long double expl(long double x)
{
long double px, xx;
int k;
if (isnan(x))
return x;
if (x > 11356.5234062941439488L) /* x > ln(2^16384 - 0.5) */
return x * 0x1p16383L;
if (x < -11399.4985314888605581L) /* x < ln(2^-16446) */
return -0x1p-16445L/x;
/* Express e**x = e**f 2**k
* = e**(f + k ln(2))
*/
px = floorl(LOG2E * x + 0.5);
k = px;
x -= px * LN2HI;
x -= px * LN2LO;
/* rational approximation of the fractional part:
* e**x = 1 + 2x P(x**2)/(Q(x**2) - x P(x**2))
*/
xx = x * x;
px = x * __polevll(xx, P, 2);
x = px/(__polevll(xx, Q, 3) - px);
x = 1.0 + 2.0 * x;
return scalbnl(x, k);
}
long double
expl(long double x)
{
long double px, xx;
int n;
if( x > MAXLOGL)
return (huge*huge); /* overflow */
if( x < MINLOGL )
return (twom10000*twom10000); /* underflow */
/* Express e**x = e**g 2**n
* = e**g e**( n loge(2) )
* = e**( g + n loge(2) )
*/
px = floorl( LOG2EL * x + 0.5L ); /* floor() truncates toward -infinity. */
n = px;
x += px * C1;
x += px * C2;
/* rational approximation for exponential
* of the fractional part:
* e**x = 1 + 2x P(x**2)/( Q(x**2) - P(x**2) )
*/
xx = x * x;
px = x * __polevll( xx, P, 4 );
xx = __polevll( xx, Q, 5 );
x = px/( xx - px );
x = 1.0L + x + x;
x = ldexpl( x, n );
return(x);
}
/* Gamma function computed by Stirling's formula.
*/
static long double stirf(long double x)
{
long double y, w, v;
w = 1.0L/x;
/* For large x, use rational coefficients from the analytical expansion. */
if( x > 1024.0L )
w = (((((6.97281375836585777429E-5L * w
+ 7.84039221720066627474E-4L) * w
- 2.29472093621399176955E-4L) * w
- 2.68132716049382716049E-3L) * w
+ 3.47222222222222222222E-3L) * w
+ 8.33333333333333333333E-2L) * w
+ 1.0L;
else
w = 1.0L + w * __polevll( w, STIR, 8 );
y = expl(x);
if( x > MAXSTIR )
{ /* Avoid overflow in pow() */
v = powl( x, 0.5L * x - 0.25L );
y = v * (v / y);
}
else
{
y = powl( x, x - 0.5L ) / y;
}
y = SQTPI * y * w;
return( y );
}
long double
expl(long double x)
{
long double px, xx;
int n;
if( isnan(x) )
return(x);
if( x > MAXLOGL)
return( INFINITY );
if( x < MINLOGL )
return(0.0L);
/* Express e**x = e**g 2**n
* = e**g e**( n loge(2) )
* = e**( g + n loge(2) )
*/
px = floorl( LOG2EL * x + 0.5L ); /* floor() truncates toward -infinity. */
n = px;
x -= px * C1;
x -= px * C2;
/* rational approximation for exponential
* of the fractional part:
* e**x = 1 + 2x P(x**2)/( Q(x**2) - P(x**2) )
*/
xx = x * x;
px = x * __polevll( xx, P, 2 );
x = px/( __polevll( xx, Q, 3 ) - px );
x = 1.0L + ldexpl( x, 1 );
x = ldexpl( x, n );
return(x);
}
long double log10l(long double x)
{
long double y, z;
int e;
if (isnan(x))
return x;
if(x <= 0.0) {
if(x == 0.0)
return -1.0 / (x*x);
return (x - x) / 0.0;
}
if (x == INFINITY)
return INFINITY;
/* separate mantissa from exponent */
/* Note, frexp is used so that denormal numbers
* will be handled properly.
