char
_cosminus1ltPiDiv4(
floatnum x,
int digits)
{
floatstruct tmp;
int reductions;
if (float_iszero(x))
return 1;
float_abs(x);
reductions = 0;
while(float_getexponent(x) >= -2)
{
float_mul(x, x, &c1Div2, digits+1);
++reductions;
}
if (!cosminus1near0(x, digits) && reductions == 0)
return !float_iszero(x);
float_create(&tmp);
for(; reductions-- > 0;)
{
float_mul(&tmp, x, x, digits);
float_add(x, x, x, digits+2);
float_add(x, x, &tmp, digits+2);
float_add(x, x, x, digits+2);
}
float_free(&tmp);
return 1;
}
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;
}
Error float_out(
p_otokens tokens,
floatnum x,
int scale,
signed char base,
char outmode)
{
t_number_desc n;
_emptytokens(tokens);
/* do some sanity checks first */
if (!_validmode(outmode) || scale < 0 || !_isvalidbase(base))
return InvalidParam;
_clearnumber(&n);
if (float_iszero(x))
n.prefix.base = IO_BASE_ZERO;
else if (!float_isnan(x))
n.prefix.base = base;
if (!_isvalidbase(n.prefix.base))
/* NaN and 0 are handled here */
return desc2str(tokens, &n, 0);
n.prefix.sign = float_getsign(x);
float_abs(x);
switch (outmode)
{
case IO_MODE_FIXPOINT:
return _outfixp(tokens, x, &n, scale);
case IO_MODE_ENG:
return _outeng(tokens, x, &n, scale);
case IO_MODE_COMPLEMENT:
return _outcompl(tokens, x, &n, 0);
default:
return _outsci(tokens, x, &n, scale);
}
}
char
_floatnum2logic(
t_longint* longint,
cfloatnum x)
{
floatstruct tmp;
int digits;
digits = float_getexponent(x)+1;
if (float_iszero(x) || digits <= 0)
{
longint->length = 1;
longint->value[0] = 0;
}
else
{
if (digits > MATHPRECISION)
return 0;
float_create(&tmp);
/* floatnum2longint rounds, we have to truncate first */
float_copy(&tmp, x, digits);
if (float_getsign(x) < 0)
float_add(&tmp, &tmp, &c1, EXACT);
_floatnum2longint(longint, &tmp);
float_free(&tmp);
if (_bitlength(longint) > LOGICRANGE)
return 0;
}
_zeroextend(longint);
if (float_getsign(x) < 0)
_not(longint);
return 1;
}
char
erfseries(
floatnum x,
int digits)
{
floatstruct xsqr, smd, pwr;
int i, workprec, expx;
expx = float_getexponent(x);
workprec = digits + 2*expx + 2;
if (workprec <= 0 || float_iszero(x))
/* for tiny arguments approx. == x */
return 1;
float_create(&xsqr);
float_create(&smd);
float_create(&pwr);
float_mul(&xsqr, x, x, workprec + 1);
workprec = digits + float_getexponent(&xsqr) + 1;
float_copy(&pwr, x, workprec + 1);
i = 1;
while (workprec > 0)
{
float_mul(&pwr, &pwr, &xsqr, workprec + 1);
float_divi(&pwr, &pwr, -i, workprec + 1);
float_divi(&smd, &pwr, 2 * i++ + 1, workprec);
float_add(x, x, &smd, digits + 3);
workprec = digits + float_getexponent(&smd) + expx + 2;
}
float_free(&pwr);
float_free(&smd);
float_free(&xsqr);
return 1;
}
char
_doshift(
floatnum dest,
cfloatnum x,
cfloatnum shift,
char right)
{
int ishift;
t_longint lx;
if (float_isnan(shift))
return _seterror(dest, NoOperand);
if (!float_isinteger(shift))
return _seterror(dest, OutOfDomain);
if(!_cvtlogic(&lx, x))
return 0;
if (float_iszero(shift))
{
float_copy(dest, x, EXACT);
return 1;
}
ishift = float_asinteger(shift);
if (ishift == 0)
ishift = (3*LOGICRANGE) * float_getsign(shift);
if (!right)
ishift = -ishift;
if (ishift > 0)
_shr(&lx, ishift);
else
_shl(&lx, -ishift);
_logic2floatnum(dest, &lx);
return 1;
}
char
float_raisei(
floatnum power,
cfloatnum base,
int exponent,
int digits)
{
if (digits <= 0 || digits > maxdigits)
return _seterror(power, InvalidPrecision);
if (float_isnan(base))
return _seterror(power, NoOperand);
if (float_iszero(base))
{
if (exponent == 0)
return _seterror(power, OutOfDomain);
if (exponent < 0)
return _seterror(power, ZeroDivide);
return _setzero(power);
}
digits += 14;
if (digits > maxdigits)
digits = maxdigits;
float_copy(power, base, digits);
if (!_raisei(power, exponent, digits)
|| !float_isvalidexp(float_getexponent(power)))
{
if (float_getexponent(base) < 0)
return _seterror(power, Underflow);
return _seterror(power, Overflow);
}
return 1;
}
/* 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);
}
/* 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;
}
/* series expansion of cos/cosh - 1 used for small x,
|x| <= 0.01.
