int
gl_signbitl (long double arg)
{
#if defined LDBL_SIGNBIT_WORD && defined LDBL_SIGNBIT_BIT
/* The use of a union to extract the bits of the representation of a
'long double' is safe in practice, despite of the "aliasing rules" of
C99, because the GCC docs say
"Even with '-fstrict-aliasing', type-punning is allowed, provided the
memory is accessed through the union type."
and similarly for other compilers. */
# define NWORDS \
((sizeof (long double) + sizeof (unsigned int) - 1) / sizeof (unsigned int))
union { long double value; unsigned int word[NWORDS]; } m;
m.value = arg;
return (m.word[LDBL_SIGNBIT_WORD] >> LDBL_SIGNBIT_BIT) & 1;
#elif HAVE_COPYSIGNL_IN_LIBC
return copysignl (1.0L, arg) < 0;
#else
/* This does not do the right thing for NaN, but this is irrelevant for
most use cases. */
if (isnanl (arg))
return 0;
if (arg < 0.0L)
return 1;
else if (arg == 0.0L)
{
/* Distinguish 0.0L and -0.0L. */
static long double plus_zero = 0.0L;
long double arg_mem = arg;
return (memcmp (&plus_zero, &arg_mem, SIZEOF_LDBL) != 0);
}
else
return 0;
#endif
}
int
main ()
{
DECL_LONG_DOUBLE_ROUNDING
BEGIN_LONG_DOUBLE_ROUNDING ();
/* See IEEE 754, section 6.3:
"the sign of the result of the round floating-point number to
integral value operation is the sign of the operand. These rules
shall apply even when operands or results are zero or infinite." */
/* Zero. */
ASSERT (!signbit (truncl (0.0L)));
ASSERT (!!signbit (truncl (minus_zerol)) == !!signbit (minus_zerol));
/* Positive numbers. */
ASSERT (!signbit (truncl (0.3L)));
ASSERT (!signbit (truncl (0.7L)));
/* Negative numbers. */
ASSERT (!!signbit (truncl (-0.3L)) == !!signbit (minus_zerol));
ASSERT (!!signbit (truncl (-0.7L)) == !!signbit (minus_zerol));
/* [MX] shaded specification in POSIX. */
/* NaN. */
ASSERT (isnanl (truncl (NaNl ())));
/* Infinity. */
ASSERT (truncl (Infinityl ()) == Infinityl ());
ASSERT (truncl (- Infinityl ()) == - Infinityl ());
return 0;
}
long double
tanl (long double x)
{
long double y[2], z = 0.0L;
int n;
/* tanl(NaN) is NaN */
if (isnanl (x))
return x;
/* |x| ~< pi/4 */
if (x >= -0.7853981633974483096156608458198757210492 &&
x <= 0.7853981633974483096156608458198757210492)
return kernel_tanl (x, z, 1);
/* tanl(Inf) is NaN, tanl(0) is 0 */
else if (x + x == x)
return x - x; /* NaN */
/* argument reduction needed */
else
{
n = ieee754_rem_pio2l (x, y);
/* 1 -- n even, -1 -- n odd */
return kernel_tanl (y[0], y[1], 1 - ((n & 1) << 1));
}
}
/* A simple Newton-Raphson method. */
long double
sqrtl (long double x)
{
long double delta, y;
int exponent;
/* Check for NaN */
if (isnanl (x))
return x;
/* Check for negative numbers */
if (x < 0.0L)
return (long double) sqrt (-1);
/* Check for zero and infinites */
if (x + x == x)
return x;
frexpl (x, &exponent);
y = ldexpl (x, -exponent / 2);
do
{
delta = y;
y = (y + x / y) * 0.5L;
delta -= y;
}
while (delta != 0.0L);
return y;
}
long double
tanhl(long double x) {
long double t, y, z;
int signx;
#ifndef lint
volatile long double dummy;
#endif
if (isnanl(x))
return (x + x); /* x is NaN */
signx = signbitl(x);
t = fabsl(x);
z = one;
if (t <= threshold) {
if (t > one)