*/
x = frexpl(x, &e);
/* logarithm using log(x) = z + z**3 P(z)/Q(z),
* where z = 2(x-1)/x+1)
*/
if (e > 2 || e < -2) {
if (x < SQRTH) { /* 2(2x-1)/(2x+1) */
e -= 1;
z = x - 0.5;
y = 0.5 * z + 0.5;
} else { /* 2 (x-1)/(x+1) */
z = x - 0.5;
z -= 0.5;
y = 0.5 * x + 0.5;
}
x = z / y;
z = x*x;
y = x * (z * __polevll(z, R, 3) / __p1evll(z, S, 3));
goto done;
}
/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
if (x < SQRTH) {
e -= 1;
x = 2.0*x - 1.0;
} else {
x = x - 1.0;
}
z = x*x;
y = x * (z * __polevll(x, P, 6) / __p1evll(x, Q, 7));
y = y - 0.5*z;
done:
/* Multiply log of fraction by log10(e)
* and base 2 exponent by log10(2).
*
* ***CAUTION***
*
* This sequence of operations is critical and it may
* be horribly defeated by some compiler optimizers.
*/
z = y * (L10EB);
z += x * (L10EB);
z += e * (L102B);
z += y * (L10EA);
z += x * (L10EA);
z += e * (L102A);
return z;
}
long double
log10l(long double x)
{
long double y;
volatile long double z;
int e;
if( isnan(x) )
return(x);
/* Test for domain */
if( x <= 0.0L )
{
if( x == 0.0L )
return (-1.0L / (x - x));
else
return (x - x) / (x - x);
}
if( x == INFINITY )
return(INFINITY);
/* separate mantissa from exponent */
/* Note, frexp is used so that denormal numbers
* will be handled properly.
*/
x = frexpl( x, &e );
/* logarithm using log(x) = z + z**3 P(z)/Q(z),
* where z = 2(x-1)/x+1)
*/
if( (e > 2) || (e < -2) )
{
if( x < SQRTH )
{ /* 2( 2x-1 )/( 2x+1 ) */
e -= 1;
z = x - 0.5L;
y = 0.5L * z + 0.5L;
}
else
{ /* 2 (x-1)/(x+1) */
z = x - 0.5L;
z -= 0.5L;
y = 0.5L * x + 0.5L;
}
x = z / y;
z = x*x;
y = x * ( z * __polevll( z, R, 3 ) / __p1evll( z, S, 3 ) );
goto done;
}
/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
if( x < SQRTH )
{
e -= 1;
x = ldexpl( x, 1 ) - 1.0L; /* 2x - 1 */
}
else
{
x = x - 1.0L;
}
z = x*x;
y = x * ( z * __polevll( x, P, 6 ) / __p1evll( x, Q, 7 ) );
y = y - ldexpl( z, -1 ); /* -0.5x^2 + ... */
done:
/* Multiply log of fraction by log10(e)
* and base 2 exponent by log10(2).
*
* ***CAUTION***
*
* This sequence of operations is critical and it may
* be horribly defeated by some compiler optimizers.
*/
z = y * (L10EB);
z += x * (L10EB);
z += e * (L102B);
z += y * (L10EA);
z += x * (L10EA);
z += e * (L102A);
return( z );
}
long double powl(long double x, long double y)
{
/* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */
int i, nflg, iyflg, yoddint;
long e;
volatile long double z=0;
long double w=0, W=0, Wa=0, Wb=0, ya=0, yb=0, u=0;
/* make sure no invalid exception is raised by nan comparision */
if (isnan(x)) {
if (!isnan(y) && y == 0.0)
return 1.0;
return x;
}
if (isnan(y)) {
if (x == 1.0)
return 1.0;
return y;
}
if (x == 1.0)
return 1.0; /* 1**y = 1, even if y is nan */
if (x == -1.0 && !isfinite(y))
return 1.0; /* -1**inf = 1 */
if (y == 0.0)
return 1.0; /* x**0 = 1, even if x is nan */
if (y == 1.0)
return x;
if (y >= LDBL_MAX) {
if (x > 1.0 || x < -1.0)
