/* 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);
}
