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
_lngamma_prim(
floatnum x,
floatnum revfactor,
int* infinity,
int digits)
{
floatstruct tmp;
char result;
char odd;
*infinity = 0;
if (float_getsign(x) > 0)
return _lngamma_prim_xgt0(x, revfactor, digits);
float_copy(revfactor, x, digits + 2);
float_sub(x, &c1, x, digits+2);
float_create(&tmp);
result = _lngamma_prim_xgt0(x, &tmp, digits);
if (result)
{
float_neg(x);
odd = float_isodd(revfactor);
_sinpix(revfactor, digits);
if (float_iszero(revfactor))
{
*infinity = 1;
float_setinteger(revfactor, odd? -1 : 1);
}
else
float_mul(&tmp, &tmp, &cPi, digits+2);
float_div(revfactor, revfactor, &tmp, digits+2);
}
float_free(&tmp);
return result;
}
char
_gamma(
floatnum x,
int digits)
{
floatstruct tmp;
int infinity;
char result;
if (float_cmp(&cMinus20, x) > 0)
{
float_create(&tmp);
result = _lngamma_prim(x, &tmp, &infinity, digits)
&& infinity == 0
&& _exp(x, digits)
&& float_div(x, x, &tmp, digits + 1);
float_free(&tmp);
if (infinity != 0)
return _seterror(x, ZeroDivide);
if (!result)
float_setnan(x);
return result;
}
return _gammagtminus20(x, digits);
}
char
_gamma0_5(
floatnum x,
int digits)
{
floatstruct tmp;
int ofs;
if (float_getexponent(x) >= 2)
return _gamma(x, digits);
float_create(&tmp);
float_sub(&tmp, x, &c1Div2, EXACT);
ofs = float_asinteger(&tmp);
float_free(&tmp);
if (ofs >= 0)
{
float_copy(x, &c1Div2, EXACT);
if(!_pochhammer_su(x, ofs, digits))
return 0;
return float_mul(x, x, &cSqrtPi, digits);
}
if(!_pochhammer_su(x, -ofs, digits))
return 0;
return float_div(x, &cSqrtPi, x, digits);
}
/* evaluates arctan x for |x| <= 1
relative error for a 100 digit result is 6e-100 */
void
_arctanlt1(
floatnum x,
int digits)
{
floatstruct tmp;
int reductions;
if (float_iszero(x))
return;
float_create(&tmp);
reductions = 0;
while(float_getexponent(x) >= -2)
{
float_mul(&tmp, x, x, digits);
float_add(&tmp, &tmp, &c1, digits+2);
float_sqrt(&tmp, digits);
float_add(&tmp, &tmp, &c1, digits+1);
float_div(x, x, &tmp, digits);
++reductions;
}
arctannear0(x, digits);
for (;reductions-- > 0;)
float_add(x, x, x, digits+1);
float_free(&tmp);
}
static Error
_pack2frac(
floatnum x,
p_ext_seq_desc n,
int digits)
{
floatstruct tmp;
int exp;
Error result;
n->seq.digits -= n->seq.trailing0;
n->seq.trailing0 = 0;
switch(n->seq.base)
{
case IO_BASE_NAN:
float_setnan(x);
break;
case IO_BASE_ZERO:
float_setzero(x);
break;
default:
if ((result = _pack2int(x, n)) != Success)
return result;
float_create(&tmp);
float_setinteger(&tmp, n->seq.base);
_raiseposi(&tmp, &exp, n->seq.digits, digits+2);
float_div(x, x, &tmp, digits + 2);
float_setexponent(x, float_getexponent(x) - exp);
float_free(&tmp);
}
n->seq.digits += n->seq.trailing0;
return Success;
}
char
float_lb(
floatnum x,
int digits)
{
if (!chckmathparam(x, digits))
return 0;
if (float_getsign(x) <= 0)
return _seterror(x, OutOfDomain);
_ln(x, digits);
float_div(x, x, &cLn2, digits);
return 1;
}
Error
pack2floatnum(
floatnum x,
p_number_desc n)
{
floatstruct tmp;
int digits;
int saveerr;
int saverange;
Error result;
signed char base;
if ((result = _pack2int(x, &n->intpart)) != Success)
return result;
if (float_isnan(x))
return Success;
saveerr = float_geterror();
saverange = float_setrange(MAXEXP);
float_create(&tmp);
float_move(&tmp, x);
float_setzero(x);
digits = DECPRECISION - float_getexponent(&tmp);
if (digits <= 0
|| (result = _pack2frac(x, &n->fracpart, digits)) == Success)
float_add(x, x, &tmp, DECPRECISION);
if (result != Success)