return INFINITY;
if (x != 0.0)
return 0.0;
}
if (y <= -LDBL_MAX) {
if (x > 1.0 || x < -1.0)
return 0.0;
if (x != 0.0 || y == -INFINITY)
return INFINITY;
}
if (x >= LDBL_MAX) {
if (y > 0.0)
return INFINITY;
return 0.0;
}
w = floorl(y);
/* Set iyflg to 1 if y is an integer. */
iyflg = 0;
if (w == y)
iyflg = 1;
/* Test for odd integer y. */
yoddint = 0;
if (iyflg) {
ya = fabsl(y);
ya = floorl(0.5 * ya);
yb = 0.5 * fabsl(w);
if( ya != yb )
yoddint = 1;
}
if (x <= -LDBL_MAX) {
if (y > 0.0) {
if (yoddint)
return -INFINITY;
return INFINITY;
}
if (y < 0.0) {
if (yoddint)
return -0.0;
return 0.0;
}
}
nflg = 0; /* (x<0)**(odd int) */
if (x <= 0.0) {
if (x == 0.0) {
if (y < 0.0) {
if (signbit(x) && yoddint)
/* (-0.0)**(-odd int) = -inf, divbyzero */
return -1.0/0.0;
/* (+-0.0)**(negative) = inf, divbyzero */
return 1.0/0.0;
}
if (signbit(x) && yoddint)
return -0.0;
return 0.0;
}
if (iyflg == 0)
return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */
/* (x<0)**(integer) */
if (yoddint)
nflg = 1; /* negate result */
x = -x;
}
/* (+integer)**(integer) */
if (iyflg && floorl(x) == x && fabsl(y) < 32768.0) {
w = powil(x, (int)y);
return nflg ? -w : w;
}
/* separate significand from exponent */
x = frexpl(x, &i);
e = i;
/* find significand in antilog table A[] */
i = 1;
if (x <= A[17])
i = 17;
if (x <= A[i+8])
i += 8;
if (x <= A[i+4])
i += 4;
if (x <= A[i+2])
i += 2;
if (x >= A[1])
i = -1;
i += 1;
/* Find (x - A[i])/A[i]
* in order to compute log(x/A[i]):
*
* log(x) = log( a x/a ) = log(a) + log(x/a)
*
* log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a
*/
x -= A[i];
x -= B[i/2];
x /= A[i];
/* rational approximation for log(1+v):
*
* log(1+v) = v - v**2/2 + v**3 P(v) / Q(v)
*/
z = x*x;
w = x * (z * __polevll(x, P, 3) / __p1evll(x, Q, 3));
w = w - 0.5*z;
/* Convert to base 2 logarithm:
* multiply by log2(e) = 1 + LOG2EA
*/
z = LOG2EA * w;
z += w;
z += LOG2EA * x;
z += x;
/* Compute exponent term of the base 2 logarithm. */
w = -i;
w /= NXT;
w += e;
/* Now base 2 log of x is w + z. */
/* Multiply base 2 log by y, in extended precision. */
/* separate y into large part ya
* and small part yb less than 1/NXT
*/
ya = reducl(y);
yb = y - ya;
/* (w+z)(ya+yb)
* = w*ya + w*yb + z*y
*/
F = z * y + w * yb;
Fa = reducl(F);
Fb = F - Fa;
G = Fa + w * ya;
Ga = reducl(G);
Gb = G - Ga;
H = Fb + Gb;
Ha = reducl(H);
w = (Ga + Ha) * NXT;
/* Test the power of 2 for overflow */
if (w > MEXP)
return huge * huge; /* overflow */
if (w < MNEXP)
return twom10000 * twom10000; /* underflow */
e = w;
Hb = H - Ha;
if (Hb > 0.0) {
e += 1;
Hb -= 1.0/NXT; /*0.0625L;*/
}
/* Now the product y * log2(x) = Hb + e/NXT.
*
* Compute base 2 exponential of Hb,
* where -0.0625 <= Hb <= 0.
*/
z = Hb * __polevll(Hb, R, 6); /* z = 2**Hb - 1 */
/* Express e/NXT as an integer plus a negative number of (1/NXT)ths.
* Find lookup table entry for the fractional power of 2.