z = one - two / (expm1l(t + t) + two);
else if (t > small) {
y = expm1l(-t - t);
z = -y / (y + two);
} else {
#ifndef lint
dummy = t + big;
/* inexact if t != 0 */
#endif
return (x);
}
} else if (!finitel(t))
return (copysignl(one, x));
else
return (signx ? -z + small * small : z - small * small);
return (signx ? -z : z);
}
long double
asinl(long double x) {
long double t, w;
#ifndef lint
volatile long double dummy;
#endif
w = fabsl(x);
if (isnanl(x))
return (x + x);
else if (w <= half) {
if (w < small) {
#ifndef lint
dummy = w + big;
/* inexact if w != 0 */
#endif
return (x);
} else
return (atanl(x / sqrtl(one - x * x)));
} else if (w < one) {
t = one - w;
w = t + t;
return (atanl(x / sqrtl(w - t * t)));
} else if (w == one)
return (atan2l(x, zero)); /* asin(+-1) = +- PI/2 */
else
return (zero / zero); /* |x| > 1: invalid */
}
long double
acoshl(long double x) {
long double t;
if (isnanl(x))
return (x + x);
else if (x > big)
return (logl(x) + ln2);
else if (x > one) {
t = sqrtl(x - one);
return (log1pl(t * (t + sqrtl(x + one))));
} else if (x == one)
return (zero);
else
return ((x - x) / (x - x));
}
long double
asinhl(long double x) {
long double t, w;
w = fabsl(x);
if (isnanl(x))
return (x + x); /* x is NaN */
if (w < tiny) {
#ifndef lint
volatile long double dummy = x + big; /* inexact if x != 0 */
#endif
return (x); /* tiny x */
} else if (w < big) {
t = one / w;
return (copysignl(log1pl(w + w / (t + sqrtl(one + t * t))), x));
} else
return (copysignl(logl(w) + ln2, x));
}
int
main ()
{
DECL_LONG_DOUBLE_ROUNDING
BEGIN_LONG_DOUBLE_ROUNDING ();
/* Zero. */
ASSERT (roundl (0.0L) == 0.0L);
ASSERT (roundl (minus_zerol) == 0.0L);
/* Positive numbers. */
ASSERT (roundl (0.3L) == 0.0L);
ASSERT (roundl (0.5L) == 1.0L);
ASSERT (roundl (0.7L) == 1.0L);
ASSERT (roundl (1.0L) == 1.0L);
ASSERT (roundl (1.5L) == 2.0L);
ASSERT (roundl (2.5L) == 3.0L);
ASSERT (roundl (1.999L) == 2.0L);
ASSERT (roundl (2.0L) == 2.0L);
ASSERT (roundl (65535.999L) == 65536.0L);
ASSERT (roundl (65536.0L) == 65536.0L);
ASSERT (roundl (65536.001L) == 65536.0L);
ASSERT (roundl (2.341e31L) == 2.341e31L);
/* Negative numbers. */
ASSERT (roundl (-0.3L) == 0.0L);
ASSERT (roundl (-0.5L) == -1.0L);
ASSERT (roundl (-0.7L) == -1.0L);
ASSERT (roundl (-1.0L) == -1.0L);
ASSERT (roundl (-1.5L) == -2.0L);
ASSERT (roundl (-2.5L) == -3.0L);
ASSERT (roundl (-1.999L) == -2.0L);
ASSERT (roundl (-2.0L) == -2.0L);
ASSERT (roundl (-65535.999L) == -65536.0L);
ASSERT (roundl (-65536.0L) == -65536.0L);
ASSERT (roundl (-65536.001L) == -65536.0L);
ASSERT (roundl (-2.341e31L) == -2.341e31L);
/* Infinite numbers. */
ASSERT (roundl (Infinityl ()) == Infinityl ());
ASSERT (roundl (- Infinityl ()) == - Infinityl ());
/* NaNs. */
ASSERT (isnanl (roundl (NaNl ())));
return 0;
}
long double
log2l (long double x)
{
if (isnanl (x))
return x;
if (x <= 0.0L)
{
if (x == 0.0L)
/* Return -Infinity. */
return - HUGE_VALL;
else
{
/* Return NaN. */
#if defined _MSC_VER || (defined __sgi && !defined __GNUC__)
static long double zero;
return zero / zero;
#else
return 0.0L / 0.0L;
#endif
}
}
/* Decompose x into
x = 2^e * y
where
e is an integer,
1/2 < y < 2.