return result;
if ((!float_getlength(x)) == 0) /* no zero, no NaN? */
{
base = n->prefix.base;
float_setinteger(&tmp, base);
if (n->exp >= 0)
{
_raiseposi_(&tmp, n->exp, DECPRECISION + 2);
float_mul(x, x, &tmp, DECPRECISION + 2);
}
else
{
_raiseposi_(&tmp, -n->exp, DECPRECISION + 2);
float_div(x, x, &tmp, DECPRECISION + 2);
}
}
float_free(&tmp);
float_setsign(x, n->prefix.sign == IO_SIGN_COMPLEMENT? -1 : n->prefix.sign);
float_geterror();
float_seterror(saveerr);
float_setrange(saverange);
if (!float_isvalidexp(float_getexponent(x)))
float_setnan(x);
return float_isnan(x)? IOExpOverflow : Success;
}
/* evaluates arccos(1+x) for -0.5 <= x <= 0
arccos(1+x) = arctan(sqrt(-x*(2+x))/(1+x))
the relative error of a 100 digit result is < 5e-100 */
void
_arccosxplus1lt0_5(
floatnum x,
int digits)
{
floatstruct tmp;
float_create(&tmp);
float_add(&tmp, x, &c2, digits+1);
float_mul(x, x, &tmp, digits+1);
float_setsign(x, 1);
float_sqrt(x, digits);
float_sub(&tmp, &tmp, &c1, digits);
float_div(x, x, &tmp, digits+1);
_arctan(x, digits);
float_free(&tmp);
}
/* evaluates arcsin x for -0.5 <= x <= 0.5
arcsin x = arctan(x/sqrt(1-x*x))
the relative error of a 100 digit result is < 5e-100 */
void
_arcsinlt0_5(
floatnum x,
int digits)
{
floatstruct tmp;
if (2*float_getexponent(x) < -digits)
return;
float_create(&tmp);
float_mul(&tmp, x, x, digits);
float_sub(&tmp, &c1, &tmp, digits);
float_sqrt(&tmp, digits);
float_div(x, x, &tmp, digits+1);
_arctanlt1(x, digits);
float_free(&tmp);
}
static int
_extractexp(
floatnum x,
int scale,
signed char base)
{
floatstruct pwr;
floatstruct fbase;
int decprec;
int pwrexp;
int exp;
int logbase;
(void)scale;
logbase = lgbase(base);
decprec = DECPRECISION + 3;
exp = (int)(aprxlog10fn(x) * 3.321928095f);
if (float_getexponent(x) < 0)
exp -= 3;
exp /= logbase;
if (exp != 0)
{
float_create(&fbase);
float_setinteger(&fbase, base);
float_create(&pwr);
float_copy(&pwr, &fbase, EXACT);
_raiseposi(&pwr, &pwrexp, exp < 0? -exp : exp, decprec);
if (float_getexponent(x) < 0)
{
float_addexp(x, pwrexp);
float_mul(x, x, &pwr, decprec);
}
else
{
float_addexp(x, -pwrexp);
float_div(x, x, &pwr, decprec);
}
float_free(&pwr);
float_free(&fbase);
}
exp += _checkbounds(x, decprec, base);
return exp;
}
char
float_lg(
floatnum x,
int digits)
{
floatstruct tmp;
int expx;
if (!chckmathparam(x, digits))
return 0;
if (float_getsign(x) <= 0)
return _seterror(x, OutOfDomain);
float_create(&tmp);
expx = float_getexponent(x);
float_setexponent(x, 0);
_ln(x, digits);
float_div(x, x, &cLn10, digits);
float_setinteger(&tmp, expx);
float_add(x, x, &tmp, digits);
float_free(&tmp);
return 1;
}
char
_gammagtminus20(
floatnum x,
int digits)
{
floatstruct factor;
int ofs;
char result;
float_create(&factor);
ofs = _ofs(x, digits+1);
float_copy(&factor, x, digits+1);
_pochhammer_su(&factor, ofs, digits);
float_addi(x, x, ofs, digits+2);
result = _lngammabigx(x, digits)
&& _exp(x, digits)
&& float_div(x, x, &factor, digits+1);
float_free(&factor);
if (!result)
float_setnan(x);
return result;
}
void
_tanltPiDiv4(
floatnum x,
int digits)
{
floatstruct tmp;
signed char sgn;
if (2*float_getexponent(x)+2 < -digits)
/* for small x: tan x approx.== x */
return;
float_create(&tmp);
sgn = float_getsign(x);
_cosminus1ltPiDiv4(x, digits);
float_add(&tmp, x, &c2, digits+1);
float_mul(x, x, &tmp, digits+1);
float_abs(x);
float_sqrt(x, digits);
float_sub(&tmp, &tmp, &c1, digits);
float_div(x, x, &tmp, digits+1);
float_setsign(x, sgn);
float_free(&tmp);
}