The function returns 0, if an underflow occurs.
The relative error seems to be less than 5e-100 for
a 100-digit calculation with |x| < 0.01 */
char
cosminus1series(
floatnum x,
int digits,
char alternating)
{
floatstruct sum, smd;
int expsqrx, pwrsz, addsz, i;
expsqrx = 2 * float_getexponent(x);
float_setexponent(x, 0);
float_mul(x, x, x, digits+1);
float_mul(x, x, &c1Div2, digits+1);
float_setsign(x, alternating? -1 : 1);
expsqrx += float_getexponent(x);
if (float_iszero(x) || expsqrx < EXPMIN)
{
/* underflow */
float_setzero(x);
return expsqrx == 0;
}
float_setexponent(x, expsqrx);
pwrsz = digits + expsqrx + 2;
if (pwrsz <= 0)
/* for very small x, cos/cosh(x) - 1 = (-/+)0.5*x*x */
return 1;
addsz = pwrsz;
float_create(&sum);
float_create(&smd);
float_copy(&smd, x, pwrsz);
float_setzero(&sum);
i = 2;
while (pwrsz > 0)
{
float_mul(&smd, &smd, x, pwrsz+1);
float_divi(&smd, &smd, i*(2*i-1), pwrsz);
float_add(&sum, &sum, &smd, addsz);
++i;
pwrsz = digits + float_getexponent(&smd);
}
float_add(x, x, &sum, digits+1);
float_free(&sum);
float_free(&smd);
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;
}
char
float_tanhminus1(
floatnum x,
int digits)
{
if (!chckmathparam(x, digits))
return 0;
if (float_cmp(x, &c1Div2) >= 0)
return _tanhminus1gt0(x, digits)? 1 : _seterror(x, Underflow);
if (!float_iszero(x))
{
if (float_abscmp(x, &c1Div2) <= 0)
_tanhlt0_5(x, digits);
else
{
float_setsign(x, 1);
_tanhgt0_5(x, digits);
float_setsign(x, -1);
}
}
return float_sub(x, x, &c1, digits);
}
static char
_pochhammer_i(
floatnum x,
cfloatnum n,
int digits)
{
/* do not use the expensive Gamma function when a few
multiplications do the same */
/* pre: n is an integer */
int ni;
signed char result;
if (float_iszero(n))
return float_copy(x, &c1, EXACT);
if (float_isinteger(x))
{
result = -1;
float_neg((floatnum)n);
if (float_getsign(x) <= 0 && float_cmp(x, n) > 0)
/* x and x+n have opposite signs, meaning 0 is
among the factors */
result = _setzero(x);
else if (float_getsign(x) > 0 && float_cmp(x, n) <= 0)
/* x and x+n have opposite signs, meaning at one point
you have to divide by 0 */
result = _seterror(x, ZeroDivide);
float_neg((floatnum)n);
if (result >= 0)
return result;
}
if (float_getexponent(x) < EXPMAX/100)
{
ni = float_asinteger(n);
if (ni != 0 && ni < 50 && ni > -50)
return _pochhammer_si(x, ni, digits+2);
}
return _pochhammer_g(x, n, digits);
}
/* 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);
}
