From 02b4fa78e51ddfd9ba15ba481f796083e8633a30 Mon Sep 17 00:00:00 2001 From: zimmerma Date: Tue, 29 Sep 2009 11:37:17 +0000 Subject: [atan.c] fixed the general case git-svn-id: svn://scm.gforge.inria.fr/svn/mpc/branches/feature-inverse-trigo@679 211d60ee-9f03-0410-a15a-8952a2c7a4e4 --- src/atan.c | 66 +++++++++++++++++++++++++++++++++----------------------------- 1 file changed, 35 insertions(+), 31 deletions(-) diff --git a/src/atan.c b/src/atan.c index 96da190..6a2ddbd 100644 --- a/src/atan.c +++ b/src/atan.c @@ -1,6 +1,6 @@ /* mpc_atan -- arctangent of a complex number. -Copyright (C) 2009 Philippe Th\'eveny +Copyright (C) 2009 Philippe Th\'eveny, Paul Zimmermann This file is part of the MPC Library. @@ -142,7 +142,7 @@ mpc_atan (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd) { /* atan(+0+iy) = +pi/2 +i*atanh(1/y), if |y| > 1 atan(-0+iy) = -pi/2 +i*atanh(1/y), if |y| > 1 */ - mp_rnd_t rnd_im; + mp_rnd_t rnd_im, rnd_away; mpfr_t y; mp_prec_t p, p_im; int ok; @@ -172,8 +172,9 @@ mpc_atan (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd) /* atanh cannot underflow: |atanh(x)| > |x| for |x| < 1 */ inex_im |= mpfr_atanh (y, y, GMP_RNDZ); + rnd_away = mpfr_sgn (y) > 0 ? GMP_RNDU : GMP_RNDD; ok = inex_im == 0 - || mpfr_can_round (y, p-2, GMP_RNDZ, rnd_im, + || mpfr_can_round (y, p - 2, GMP_RNDZ, rnd_away, p_im + (rnd_im == GMP_RNDN)); } while (ok == 0); @@ -209,13 +210,15 @@ mpc_atan (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd) b = o(atan2(x,a)) error(b) < [1+2^{3+Exp(x)-Exp(a)-Exp(b)}] ulp(b) = kb ulp(b) c = o(1+y) error(c) < 1 ulp(c) - d = o(atan2(-x,a)) error(d) < [1+2^{3+Exp(x)-Exp(c)-Exp(d)}] ulp(d) + d = o(atan2(-x,c)) error(d) < [1+2^{3+Exp(x)-Exp(c)-Exp(d)}] ulp(d) = kd ulp(d) e = o(b - d) error(e) < [1 + kb*2^{Exp(b}-Exp(e)} + kd*2^{Exp(d)-Exp(e)}] ulp(e) error(e) < [1 + 2^{4+Exp(x)-Exp(a)-Exp(e)} + 2^{4+Exp(x)-Exp(c)-Exp(e)}] ulp(e) because |atan(u)| < |u| + < [1 + 2^{5+Exp(x)-min(Exp(a),Exp(c)) + -Exp(e)}] ulp(e) f = e/2 exact */ @@ -223,12 +226,8 @@ mpc_atan (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd) /* p: working precision */ p = (op_im_exp > 0 || prec > SAFE_ABS (mp_prec_t, op_im_exp)) ? prec : (prec - op_im_exp); - rnd1 = - mpfr_sgn (MPC_RE (op)) * mpfr_cmp_ui (MPC_IM (op), 1) > 0 ? GMP_RNDZ - : mpfr_cmp_ui (MPC_IM (op), 1) > 0 ? GMP_RNDD : GMP_RNDU; - rnd2 = - mpfr_sgn (MPC_RE (op)) * mpfr_cmp_si (MPC_IM (op), -1) < 0 ? GMP_RNDZ - : mpfr_cmp_si (MPC_IM (op), -1) > 0 ? GMP_RNDU : GMP_RNDD; + rnd1 = mpfr_sgn (MPC_RE (op)) > 0 ? GMP_RNDD : GMP_RNDU; + rnd2 = mpfr_sgn (MPC_RE (op)) < 0 ? GMP_RNDU : GMP_RNDD; do { @@ -237,29 +236,31 @@ mpc_atan (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd) mpfr_set_prec (b, p); mpfr_set_prec (x, p); - /* x = upper bound for atan (x/1-y) */ + /* x = upper bound for atan (x/(1-y)). Since atan is increasing, we + need an upper bound on x/(1-y), i.e., a lower bound on 1-y for + x positive, and an upper bound on 1-y for x negative */ mpfr_ui_sub (a, 1, MPC_IM (op), rnd1); + err = MPFR_EXP (a); /* err = Exp(a) with the notations above */ mpfr_atan2 (x, MPC_RE (op), a, GMP_RNDU); - err = MPFR_EXP (a); - /* b = lower bound for atan (-x/1+y) */ + /* b = lower bound for atan (-x/(1+y)): for x negative, we need a + lower bound on -x/(1+y), i.e., an upper bound on 1+y */ mpfr_add_ui (a, MPC_IM(op), 1, rnd2); + expo = MPFR_EXP (a); /* expo = Exp(d) with the notations above */ mpfr_atan2 (b, minus_op_re, a, GMP_RNDD); - expo = MPFR_EXP (a); mpfr_sub (x, x, b, GMP_RNDU); - if (err < expo) - err = 4 + op_re_exp - err - MPFR_EXP (x); /* FIXME: may overflow */ - else if (err == expo) - err = 5 + op_re_exp - err - MPFR_EXP (x); /* FIXME: may overflow */ - else - err = 4 + op_re_exp - expo - MPFR_EXP (x); /* FIXME: may overflow */ - err = err > 0 ? (err + 1) : 1; + err = err < expo ? err : expo; /* err = min(Exp(a),Exp(c)) */ + err = 5 + op_re_exp - err - MPFR_EXP (x); + /* error is bounded by [1 + 2^err] ulp(e) */ + err = err < 0 ? 1 : err + 1; mpfr_div_2ui (x, x, 1, GMP_RNDU); - ok = mpfr_can_round (x, p - err, GMP_RNDU, GMP_RNDZ, + /* Note: using RND2=RNDD guarantees that if x is exactly representable + on prec + ... bits, mpfr_can_round will return 0 */ + ok = mpfr_can_round (x, p - err, GMP_RNDU, GMP_RNDD, prec + (MPC_RND_RE (rnd) == GMP_RNDN)); } while (ok == 0); @@ -275,12 +276,14 @@ mpc_atan (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd) e = o(log(d)) error(e) < [1 + 7*2^{2-Exp(e)}] ulp(e) = ke ulp(e) f = o(1-y) error(f) < 1 ulp(f) g = o(f^2) error(g) < 5 ulp(g) - h = o(g-e) error(h) < 7 ulp(h) + h = o(c+f) error(h) < 7 ulp(h) i = o(log(h)) error(i) < [1 + 7*2^{2-Exp(i)}] ulp(i) = ki ulp(i) j = o(e-i) error(j) < [1 + ke*2^{Exp(e)-Exp(j)} + ki*2^{Exp(i)-Exp(j)}] ulp(j) error(j) < [1 + 2^{Exp(e)-Exp(j)} + 2^{Exp(i)-Exp(j)} + 7*2^{3-Exp(j)}] ulp(j) + < [1 + 2^{max(Exp(e),Exp(i))-Exp(j)+1} + + 7*2^{3-Exp(j)}] ulp(j) k = j/4 exact */ err = 2; @@ -304,29 +307,30 @@ mpc_atan (mpc_ptr rop, mpc_srcptr op, mpc_rnd_t rnd) /* b = lower bound for log(x^2 + (1-y)^2) */ mpfr_ui_sub (b, 1, MPC_IM (op), GMP_RNDZ); mpfr_sqr (b, b, GMP_RNDU); - mpfr_sqr (y, MPC_RE (op), GMP_RNDZ); + /* mpfr_sqr (y, MPC_RE (op), GMP_RNDZ); */ + mpfr_nextbelow (y); mpfr_add (b, b, y, GMP_RNDZ); mpfr_log (b, b, GMP_RNDZ); mpfr_sub (y, a, b, GMP_RNDU); expo = MPFR_EXP (a) < MPFR_EXP (b) ? MPFR_EXP (b) : MPFR_EXP (a); - if (expo < 6) - expo = 6; - if (MPFR_EXP (y) < expo) - err = 4 + expo - MPFR_EXP (y); /* FIXME: possible overflow */ + expo = expo - MPFR_EXP (y) + 1; + err = 3 - MPFR_EXP (y); + /* error(j) <= [1 + 2^expo + 7*2^err] ulp(j) */ + if (expo <= err) /* error(j) <= [1 + 2^{err+1}] ulp(j) */ + err = (err < 0) ? 1 : err + 2; else - err = 2; + err = (expo < 0) ? 1 : expo + 2; mpfr_div_2ui (y, y, 2, GMP_RNDN); if (mpfr_zero_p (y)) err = 2; /* underflow */ - ok = mpfr_can_round (y, p - err, GMP_RNDU, GMP_RNDZ, + ok = mpfr_can_round (y, p - err, GMP_RNDU, GMP_RNDD, prec + (MPC_RND_IM (rnd) == GMP_RNDN)); } while (ok == 0); - inex = mpc_set_fr_fr (rop, x, y, rnd); mpfr_clears (a, b, x, y, (mpfr_ptr)0); -- cgit v1.2.1