*/
if (e < 0)
i = 0;
else
i = 1;
i = e/NXT + i;
e = NXT*i - e;
w = A[e];
z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */
z = z + w;
z = scalbnl(z, i); /* multiply by integer power of 2 */
if (nflg)
z = -z;
return z;
}
long double log1pl(long double xm1)
{
long double x, y, z;
int e;
if (isnan(xm1))
return xm1;
if (xm1 == INFINITY)
return xm1;
if (xm1 == 0.0)
return xm1;
x = xm1 + 1.0;
/* Test for domain errors. */
if (x <= 0.0) {
if (x == 0.0)
return -1/x; /* -inf with divbyzero */
return 0/0.0f; /* nan with invalid */
}
/* Separate mantissa from exponent.
Use frexp so that denormal numbers will be handled properly. */
x = frexpl(x, &e);
/* logarithm using log(x) = z + z^3 P(z)/Q(z),
where z = 2(x-1)/x+1) */
if (e > 2 || e < -2) {
if (x < SQRTH) { /* 2(2x-1)/(2x+1) */
e -= 1;
z = x - 0.5;
y = 0.5 * z + 0.5;
} else { /* 2 (x-1)/(x+1) */
z = x - 0.5;
z -= 0.5;
y = 0.5 * x + 0.5;
}
x = z / y;
z = x*x;
z = x * (z * __polevll(z, R, 3) / __p1evll(z, S, 3));
z = z + e * C2;
z = z + x;
z = z + e * C1;
return z;
}
/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
if (x < SQRTH) {
e -= 1;
if (e != 0)
x = 2.0 * x - 1.0;
else
x = xm1;
} else {
if (e != 0)
x = x - 1.0;
else
x = xm1;
}
z = x*x;
y = x * (z * __polevll(x, P, 6) / __p1evll(x, Q, 6));
y = y + e * C2;
z = y - 0.5 * z;
z = z + x;
z = z + e * C1;
return z;
}
long double logl(long double x)
{
long double y, z;
int e;
if (isnan(x))
return x;
if (x == INFINITY)
return x;
if (x <= 0.0) {
if (x == 0.0)
return -INFINITY;
return NAN;
}
/* separate mantissa from exponent */
/* Note, frexp is used so that denormal numbers
* will be handled properly.
*/
x = frexpl(x, &e);
/* logarithm using log(x) = z + z**3 P(z)/Q(z),
* where z = 2(x-1)/x+1)
*/
if (e > 2 || e < -2) {
if (x < SQRTH) { /* 2(2x-1)/(2x+1) */
e -= 1;
z = x - 0.5;
y = 0.5 * z + 0.5;
} else { /* 2 (x-1)/(x+1) */
z = x - 0.5;
z -= 0.5;
y = 0.5 * x + 0.5;
}
x = z / y;
z = x*x;
z = x * (z * __polevll(z, R, 3) / __p1evll(z, S, 3));
z = z + e * C2;
z = z + x;
z = z + e * C1;
return z;
}
/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
if (x < SQRTH) {
e -= 1;
x = 2.0*x - 1.0;
} else {
x = x - 1.0;
}
z = x*x;
y = x * (z * __polevll(x, P, 6) / __p1evll(x, Q, 6));
y = y + e * C2;
z = y - 0.5*z;
/* Note, the sum of above terms does not exceed x/4,
* so it contributes at most about 1/4 lsb to the error.
*/
z = z + x;
z = z + e * C1; /* This sum has an error of 1/2 lsb. */
return z;
}
long double
logl(long double x)
{
long double y, z;
int e;
if( isnan(x) )
return(x);
if( x == INFINITY )
return(x);
/* Test for domain */
if( x <= 0.0L )
{
if( x == 0.0L )
return( -INFINITY );
else
return( NAN );
}
/* separate mantissa from exponent */
/* Note, frexp is used so that denormal numbers
* will be handled properly.
