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Diffstat (limited to 'src/atan.c')
| -rw-r--r-- | src/atan.c | 437 |
1 files changed, 437 insertions, 0 deletions
diff --git a/src/atan.c b/src/atan.c new file mode 100644 index 0000000..45e8e75 --- /dev/null +++ b/src/atan.c @@ -0,0 +1,437 @@ +/* mpfr_atan -- arc-tangent of a floating-point number + +Copyright 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. +Contributed by the AriC and Caramel 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 +http://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" + +/* If x = p/2^r, put in y an approximation of atan(x)/x using 2^m terms + for the series expansion, with an error of at most 1 ulp. + Assumes |x| < 1. + + If X=x^2, we want 1 - X/3 + X^2/5 - ... + (-1)^k*X^k/(2k+1) + ... + + Assume p is non-zero. + + When we sum terms up to x^k/(2k+1), the denominator Q[0] is + 3*5*7*...*(2k+1) ~ (2k/e)^k. +*/ +static void +mpfr_atan_aux (mpfr_ptr y, mpz_ptr p, long r, int m, mpz_t *tab) +{ + mpz_t *S, *Q, *ptoj; + unsigned long n, i, k, j, l; + mpfr_exp_t diff, expo; + int im, done; + mpfr_prec_t mult, *accu, *log2_nb_terms; + mpfr_prec_t precy = MPFR_PREC(y); + + MPFR_ASSERTD(mpz_cmp_ui (p, 0) != 0); + + accu = (mpfr_prec_t*) (*__gmp_allocate_func) ((2 * m + 2) * sizeof (mpfr_prec_t)); + log2_nb_terms = accu + m + 1; + + /* Set Tables */ + S = tab; /* S */ + ptoj = S + 1*(m+1); /* p^2^j Precomputed table */ + Q = S + 2*(m+1); /* Product of Odd integer table */ + + /* From p to p^2, and r to 2r */ + mpz_mul (p, p, p); + MPFR_ASSERTD (2 * r > r); + r = 2 * r; + + /* Normalize p */ + n = mpz_scan1 (p, 0); + mpz_tdiv_q_2exp (p, p, n); /* exact */ + MPFR_ASSERTD (r > n); + r -= n; + /* since |p/2^r| < 1, and p is a non-zero integer, necessarily r > 0 */ + + MPFR_ASSERTD (mpz_sgn (p) > 0); + MPFR_ASSERTD (m > 0); + + /* check if p=1 (special case) */ + l = 0; + /* + We compute by binary splitting, with X = x^2 = p/2^r: + P(a,b) = p if a+1=b, P(a,c)*P(c,b) otherwise + Q(a,b) = (2a+1)*2^r if a+1=b [except Q(0,1)=1], Q(a,c)*Q(c,b) otherwise + S(a,b) = p*(2a+1) if a+1=b, Q(c,b)*S(a,c)+Q(a,c)*P(a,c)*S(c,b) otherwise + Then atan(x)/x ~ S(0,i)/Q(0,i) for i so that (p/2^r)^i/i is small enough. + The factor 2^(r*(b-a)) in Q(a,b) is implicit, thus we have to take it + into account when we compute with Q. + */ + accu[0] = 0; /* accu[k] = Mult[0] + ... + Mult[k], where Mult[j] is the + number of bits of the corresponding term S[j]/Q[j] */ + if (mpz_cmp_ui (p, 1) != 0) + { + /* p <> 1: precompute ptoj table */ + mpz_set (ptoj[0], p); + for (im = 1 ; im <= m ; im ++) + mpz_mul (ptoj[im], ptoj[im - 1], ptoj[im - 1]); + /* main loop */ + n = 1UL << m; + /* the ith term being X^i/(2i+1) with X=p/2^r, we can stop when + p^i/2^(r*i) < 2^(-precy), i.e. r*i > precy + log2(p^i) */ + for (i = k = done = 0; (i < n) && (done == 0); i += 2, k ++) + { + /* initialize both S[k],Q[k] and S[k+1],Q[k+1] */ + mpz_set_ui (Q[k+1], 2 * i + 3); /* Q(i+1,i+2) */ + mpz_mul_ui (S[k+1], p, 2 * i + 1); /* S(i+1,i+2) */ + mpz_mul_2exp (S[k], Q[k+1], r); + mpz_sub (S[k], S[k], S[k+1]); /* S(i,i+2) */ + mpz_mul_ui (Q[k], Q[k+1], 2 * i + 1); /* Q(i,i+2) */ + log2_nb_terms[k] = 1; /* S[k]/Q[k] corresponds to 2 terms */ + for (j = (i + 2) >> 1, l = 1; (j & 1) == 0; l ++, j >>= 1, k --) + { + /* invariant: S[k-1]/Q[k-1] and S[k]/Q[k] correspond + to 2^l terms each. We combine them into S[k-1]/Q[k-1] */ + MPFR_ASSERTD (k > 0); + mpz_mul (S[k], S[k], Q[k-1]); + mpz_mul (S[k], S[k], ptoj[l]); + mpz_mul (S[k-1], S[k-1], Q[k]); + mpz_mul_2exp (S[k-1], S[k-1], r << l); + mpz_add (S[k-1], S[k-1], S[k]); + mpz_mul (Q[k-1], Q[k-1], Q[k]); + log2_nb_terms[k-1] = l + 1; + /* now S[k-1]/Q[k-1] corresponds to 2^(l+1) terms */ + MPFR_MPZ_SIZEINBASE2(mult, ptoj[l+1]); + /* FIXME: precompute bits(ptoj[l+1]) outside the loop? */ + mult = (r << (l + 1)) - mult - 1; + accu[k-1] = (k == 1) ? mult : accu[k-2] + mult; + if (accu[k-1] > precy) + done = 1; + } + } + } + else /* special case p=1: the ith term being X^i/(2i+1) with X=1/2^r, + we can stop when r*i > precy i.e. i > precy/r */ + { + n = 1UL << m; + for (i = k = 0; (i < n) && (i <= precy / r); i += 2, k ++) + { + mpz_set_ui (Q[k + 1], 2 * i + 3); + mpz_mul_2exp (S[k], Q[k+1], r); + mpz_sub_ui (S[k], S[k], 1 + 2 * i); + mpz_mul_ui (Q[k], Q[k + 1], 1 + 2 * i); + log2_nb_terms[k] = 1; /* S[k]/Q[k] corresponds to 2 terms */ + for (j = (i + 2) >> 1, l = 1; (j & 1) == 0; l++, j >>= 1, k --) + { + MPFR_ASSERTD (k > 0); + mpz_mul (S[k], S[k], Q[k-1]); + mpz_mul (S[k-1], S[k-1], Q[k]); + mpz_mul_2exp (S[k-1], S[k-1], r << l); + mpz_add (S[k-1], S[k-1], S[k]); + mpz_mul (Q[k-1], Q[k-1], Q[k]); + log2_nb_terms[k-1] = l + 1; + } + } + } + + /* we need to combine S[0]/Q[0]...S[k-1]/Q[k-1] */ + l = 0; /* number of terms accumulated in S[k]/Q[k] */ + while (k > 1) + { + k --; + /* combine S[k-1]/Q[k-1] and S[k]/Q[k] */ + j = log2_nb_terms[k-1]; + mpz_mul (S[k], S[k], Q[k-1]); + if (mpz_cmp_ui (p, 1) != 0) + mpz_mul (S[k], S[k], ptoj[j]); + mpz_mul (S[k-1], S[k-1], Q[k]); + l += 1 << log2_nb_terms[k]; + mpz_mul_2exp (S[k-1], S[k-1], r * l); + mpz_add (S[k-1], S[k-1], S[k]); + mpz_mul (Q[k-1], Q[k-1], Q[k]); + } + (*__gmp_free_func) (accu, (2 * m + 2) * sizeof (mpfr_prec_t)); + + MPFR_MPZ_SIZEINBASE2 (diff, S[0]); + diff -= 2 * precy; + expo = diff; + if (diff >= 0) + mpz_tdiv_q_2exp (S[0], S[0], diff); + else + mpz_mul_2exp (S[0], S[0], -diff); + + MPFR_MPZ_SIZEINBASE2 (diff, Q[0]); + diff -= precy; + expo -= diff; + if (diff >= 0) + mpz_tdiv_q_2exp (Q[0], Q[0], diff); + else + mpz_mul_2exp (Q[0], Q[0], -diff); + + mpz_tdiv_q (S[0], S[0], Q[0]); + mpfr_set_z (y, S[0], MPFR_RNDD); + MPFR_SET_EXP (y, MPFR_EXP(y) + expo - r * (i - 1)); +} + +int +mpfr_atan (mpfr_ptr atan, mpfr_srcptr x, mpfr_rnd_t rnd_mode) +{ + mpfr_t xp, arctgt, sk, tmp, tmp2; + mpz_t ukz; + mpz_t *tabz; + mpfr_exp_t exptol; + mpfr_prec_t prec, realprec, est_lost, lost; + unsigned long twopoweri, log2p, red; + int comparaison, inexact; + int i, n0, oldn0; + MPFR_GROUP_DECL (group); + MPFR_SAVE_EXPO_DECL (expo); + MPFR_ZIV_DECL (loop); + + MPFR_LOG_FUNC + (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (x), mpfr_log_prec, x, rnd_mode), + ("atan[%Pu]=%.*Rg inexact=%d", + mpfr_get_prec (atan), mpfr_log_prec, atan, inexact)); + + /* Singular cases */ + if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (x))) + { + if (MPFR_IS_NAN (x)) + { + MPFR_SET_NAN (atan); + MPFR_RET_NAN; + } + else if (MPFR_IS_INF (x)) + { + MPFR_SAVE_EXPO_MARK (expo); + if (MPFR_IS_POS (x)) /* arctan(+inf) = Pi/2 */ + inexact = mpfr_const_pi (atan, rnd_mode); + else /* arctan(-inf) = -Pi/2 */ + { + inexact = -mpfr_const_pi (atan, + MPFR_INVERT_RND (rnd_mode)); + MPFR_CHANGE_SIGN (atan); + } + mpfr_div_2ui (atan, atan, 1, rnd_mode); /* exact (no exceptions) */ + MPFR_SAVE_EXPO_FREE (expo); + return mpfr_check_range (atan, inexact, rnd_mode); + } + else /* x is necessarily 0 */ + { + MPFR_ASSERTD (MPFR_IS_ZERO (x)); + MPFR_SET_ZERO (atan); + MPFR_SET_SAME_SIGN (atan, x); + MPFR_RET (0); + } + } + + /* atan(x) = x - x^3/3 + x^5/5... + so the error is < 2^(3*EXP(x)-1) + so `EXP(x)-(3*EXP(x)-1)` = -2*EXP(x)+1 */ + MPFR_FAST_COMPUTE_IF_SMALL_INPUT (atan, x, -2 * MPFR_GET_EXP (x), 1, 0, + rnd_mode, {}); + + /* Set x_p=|x| */ + MPFR_TMP_INIT_ABS (xp, x); + + MPFR_SAVE_EXPO_MARK (expo); + + /* Other simple case arctan(-+1)=-+pi/4 */ + comparaison = mpfr_cmp_ui (xp, 1); + if (MPFR_UNLIKELY (comparaison == 0)) + { + int neg = MPFR_IS_NEG (x); + inexact = mpfr_const_pi (atan, MPFR_IS_POS (x) ? rnd_mode + : MPFR_INVERT_RND (rnd_mode)); + if (neg) + { + inexact = -inexact; + MPFR_CHANGE_SIGN (atan); + } + mpfr_div_2ui (atan, atan, 2, rnd_mode); /* exact (no exceptions) */ + MPFR_SAVE_EXPO_FREE (expo); + return mpfr_check_range (atan, inexact, rnd_mode); + } + + realprec = MPFR_PREC (atan) + MPFR_INT_CEIL_LOG2 (MPFR_PREC (atan)) + 4; + prec = realprec + GMP_NUMB_BITS; + + /* Initialisation */ + mpz_init (ukz); + MPFR_GROUP_INIT_4 (group, prec, sk, tmp, tmp2, arctgt); + oldn0 = 0; + tabz = (mpz_t *) 0; + + MPFR_ZIV_INIT (loop, prec); + for (;;) + { + /* First, if |x| < 1, we need to have more prec to be able to round (sup) + n0 = ceil(log(prec_requested + 2 + 1+ln(2.4)/ln(2))/log(2)) */ + mpfr_prec_t sup; + sup = MPFR_GET_EXP (xp) < 0 ? 