char
erfcasymptotic(
floatnum x,
int digits)
{
floatstruct smd, fct;
int i, workprec, newprec;
float_create(&smd);
float_create(&fct);
workprec = digits - 2 * float_getexponent(x) + 1;
if (workprec <= 0)
{
float_copy(x, &c1, EXACT);
return 1;
}
float_mul(&fct, x, x, digits + 1);
float_div(&fct, &c1Div2, &fct, digits);
float_neg(&fct);
float_copy(&smd, &c1, EXACT);
float_setzero(x);
newprec = digits;
workprec = newprec;
i = 1;
while (newprec > 0 && newprec <= workprec)
{
workprec = newprec;
float_add(x, x, &smd, digits + 4);
float_muli(&smd, &smd, i, workprec + 1);
float_mul(&smd, &smd, &fct, workprec + 2);
newprec = digits + float_getexponent(&smd) + 1;
i += 2;
}
float_free(&fct);
float_free(&smd);
return newprec <= workprec;
}
static char
_pochhammer_g(
floatnum x,
cfloatnum n,
int digits)
{
/* this generalizes the rising Pochhammer symbol using the
formula pochhammer(x,n) = Gamma(x+1)/Gamma(x-n+1) */
floatstruct tmp, factor1, factor2;
int inf1, inf2;
char result;
float_create(&tmp);
float_create(&factor1);
float_create(&factor2);
inf2 = 0;
float_add(&tmp, x, n, digits+1);
result = _lngamma_prim(x, &factor1, &inf1, digits)
&& _lngamma_prim(&tmp, &factor2, &inf2, digits)
&& (inf2 -= inf1) <= 0;
if (inf2 > 0)
float_seterror(ZeroDivide);
if (result && inf2 < 0)
float_setzero(x);
if (result && inf2 == 0)
result = float_div(&factor1, &factor1, &factor2, digits+1)
&& float_sub(x, &tmp, x, digits+1)
&& _exp(x, digits)
&& float_mul(x, x, &factor1, digits+1);
float_free(&tmp);
float_free(&factor2);
float_free(&factor1);
if (!result)
float_setnan(x);
return result;
}
char
erfcsum(
floatnum x, /* should be the square of the parameter to erfc */
int digits)
{
int i, workprec;
floatstruct sum, smd;
floatnum Ei;
if (digits > erfcdigits)
{
/* cannot re-use last evaluation's intermediate results */
for (i = MAXERFCIDX; --i >= 0;)
/* clear all exp(-k*k*alpha*alpha) to indicate their absence */
float_free(&erfccoeff[i]);
/* current precision */
erfcdigits = digits;
/* create new alpha appropriate for the desired precision
This alpha need not be high precision, any alpha near the
one evaluated here would do */
float_muli(&erfcalpha, &cLn10, digits + 4, 3);
float_sqrt(&erfcalpha, 3);
float_div(&erfcalpha, &cPi, &erfcalpha, 3);
float_mul(&erfcalphasqr, &erfcalpha, &erfcalpha, EXACT);
/* the exp(-k*k*alpha*alpha) are later evaluated iteratively.
Initiate the iteration here */
float_copy(&erfct2, &erfcalphasqr, EXACT);
float_neg(&erfct2);
_exp(&erfct2, digits + 3); /* exp(-alpha*alpha) */
float_copy(erfccoeff, &erfct2, EXACT); /* start value */
float_mul(&erfct3, &erfct2, &erfct2, digits + 3); /* exp(-2*alpha*alpha) */
}
float_create(&sum);
float_create(&smd);
float_setzero(&sum);
for (i = 0; ++i < MAXERFCIDX;)
{
Ei = &erfccoeff[i-1];
if (float_isnan(Ei))
{
/* if exp(-i*i*alpha*alpha) is not available, evaluate it from
the coefficient of the last summand */
float_mul(&erfct2, &erfct2, &erfct3, workprec + 3);
float_mul(Ei, &erfct2, &erfccoeff[i-2], workprec + 3);
}
/* Ei finally decays rapidly. save some time by adjusting the
working precision */
workprec = digits + float_getexponent(Ei) + 1;
if (workprec <= 0)
break;
/* evaluate the summand exp(-i*i*alpha*alpha)/(i*i*alpha*alpha+x) */
float_muli(&smd, &erfcalphasqr, i*i, workprec);
float_add(&smd, x, &smd, workprec + 2);
float_div(&smd, Ei, &smd, workprec + 1);
/* add summand to the series */
float_add(&sum, &sum, &smd, digits + 3);
}
float_move(x, &sum);
float_free(&smd);
return 1;
}