*/
x = frexpl( x, &e );
/* logarithm using log(x) = z + z**3 P(z)/Q(z),
* where z = 2(x-1)/x+1)
*/
if( (e > 2) || (e < -2) )
{
if( x < SQRTH )
{ /* 2( 2x-1 )/( 2x+1 ) */
e -= 1;
z = x - 0.5L;
y = 0.5L * z + 0.5L;
}
else
{ /* 2 (x-1)/(x+1) */
z = x - 0.5L;
z -= 0.5L;
y = 0.5L * x + 0.5L;
}
x = z / y;
z = x*x;
z = x * ( z * __polevll( z, (void *)R, 3 ) / __p1evll( z, (void *)S, 3 ) );
z = z + e * C2;
z = z + x;
z = z + e * C1;
return( z );
}
/* logarithm using log(1+x) = x - .5x**2 + x**3 P(x)/Q(x) */
if( x < SQRTH )
{
e -= 1;
x = ldexpl( x, 1 ) - 1.0L; /* 2x - 1 */
}
else
{
x = x - 1.0L;
}
z = x*x;
y = x * ( z * __polevll( x, (void *)P, 6 ) / __p1evll( x, (void *)Q, 6 ) );
y = y + e * C2;
z = y - ldexpl( z, -1 ); /* y - 0.5 * z */
/* Note, the sum of above terms does not exceed x/4,
* so it contributes at most about 1/4 lsb to the error.
*/
z = z + x;
z = z + e * C1; /* This sum has an error of 1/2 lsb. */
return( z );
}
long double
tgammal(long double x)
{
long double p, q, z;
int i;
signgam = 1;
if( isnan(x) )
return(NAN);
if(x == INFINITY)
return(INFINITY);
if(x == -INFINITY)
return(x - x);
q = fabsl(x);
if( q > 13.0L )
{
if( q > MAXGAML )
goto goverf;
if( x < 0.0L )
{
p = floorl(q);
if( p == q )
return (x - x) / (x - x);
i = p;
if( (i & 1) == 0 )
signgam = -1;
z = q - p;
if( z > 0.5L )
{
p += 1.0L;
z = q - p;
}
z = q * sinl( PIL * z );
z = fabsl(z) * stirf(q);
if( z <= PIL/LDBL_MAX )
{
goverf:
return( signgam * INFINITY);
}
z = PIL/z;
}
else
{
z = stirf(x);
}
return( signgam * z );
}
z = 1.0L;
while( x >= 3.0L )
{
x -= 1.0L;
z *= x;
}
while( x < -0.03125L )
{
z /= x;
x += 1.0L;
}
if( x <= 0.03125L )
goto small;
while( x < 2.0L )
{
z /= x;
x += 1.0L;
}
if( x == 2.0L )
return(z);
x -= 2.0L;
p = __polevll( x, P, 7 );
q = __polevll( x, Q, 8 );
z = z * p / q;
if( z < 0 )
signgam = -1;
return z;
small:
if( x == 0.0L )
return (x - x) / (x - x);
else
{
if( x < 0.0L )
{
x = -x;
q = z / (x * __polevll( x, SN, 8 ));
signgam = -1;
}
else
q = z / (x * __polevll( x, S, 8 ));
}
return q;
}
long double tgammal(long double x)
{
long double p, q, z;
if (!isfinite(x))
return x + INFINITY;
q = fabsl(x);
if (q > 13.0) {
if (x < 0.0) {
p = floorl(q);
z = q - p;
if (z == 0)
return 0 / z;
if (q > MAXGAML) {
z = 0;
} else {
if (z > 0.5) {
p += 1.0;
z = q - p;
}
z = q * sinl(PIL * z);
z = fabsl(z) * stirf(q);
z = PIL/z;
}
if (0.5 * p == floorl(q * 0.5))
z = -z;
} else if (x > MAXGAML) {
z = x * 0x1p16383L;
} else {
z = stirf(x);
}
return z;
}
z = 1.0;
while (x >= 3.0) {
x -= 1.0;
z *= x;
}
while (x < -0.03125L) {
z /= x;
x += 1.0;
}
if (x <= 0.03125L)
goto small;
while (x < 2.0) {
z /= x;
x += 1.0;
}
if (x == 2.0)
return z;
x -= 2.0;
p = __polevll(x, P, 7);
q = __polevll(x, Q, 8);
z = z * p / q;
return z;
small:
/* z==1 if x was originally +-0 */
if (x == 0 && z != 1)
return x / x;
if (x < 0.0) {
x = -x;
q = z / (x * __polevll(x, SN, 8));
} else
q = z / (x * __polevll(x, S, 8));
return q;
}
