char
float_tanh(
floatnum x,
int digits)
{
signed char sgn;
if (!chckmathparam(x, digits))
return 0;
sgn = float_getsign(x);
float_abs(x);
if (float_cmp(x, &c1Div2) >= 0)
_tanhgt0_5(x, digits);
else
_tanhlt0_5(x, digits);
float_setsign(x, sgn);
return 1;
}
char
float_raise(
floatnum power,
cfloatnum base,
cfloatnum exponent,
int digits)
{
signed char sgn;
if (float_isnan(exponent) || float_isnan(base))
return _seterror(power, NoOperand);
if (digits <= 0 || digits > MATHPRECISION)
return _seterror(power, InvalidPrecision);
if (float_iszero(base))
{
switch(float_getsign(exponent))
{
case 0:
return _seterror(power, OutOfDomain);
case -1:
return _seterror(power, ZeroDivide);
}
return _setzero(power);
}
sgn = float_getsign(base);
if (sgn < 0)
{
if (!float_isinteger(exponent))
return _seterror(power, OutOfDomain);
if ((float_getdigit(exponent, float_getexponent(exponent)) & 1) == 0)
sgn = 1;
}
float_copy(power, base, digits+1);
float_abs(power);
if (!_raise(power, exponent, digits))
{
float_seterror(Overflow);
if (float_getexponent(base) * float_getsign(exponent) < 0)
float_seterror(Underflow);
return _setnan(power);
}
float_setsign(power, sgn);
return 1;
}
/* 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);
}
void
_logic2floatnum(
floatnum f,
t_longint* longint)
{
int idx;
signed char sign;
sign = _signextend(longint);
if (sign < 0)
_neg(longint);
idx = MAXIDX;
while (idx >= 0 && longint->value[idx] == 0)
--idx;
if (idx < 0)
longint->length = 0;
else
longint->length = idx + 1;
_longint2floatnum(f, longint);
float_setsign(f, sign);
}
/* evaluate sin x for |x| <= pi/4,
using |sin x| = sqrt((1-cos x)*(2 + cos x-1))
relative error for 100 digit results is < 6e-100*/
void
_sinltPiDiv4(
floatnum x,
int digits)
{
floatstruct tmp;
signed char sgn;
if (2*float_getexponent(x)+2 < -digits)
/* for small x: sin 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_setsign(x, sgn);
float_free(&tmp);
}
/* evaluates sin x for |x| <= pi. */
void
_sin(
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);
if (float_cmp(x, &cPiDiv4) <= 0)
_sinltPiDiv4(x, digits);
else
{
float_sub(x, &cPiDiv2, x, digits+1);
if (2*float_getexponent(x)+2 < - digits)
float_setzero(x);
else
_cosminus1ltPiDiv4(x, digits);
float_add(x, x, &c1, digits);
}
float_setsign(x, sgn);
}
/* the Taylor series of arctan/arctanh x at x == 0. For small
|x| < 0.01 this series converges very fast, yielding 4 or
more digits of the result with every summand. The working
precision is adjusted, so that the relative error for
100-digit arguments is around 5.0e-100. This means, the error
is 1 in the 100-th place (or less) */
void
arctanseries(
floatnum x,
int digits,
char alternating)
{
int expx;
int expsqrx;
int pwrsz;
int addsz;
int i;
floatstruct xsqr;
floatstruct pwr;
floatstruct smd;
floatstruct sum;
/* upper limit of log(x) and log(result) */
expx = float_getexponent(x)+1;
/* the summands of the series from the second on are
bounded by x^(2*i-1)/3. So the summation yields a
result bounded by (x^3/(1-x*x))/3.
For x < sqrt(1/3) approx.= 0.5, this is less than 0.5*x^3.
We need to sum up only, if the first <digits> places
of the result (roughly x) are touched. Ignoring the effect of
a possile carry, this is only the case, if
x*x >= 2*10^(-digits) > 10^(-digits)
Example: for x = 9e-51, a 100-digits result covers
the decimal places from 1e-51 to 1e-150. x^3/3
is roughly 3e-151, and so is the sum of the series.
So we can ignore the sum, but we couldn't for x = 9e-50 */
if (float_iszero(x) || 2*expx < -digits)
/* for very tiny arguments arctan/arctanh x is approx.== x */
return;
float_create(&xsqr);
float_create(&pwr);
float_create(&smd);
float_create(&sum);
/* we adapt the working precision to the decreasing
summands, saving time when multiplying. Unfortunately,
there is no error bound given for the operations of
bc_num. Tests show, that the last digit in an incomplete
multiplication is usually not correct up to 5 ULP's. */
pwrsz = digits + 2*expx + 1;
/* the precision of the addition must not decrease, of course */
addsz = pwrsz;
i = 3;
float_mul(&xsqr, x, x, pwrsz);
float_setsign(&xsqr, alternating? -1 : 1);
expsqrx = float_getexponent(&xsqr);
float_copy(&pwr, x, pwrsz);
float_setzero(&sum);
for(; pwrsz > 0; )
{
/* x^i */
float_mul(&pwr, &pwr, &xsqr, pwrsz+1);
/* x^i/i */
float_divi(&smd, &pwr, i, pwrsz);
/* The addition virtually does not introduce errors */
float_add(&sum, &sum, &smd, addsz);
/* reduce the working precision according to the decreasing powers */
pwrsz = digits - expx + float_getexponent(&smd) + expsqrx + 3;
i += 2;
}
/* add the first summand */
float_add(x, x, &sum, digits+1);
float_free(&xsqr);
float_free(&pwr);
float_free(&smd);
float_free(&sum);
}
