mpfr_mul_2ui (v, mpc_imagref (b), scale, MPFR_RNDN); mpfr_sqr (v, v, MPFR_RNDN); } else mpfr_mul_2ui (v, v, 2*scale, MPFR_RNDN); inexact = mpfr_add (a, u, v, rnd); mpfr_clear_underflow (); inex_underflow = mpfr_div_2ui (a, a, 2*scale, rnd); if (mpfr_underflow_p ()) inexact = inex_underflow; } } else /* no problems, ternary value due to mpfr_can_round trick */ inexact = mpfr_set (a, res, rnd); /* restore underflow and overflow flags from MPFR */ if (saved_underflow) mpfr_set_underflow (); if (saved_overflow) mpfr_set_overflow (); mpfr_clear (u); mpfr_clear (v); mpfr_clear (res); } return inexact; }