2 - MPFR_GET_EXP (xp) : 1; /* sup >= 1 */ + + n0 = MPFR_INT_CEIL_LOG2 ((realprec + sup) + 3); + /* since realprec >= 4, n0 >= ceil(log2(8)) >= 3, thus 3*n0 > 2 */ + prec = (realprec + sup) + 1 + MPFR_INT_CEIL_LOG2 (3*n0-2); + + /* the number of lost bits due to argument reduction is + 9 - 2 * EXP(sk), which we estimate by 9 + 2*ceil(log2(p)) + since we manage that sk < 1/p */ + if (MPFR_PREC (atan) > 100) + { + log2p = MPFR_INT_CEIL_LOG2(prec) / 2 - 3; + est_lost = 9 + 2 * log2p; + prec += est_lost; + } + else + log2p = est_lost = 0; /* don't reduce the argument */ + + /* Initialisation */ + MPFR_GROUP_REPREC_4 (group, prec, sk, tmp, tmp2, arctgt); + if (MPFR_LIKELY (oldn0 == 0)) + { + oldn0 = 3 * (n0 + 1); + tabz = (mpz_t *) (*__gmp_allocate_func) (oldn0 * sizeof (mpz_t)); + for (i = 0; i < oldn0; i++) + mpz_init (tabz[i]); + } + else if (MPFR_UNLIKELY (oldn0 < 3 * (n0 + 1))) + { + tabz = (mpz_t *) (*__gmp_reallocate_func) + (tabz, oldn0 * sizeof (mpz_t), 3 * (n0 + 1)*sizeof (mpz_t)); + for (i = oldn0; i < 3 * (n0 + 1); i++) + mpz_init (tabz[i]); + oldn0 = 3 * (n0 + 1); + } + + /* The mpfr_ui_div below mustn't underflow. This is guaranteed by + MPFR_SAVE_EXPO_MARK, but let's check that for maintainability. */ + MPFR_ASSERTD (__gmpfr_emax <= 1 - __gmpfr_emin); + + if (comparaison > 0) /* use atan(xp) = Pi/2 - atan(1/xp) */ + mpfr_ui_div (sk, 1, xp, MPFR_RNDN); + else + mpfr_set (sk, xp, MPFR_RNDN); + + /* now 0 < sk <= 1 */ + + /* Argument reduction: atan(x) = 2 atan((sqrt(1+x^2)-1)/x). + We want |sk| < k/sqrt(p) where p is the target precision. */ + lost = 0; + for (red = 0; MPFR_GET_EXP(sk) > - (mpfr_exp_t) log2p; red ++) + { + lost = 9 - 2 * MPFR_EXP(sk); + mpfr_mul (tmp, sk, sk, MPFR_RNDN); + mpfr_add_ui (tmp, tmp, 1, MPFR_RNDN); + mpfr_sqrt (tmp, tmp, MPFR_RNDN); + mpfr_sub_ui (tmp, tmp, 1, MPFR_RNDN); + if (red == 0 && comparaison > 0) + /* use xp = 1/sk */ + mpfr_mul (sk, tmp, xp, MPFR_RNDN); + else + mpfr_div (sk, tmp, sk, MPFR_RNDN); + } + + /* we started from x0 = 1/|x| if |x| > 1, and |x| otherwise, thus + we had x0 = min(|x|, 1/|x|) <= 1, and applied 'red' times the + argument reduction x -> (sqrt(1+x^2)-1)/x, which keeps 0 < x < 1, + thus 0 < sk <= 1, and sk=1 can occur only if red=0 */ + + /* If