/* evaluates tan x for |x| <= pi. A return value of 0 indicates that x = +/- pi/2 within small tolerances, so that tan x cannot be reliable computed */ char _tan( floatnum x, int digits) { signed char sgn; sgn = float_getsign(x); float_abs(x); if (float_cmp(x, &cPiDiv2) > 0) { float_sub(x, &cPi, x, digits+1); sgn = -sgn; } if (float_cmp(x, &cPiDiv4) <= 0) _tanltPiDiv4(x, digits); else { float_sub(x, &cPiDiv2, x, digits+1); if (float_iszero(x) || float_getexponent(x) < -digits) return 0; _tanltPiDiv4(x, digits); float_reciprocal(x, digits); } float_setsign(x, sgn); return 1; }
char binetasymptotic(floatnum x, int digits) { floatstruct recsqr; floatstruct sum; floatstruct smd; floatstruct pwr; int i, workprec; if (float_getexponent(x) >= digits) { /* if x is very big, ln(gamma(x)) is dominated by x*ln x and the Binet function does not contribute anything substantial to the final result */ float_setzero(x); return 1; } float_create(&recsqr); float_create(&sum); float_create(&smd); float_create(&pwr); float_copy(&pwr, &c1, EXACT); float_setzero(&sum); float_div(&smd, &c1, &c12, digits+1); workprec = digits - 2*float_getexponent(x)+3; i = 1; if (workprec > 0) { float_mul(&recsqr, x, x, workprec); float_reciprocal(&recsqr, workprec); while (float_getexponent(&smd) > -digits-1 && ++i <= MAXBERNOULLIIDX) { workprec = digits + float_getexponent(&smd) + 3; float_add(&sum, &sum, &smd, digits+1); float_mul(&pwr, &recsqr, &pwr, workprec); float_muli(&smd, &cBernoulliDen[i-1], 2*i*(2*i-1), workprec); float_div(&smd, &pwr, &smd, workprec); float_mul(&smd, &smd, &cBernoulliNum[i-1], workprec); } } else /* sum reduces to the first summand*/ float_move(&sum, &smd); if (i > MAXBERNOULLIIDX) /* x was not big enough for the asymptotic series to converge sufficiently */ float_setnan(x); else float_div(x, &sum, x, digits); float_free(&pwr); float_free(&smd); float_free(&sum); float_free(&recsqr); return i <= MAXBERNOULLIIDX; }
static char _pochhammer_si( floatnum x, int n, int digits) { /* this extends the rising Pochhammer symbol to negative integer offsets following the formula pochhammer(x,n-1) = pochhammer(x,n)/(x-n+1) */ if (n >= 0) return _pochhammer_su(x, n, digits); return float_addi(x, x, n, digits) && _pochhammer_su(x, -n, digits) && float_reciprocal(x, digits); }
/* evaluates arctan x for all x. The result is in the range -pi/2 < result < pi/2 relative error for a 100 digit result is 9e-100 */ void _arctan( floatnum x, int digits) { signed char sgn; if (float_abscmp(x, &c1) > 0) { sgn = float_getsign(x); float_abs(x); float_reciprocal(x, digits); _arctanlt1(x, digits); float_sub(x, &cPiDiv2, x, digits+1); float_setsign(x, sgn); } else _arctanlt1(x, digits); }