Then log2(x) = e + log2(y) = e + log(y)/log(2). */
{
int e;
long double y;
y = frexpl (x, &e);
if (y < SQRT_HALF)
{
y = 2.0L * y;
e = e - 1;
}
return (long double) e + logl (y) * LOG2_INVERSE;
}
}
long double
ldexpl (long double x, int exp)
{
long double factor;
int bit;
DECL_LONG_DOUBLE_ROUNDING
BEGIN_LONG_DOUBLE_ROUNDING ();
/* Check for zero, nan and infinity. */
if (!(isnanl (x) || x + x == x))
{
if (exp < 0)
{
exp = -exp;
factor = 0.5L;
}
else
factor = 2.0L;
if (exp > 0)
for (bit = 1;;)
{
/* Invariant: Here bit = 2^i, factor = 2^-2^i or = 2^2^i,
and bit <= exp. */
if (exp & bit)
x *= factor;
bit <<= 1;
if (bit > exp)
break;
factor = factor * factor;
}
}
END_LONG_DOUBLE_ROUNDING ();
return x;
}
long double
cosl (long double x)
{
long double y[2],z=0.0L;
int n;
/* cosl(NaN) is NaN */
if (isnanl (x))
return x;
/* |x| ~< pi/4 */
if (x >= -0.7853981633974483096156608458198757210492
&& x <= 0.7853981633974483096156608458198757210492)
return kernel_cosl(x, z);
/* cosl(Inf) is NaN, cosl(0) is 1 */
else if (x + x == x && x != 0.0)
return x - x; /* NaN */
/* argument reduction needed */
else
{
n = ieee754_rem_pio2l (x, y);
switch (n & 3)
{
case 0:
return kernel_cosl (y[0], y[1]);
case 1:
return -kernel_sinl (y[0], y[1], 1);
case 2:
return -kernel_cosl (y[0], y[1]);
default:
return kernel_sinl (y[0], y[1], 1);
}
}
}
int
main ()
{
int i;
long double x;
DECL_LONG_DOUBLE_ROUNDING
BEGIN_LONG_DOUBLE_ROUNDING ();
{ /* NaN. */
int exp = -9999;
long double mantissa;
x = 0.0L / 0.0L;
mantissa = frexpl (x, &exp);
ASSERT (isnanl (mantissa));
}
{ /* Positive infinity. */
int exp = -9999;
long double mantissa;
x = 1.0L / 0.0L;
mantissa = frexpl (x, &exp);
ASSERT (mantissa == x);
}
{ /* Negative infinity. */
int exp = -9999;
long double mantissa;
x = -1.0L / 0.0L;
mantissa = frexpl (x, &exp);
ASSERT (mantissa == x);
}
{ /* Positive zero. */
int exp = -9999;
long double mantissa;
x = 0.0L;
mantissa = frexpl (x, &exp);
ASSERT (exp == 0);
ASSERT (mantissa == x);
ASSERT (!signbit (mantissa));
}
{ /* Negative zero. */
int exp = -9999;
long double mantissa;
x = minus_zero;
mantissa = frexpl (x, &exp);
ASSERT (exp == 0);
ASSERT (mantissa == x);
ASSERT (signbit (mantissa));
}
for (i = 1, x = 1.0L; i <= LDBL_MAX_EXP; i++, x *= 2.0L)
{
int exp = -9999;
long double mantissa = frexpl (x, &exp);
ASSERT (exp == i);
ASSERT (mantissa == 0.5L);
}
for (i = 1, x = 1.0L; i >= MIN_NORMAL_EXP; i--, x *= 0.5L)
{
int exp = -9999;
long double mantissa = frexpl (x, &exp);
ASSERT (exp == i);
ASSERT (mantissa == 0.5L);
}
for (; i >= LDBL_MIN_EXP - 100 && x > 0.0L; i--, x *= 0.5L)
{
int exp = -9999;
long double mantissa = frexpl (x, &exp);
ASSERT (exp == i);
ASSERT (mantissa == 0.5L);
}
for (i = 1, x = -1.0L; i <= LDBL_MAX_EXP; i++, x *= 2.0L)
{
int exp = -9999;
long double mantissa = frexpl (x, &exp);
ASSERT (exp == i);
ASSERT (mantissa == -0.5L);
}