sk=1, then if |x| < 1, we have 1 - 2^(-prec-1) <= |x| < 1, + or if |x| > 1, we have 1 - 2^(-prec-1) <= 1/|x| < 1, thus in all + cases ||x| - 1| <= 2^(-prec), from which it follows + |atan|x| - Pi/4| <= 2^(-prec), given the Taylor expansion + atan(1+x) = Pi/4 + x/2 - x^2/4 + ... + Since Pi/4 = 0.785..., the error is at most one ulp. + */ + if (MPFR_UNLIKELY(mpfr_cmp_ui (sk, 1) == 0)) + { + mpfr_const_pi (arctgt, MPFR_RNDN); /* 1/2 ulp extra error */ + mpfr_div_2ui (arctgt, arctgt, 2, MPFR_RNDN); /* exact */ + realprec = prec - 2; + goto can_round; + } + + /* Assignation */ + MPFR_SET_ZERO (arctgt); + twopoweri = 1 << 0; + MPFR_ASSERTD (n0 >= 4); + for (i = 0 ; i < n0; i++) + { + if (MPFR_UNLIKELY (MPFR_IS_ZERO (sk))) + break; + /* Calculation of trunc(tmp) --> mpz */ + mpfr_mul_2ui (tmp, sk, twopoweri, MPFR_RNDN); + mpfr_trunc (tmp, tmp); + if (!MPFR_IS_ZERO (tmp)) + { + /* tmp = ukz*2^exptol */ + exptol = mpfr_get_z_2exp (ukz, tmp); + /* since the s_k are decreasing (see algorithms.tex), + and s_0 = min(|x|, 1/|x|) < 1, we have sk < 1, + thus exptol < 0 */ + MPFR_ASSERTD (exptol < 0); + mpz_tdiv_q_2exp (ukz, ukz, (unsigned long int) (-exptol)); + /* since tmp is a non-zero integer, and tmp = ukzold*2^exptol, + we now have ukz = tmp, thus ukz is non-zero */ + /* Calculation of arctan(Ak) */ + mpfr_set_z (tmp, ukz, MPFR_RNDN); + mpfr_div_2ui (tmp, tmp, twopoweri, MPFR_RNDN); + mpfr_atan_aux (tmp2, ukz, twopoweri, n0 - i, tabz); + mpfr_mul (tmp2, tmp2, tmp, MPFR_RNDN); + /* Addition */ + mpfr_add (arctgt, arctgt, tmp2, MPFR_RNDN); + /* Next iteration */ + mpfr_sub (tmp2, sk, tmp, MPFR_RNDN); + mpfr_mul (sk, sk, tmp, MPFR_RNDN); + mpfr_add_ui (sk, sk, 1, MPFR_RNDN); + mpfr_div (sk, tmp2, sk, MPFR_RNDN); + } + twopoweri <<= 1; + } + /* Add last step (Arctan(sk) ~= sk */ + mpfr_add (arctgt, arctgt, sk, MPFR_RNDN); + + /* argument reduction */ + mpfr_mul_2exp (arctgt, arctgt, red, MPFR_RNDN); + + if (comparaison > 0) + { /* atan(x) = Pi/2-atan(1/x) for x > 0 */ + mpfr_const_pi (tmp, MPFR_RNDN); + mpfr_div_2ui (tmp, tmp, 1, MPFR_RNDN); + mpfr_sub (arctgt, tmp, arctgt, MPFR_RNDN); + } + MPFR_SET_POS (arctgt); + + can_round: + if (MPFR_LIKELY (MPFR_CAN_ROUND (arctgt, realprec + est_lost - lost, + MPFR_PREC (atan), rnd_mode))) + break; + MPFR_ZIV_NEXT (loop, realprec); + } + MPFR_ZIV_FREE (loop); + + inexact = mpfr_set4 (atan, arctgt, rnd_mode, MPFR_SIGN (x)); + + for (i = 0 ; i < oldn0 ; i++) + mpz_clear (tabz[i]); + mpz_clear (ukz); + (*__gmp_free_func) (tabz, oldn0 * sizeof (mpz_t)); + MPFR_GROUP_CLEAR (group); + + MPFR_SAVE_EXPO_FREE (expo); + return mpfr_check_range (atan, inexact, rnd_mode); +} |