int main() { slong iter; flint_rand_t state; flint_printf("atan_arf_bb...."); fflush(stdout); flint_randinit(state); for (iter = 0; iter < 5000; iter++) { arb_t x, y, z; slong prec, prec2; arb_init(x); arb_init(y); arb_init(z); prec = 2 + n_randint(state, 8000); arb_randtest(x, state, 1 + n_randint(state, 8000), 3); mag_zero(arb_radref(x)); if (n_randint(state, 2)) arb_mul_2exp_si(x, x, 1 + n_randint(state, 40)); else arb_mul_2exp_si(x, x, -n_randint(state, 1.5 * prec)); if (!arf_is_special(arb_midref(x))) prec2 = prec + 100 + 2 * (-ARF_EXP(arb_midref(x))); else prec2 = prec + 100; arb_atan_arf_via_mpfr(y, arb_midref(x), prec2); arb_atan_arf_bb(z, arb_midref(x), prec); if (!arb_contains(z, y)) { flint_printf("FAIL: containment\n\n"); flint_printf("prec = %wd\n\n", prec); flint_printf("x = "); arb_printd(x, 50); flint_printf("\n\n"); flint_printf("y = "); arb_printd(y, 50); flint_printf("\n\n"); flint_printf("z = "); arb_printd(z, 50); flint_printf("\n\n"); abort(); } if (arb_rel_accuracy_bits(z) < prec - 2) { flint_printf("FAIL: poor accuracy\n\n"); flint_printf("prec = %wd, acc = %wd\n\n", prec, arb_rel_accuracy_bits(z)); flint_printf("x = "); arb_printd(x, 50); flint_printf("\n\n"); flint_printf("y = "); arb_printd(y, 50); flint_printf("\n\n"); flint_printf("z = "); arb_printd(z, 50); flint_printf("\n\n"); abort(); } arb_atan_arf_bb(x, arb_midref(x), prec); if (!arb_overlaps(x, z)) { flint_printf("FAIL: aliasing\n\n"); abort(); } arb_clear(x); arb_clear(y); arb_clear(z); } flint_randclear(state); flint_cleanup(); flint_printf("PASS\n"); return EXIT_SUCCESS; }
void arb_atan_arf(arb_t z, const arf_t x, slong prec) { if (arf_is_special(x)) { if (arf_is_zero(x)) { arb_zero(z); } else if (arf_is_pos_inf(x)) { arb_const_pi(z, prec); arb_mul_2exp_si(z, z, -1); } else if (arf_is_neg_inf(x)) { arb_const_pi(z, prec); arb_mul_2exp_si(z, z, -1); arb_neg(z, z); } else { arb_indeterminate(z); } } else if (COEFF_IS_MPZ(*ARF_EXPREF(x))) { if (fmpz_sgn(ARF_EXPREF(x)) < 0) arb_atan_eps(z, x, prec); else arb_atan_inf_eps(z, x, prec); } else { slong exp, wp, wn, N, r; mp_srcptr xp; mp_size_t xn, tn; mp_ptr tmp, w, t, u; mp_limb_t p1, q1bits, p2, q2bits, error, error2; int negative, inexact, reciprocal; TMP_INIT; exp = ARF_EXP(x); negative = ARF_SGNBIT(x); if (exp < -(prec/2) - 2 || exp > prec + 2) { if (exp < 0) arb_atan_eps(z, x, prec); else arb_atan_inf_eps(z, x, prec); return; } ARF_GET_MPN_READONLY(xp, xn, x); /* Special case: +/- 1 (we require |x| != 1 later on) */ if (exp == 1 && xn == 1 && xp[xn-1] == LIMB_TOP) { arb_const_pi(z, prec); arb_mul_2exp_si(z, z, -2); if (negative) arb_neg(z, z); return; } /* Absolute working precision (NOT rounded to a limb multiple) */ wp = prec - FLINT_MIN(0, exp) + 4; /* Too high precision to use table */ if (wp > ARB_ATAN_TAB2_PREC) { arb_atan_arf_bb(z, x, prec); return; } /* Working precision in limbs */ wn = (wp + FLINT_BITS - 1) / FLINT_BITS; TMP_START; tmp = TMP_ALLOC_LIMBS(4 * wn + 3); w = tmp; /* requires wn+1 limbs */ t = w + wn + 1; /* requires wn+1 limbs */ u = t + wn + 1; /* requires 2wn+1 limbs */ /* ----------------------------------------------------------------- */ /* Convert x or 1/x to a fixed-point number |w| < 1 */ /* ----------------------------------------------------------------- */ if (exp <= 0) /* |x| < 1 */ { reciprocal = 0; /* todo: just zero top */ flint_mpn_zero(w, wn); /* w = x as a fixed-point number */ error = _arf_get_integer_mpn(w, xp, xn, exp + wn * FLINT_BITS); } else /* |x| > 1 */ { slong one_exp, one_limbs, one_bits; mp_ptr one; reciprocal = 1; one_exp = xn * FLINT_BITS + wn * FLINT_BITS - exp; flint_mpn_zero(w, wn); /* 1/x becomes zero */ if (one_exp >= FLINT_BITS - 1) { /* w = 1/x */ one_limbs = one_exp / FLINT_BITS; one_bits = one_exp % FLINT_BITS; if (one_limbs + 1 >= xn) { one = TMP_ALLOC_LIMBS(one_limbs + 1); flint_mpn_zero(one, one_limbs); one[one_limbs] = UWORD(1) << one_bits; /* todo: only zero necessary part */ flint_mpn_zero(w, wn); mpn_tdiv_q(w, one, one_limbs + 1, xp, xn); /* Now w must be < 1 since x > 1 and we rounded down; thus w[wn] must be zero */ } } /* todo: moderate powers of two would be exact... */ error = 1; } /* ----------------------------------------------------------------- */ /* Table-based argument reduction */ /* ----------------------------------------------------------------- */ /* choose p such that p/q <= x < (p+1)/q */ if (wp <= ARB_ATAN_TAB1_PREC) q1bits = ARB_ATAN_TAB1_BITS; else q1bits = ARB_ATAN_TAB21_BITS; p1 = w[wn-1] >> (FLINT_BITS - q1bits); /* atan(w) = atan(p/q) + atan(w2) */ /* where w2 = (q*w-p)/(q+p*w) */ if (p1 != 0) { t[wn] = (UWORD(1) << q1bits) + mpn_mul_1(t, w, wn, p1); flint_mpn_zero(u, wn); u[2 * wn] = mpn_lshift(u + wn, w, wn, q1bits) - p1; mpn_tdiv_q(w, u, 2 * wn + 1, t, wn + 1); error++; /* w2 is computed with 1 ulp error */ } /* Do a second round of argument reduction */ if (wp <= ARB_ATAN_TAB1_PREC) { p2 = 0; } else { q2bits = ARB_ATAN_TAB21_BITS + ARB_ATAN_TAB22_BITS; p2 = w[wn-1] >> (FLINT_BITS - q2bits); if (p2 != 0) { t[wn] = (UWORD(1) << q2bits) + mpn_mul_1(t, w, wn, p2); flint_mpn_zero(u, wn); u[2 * wn] = mpn_lshift(u + wn, w, wn, q2bits) - p2; mpn_tdiv_q(w, u, 2 * wn + 1, t, wn + 1); error++; } } /* |w| <= 2^-r */ r = _arb_mpn_leading_zeros(w, wn); /* N >= (wp-r)/(2r) */ N = (wp - r + (2*r-1)) / (2*r); /* Evaluate Taylor series */ _arb_atan_taylor_rs(t, &error2, w, wn, N, 1); /* Taylor series evaluation error */ error += error2; /* Size of output number */ tn = wn; /* First table lookup */ if (p1 != 0) { if (wp <= ARB_ATAN_TAB1_PREC) mpn_add_n(t, t, arb_atan_tab1[p1] + ARB_ATAN_TAB1_LIMBS - tn, tn); else mpn_add_n(t, t, arb_atan_tab21[p1] + ARB_ATAN_TAB2_LIMBS - tn, tn); error++; } /* Second table lookup */ if (p2 != 0) { mpn_add_n(t, t, arb_atan_tab22[p2] + ARB_ATAN_TAB2_LIMBS - tn, tn); error++; } /* pi/2 - atan(1/x) */ if (reciprocal) { t[tn] = LIMB_ONE - mpn_sub_n(t, arb_atan_pi2_minus_one + ARB_ATAN_TAB2_LIMBS - tn, t, tn); /* result can be >= 1 */ tn += (t[tn] != 0); /* error of pi/2 */ error++; } /* The accumulated arithmetic error */ mag_set_ui_2exp_si(arb_radref(z), error, -wn * FLINT_BITS); /* Truncation error from the Taylor series */ mag_add_ui_2exp_si(arb_radref(z), arb_radref(z), 1, -r*(2*N+1)); /* Set the midpoint */ inexact = _arf_set_mpn_fixed(arb_midref(z), t, tn, wn, negative, prec, ARB_RND); if (inexact) arf_mag_add_ulp(arb_radref(z), arb_radref(z), arb_midref(z), prec); TMP_END; } }