for (i = 1, x = -1.0L; i >= MIN_NORMAL_EXP; i--, x *= 0.5L)
{
int exp = -9999;
long double mantissa = frexpl (x, &exp);
ASSERT (exp == i);
ASSERT (mantissa == -0.5L);
}
for (; i >= LDBL_MIN_EXP - 100 && x < 0.0L; i--, x *= 0.5L)
{
int exp = -9999;
long double mantissa = frexpl (x, &exp);
ASSERT (exp == i);
ASSERT (mantissa == -0.5L);
}
for (i = 1, x = 1.01L; i <= LDBL_MAX_EXP; i++, x *= 2.0L)
{
int exp = -9999;
long double mantissa = frexpl (x, &exp);
ASSERT (exp == i);
ASSERT (mantissa == 0.505L);
}
for (i = 1, x = 1.01L; i >= MIN_NORMAL_EXP; i--, x *= 0.5L)
{
int exp = -9999;
long double mantissa = frexpl (x, &exp);
ASSERT (exp == i);
ASSERT (mantissa == 0.505L);
}
for (; i >= LDBL_MIN_EXP - 100 && x > 0.0L; i--, x *= 0.5L)
{
int exp = -9999;
long double mantissa = frexpl (x, &exp);
ASSERT (exp == i);
ASSERT (mantissa >= 0.5L);
ASSERT (mantissa < 1.0L);
ASSERT (mantissa == my_ldexp (x, - exp));
}
for (i = 1, x = 1.73205L; i <= LDBL_MAX_EXP; i++, x *= 2.0L)
{
int exp = -9999;
long double mantissa = frexpl (x, &exp);
ASSERT (exp == i);
ASSERT (mantissa == 0.866025L);
}
for (i = 1, x = 1.73205L; i >= MIN_NORMAL_EXP; i--, x *= 0.5L)
{
int exp = -9999;
long double mantissa = frexpl (x, &exp);
ASSERT (exp == i);
ASSERT (mantissa == 0.866025L);
}
for (; i >= LDBL_MIN_EXP - 100 && x > 0.0L; i--, x *= 0.5L)
{
int exp = -9999;
long double mantissa = frexpl (x, &exp);
ASSERT (exp == i || exp == i + 1);
ASSERT (mantissa >= 0.5L);
ASSERT (mantissa < 1.0L);
ASSERT (mantissa == my_ldexp (x, - exp));
}
return 0;
}
long double
exp2l (long double x)
{
/* exp2(x) = exp(x*log(2)).
If we would compute it like this, there would be rounding errors for
integer or near-integer values of x. To avoid these, we inline the
algorithm for exp(), and the multiplication with log(2) cancels a
division by log(2). */
if (isnanl (x))
return x;
if (x > (long double) LDBL_MAX_EXP)
/* x > LDBL_MAX_EXP
hence exp2(x) > 2^LDBL_MAX_EXP, overflows to Infinity. */
return HUGE_VALL;
if (x < (long double) (LDBL_MIN_EXP - 1 - LDBL_MANT_DIG))
/* x < (LDBL_MIN_EXP - 1 - LDBL_MANT_DIG)
hence exp2(x) < 2^(LDBL_MIN_EXP-1-LDBL_MANT_DIG),
underflows to zero. */
return 0.0L;
/* Decompose x into
x = n + m/256 + y/log(2)
where
n is an integer,
m is an integer, -128 <= m <= 128,
y is a number, |y| <= log(2)/512 + epsilon = 0.00135...
Then
exp2(x) = 2^n * exp(m * log(2)/256) * exp(y)
The first factor is an ldexpl() call.
The second factor is a table lookup.
The third factor is computed
- either as sinh(y) + cosh(y)
where sinh(y) is computed through the power series:
sinh(y) = y + y^3/3! + y^5/5! + ...
and cosh(y) is computed as hypot(1, sinh(y)),
- or as exp(2*z) = (1 + tanh(z)) / (1 - tanh(z))
where z = y/2
and tanh(z) is computed through its power series:
tanh(z) = z
- 1/3 * z^3
+ 2/15 * z^5
- 17/315 * z^7
+ 62/2835 * z^9
- 1382/155925 * z^11
+ 21844/6081075 * z^13
- 929569/638512875 * z^15
+ ...
Since |z| <= log(2)/1024 < 0.0007, the relative contribution of the
z^13 term is < 0.0007^12 < 2^-120 <= 2^-LDBL_MANT_DIG, therefore we
can truncate the series after the z^11 term. */
{
long double nm = roundl (x * 256.0L); /* = 256 * n + m */
long double z = (x * 256.0L - nm) * (LOG2_BY_256 * 0.5L);
/* Coefficients of the power series for tanh(z). */
#define TANH_COEFF_1 1.0L
#define TANH_COEFF_3 -0.333333333333333333333333333333333333334L
#define TANH_COEFF_5 0.133333333333333333333333333333333333334L
#define TANH_COEFF_7 -0.053968253968253968253968253968253968254L
#define TANH_COEFF_9 0.0218694885361552028218694885361552028218L
#define TANH_COEFF_11 -0.00886323552990219656886323552990219656886L
#define TANH_COEFF_13 0.00359212803657248101692546136990581435026L
#define TANH_COEFF_15 -0.00145583438705131826824948518070211191904L
long double z2 = z * z;
long double tanh_z =
(((((TANH_COEFF_11
* z2 + TANH_COEFF_9)
* z2 + TANH_COEFF_7)
* z2 + TANH_COEFF_5)
* z2 + TANH_COEFF_3)
* z2 + TANH_COEFF_1)
* z;
long double exp_y = (1.0L + tanh_z) / (1.0L - tanh_z);
int n = (int) roundl (nm * (1.0L / 256.0L));
int m = (int) nm - 256 * n;
return ldexpl (gl_expl_table[128 + m] * exp_y, n);
}
}
ldcomplex
cpowl(ldcomplex z, ldcomplex w) {
ldcomplex ans;
long double x, y, u, v, t, c, s, r;
long double t1, t2, t3, t4, x1, x2, y1, y2, u1, v1, b[4], w1, w2;
int ix, iy, hx, hy, hv, hu, iu, iv, i, j, k;
x = LD_RE(z);
y = LD_IM(z);
u = LD_RE(w);
v = LD_IM(w);
hx = HI_XWORD(x);
hy = HI_XWORD(y);
hu = HI_XWORD(u);
hv = HI_XWORD(v);
ix = hx & 0x7fffffff;
iy = hy & 0x7fffffff;
iu = hu & 0x7fffffff;
iv = hv & 0x7fffffff;
j = 0;
if (v == zero) { /* z**(real) */
if (u == one) { /* (anything) ** 1 is itself */
LD_RE(ans) = x;
LD_IM(ans) = y;
} else if (u == zero) { /* (anything) ** 0 is 1 */
LD_RE(ans) = one;
LD_IM(ans) = zero;
} else if (y == zero) { /* real ** real */
LD_IM(ans) = zero;
if (hx < 0 && ix < hiinf && iu < hiinf) {
/* -x ** u is exp(i*pi*u)*pow(x,u) */
r = powl(-x, u);
sincospil(u, &s, &c);
LD_RE(ans) = (c == zero)? c: c * r;
LD_IM(ans) = (s == zero)? s: s * r;
} else
LD_RE(ans) = powl(x, u);
} else if (x == zero || ix >= hiinf || iy >= hiinf) {
if (isnanl(x) || isnanl(y) || isnanl(u))
LD_RE(ans) = LD_IM(ans) = x + y + u;
else {
if (x == zero)
r = fabsl(y);
else
r = fabsl(x) + fabsl(y);
t = atan2pil(y, x);
sincospil(t * u, &s, &c);
LD_RE(ans) = (c == zero)? c: c * r;
LD_IM(ans) = (s == zero)? s: s * r;
}
} else if (fabsl(x) == fabsl(y)) { /* |x| = |y| */
if (hx >= 0) {
t = (hy >= 0)? 0.25L : -0.25L;
sincospil(t * u, &s, &c);
} else if ((LAST(u) & 3) == 0) {
t = (hy >= 0)? 0.75L : -0.75L;
sincospil(t * u, &s, &c);
} else {
r = (hy >= 0)? u : -u;
t = -0.25L * r;
w1 = r + t;
w2 = t - (w1 - r);
sincospil(w1, &t1, &t2);
sincospil(w2, &t3, &t4);
s = t1 * t4 + t3 * t2;
c = t2 * t4 - t1 * t3;
}
if (ix < 0x3ffe0000) /* |x| < 1/2 */
r = powl(fabsl(x + x), u) * exp2l(-0.5L * u);
else if (ix >= 0x3fff0000 || iu < 0x400cfff8)
/* |x| >= 1 or |u| < 16383 */
r = powl(fabsl(x), u) * exp2l(0.5L * u);
else /* special treatment */
j = 2;
if (j == 0) {
LD_RE(ans) = (c == zero)? c: c * r;
LD_IM(ans) = (s == zero)? s: s * r;
}
} else
j = 1;
if (j == 0)
return (ans);
}
if (iu >= hiinf || iv >= hiinf || ix >= hiinf || iy >= hiinf) {
/*
* non-zero imag part(s) with inf component(s) yields NaN
*/
t = fabsl(x) + fabsl(y) + fabsl(u) + fabsl(v);
LD_RE(ans) = LD_IM(ans) = t - t;
} else {
k = 0; /* no scaling */
if (iu > 0x7ffe0000 || iv > 0x7ffe0000) {
u *= 1.52587890625000000000e-05L;
v *= 1.52587890625000000000e-05L;
k = 1; /* scale u and v by 2**-16 */
}
/*
* Use similated higher precision arithmetic to compute:
* r = u * log(hypot(x, y)) - v * atan2(y, x)
* q = u * atan2(y, x) + v * log(hypot(x, y))
*/
t1 = __k_clog_rl(x, y, &t2);
t3 = __k_atan2l(y, x, &t4);
x1 = t1; HALF(x1);
y1 = t3; HALF(y1);
u1 = u; HALF(u1);
v1 = v; HALF(v1);
x2 = t2 - (x1 - t1); /* log(hypot(x,y)) = x1 + x2 */
y2 = t4 - (y1 - t3); /* atan2(y,x) = y1 + y2 */
/* compute q = u * atan2(y, x) + v * log(hypot(x, y)) */
if (j != 2) {
b[0] = u1 * y1;
b[1] = (u - u1) * y1 + u * y2;
if (j == 1) { /* v = 0 */
w1 = b[0] + b[1];
w2 = b[1] - (w1 - b[0]);
} else {
b[2] = v1 * x1;
b[3] = (v - v1) * x1 + v * x2;
w1 = sum4fpl(b, &w2);
}
sincosl(w1, &t1, &t2);
sincosl(w2, &t3, &t4);
s = t1 * t4 + t3 * t2;
c = t2 * t4 - t1 * t3;
if (k == 1) /* square j times */
for (i = 0; i < 10; i++) {
t1 = s * c;
c = (c + s) * (c - s);
s = t1 + t1;
}
}
/* compute r = u * (t1, t2) - v * (t3, t4) */
b[0] = u1 * x1;
b[1] = (u - u1) * x1 + u * x2;
if (j == 1) { /* v = 0 */
w1 = b[0] + b[1];
w2 = b[1] - (w1 - b[0]);
} else {
b[2] = -v1 * y1;
b[3] = (v1 - v) * y1 - v * y2;
w1 = sum4fpl(b, &w2);
}
/* scale back unless w1 is large enough to cause exception */
if (k != 0 && fabsl(w1) < 20000.0L) {
w1 *= 65536.0L; w2 *= 65536.0L;
}
hx = HI_XWORD(w1);
ix = hx & 0x7fffffff;
/* compute exp(w1 + w2) */
k = 0;
if (ix < 0x3f8c0000) /* exp(tiny < 2**-115) = 1 */
r = one;
else if (ix >= 0x400c6760) /* overflow/underflow */
r = (hx < 0)? tiny * tiny : huge * huge;
else { /* compute exp(w1 + w2) */
k = (int) (invln2 * w1 + ((hx >= 0)? 0.5L : -0.5L));
t1 = (long double) k;
t2 = w1 - t1 * ln2hil;
t3 = w2 - t1 * ln2lol;
r = expl(t2 + t3);
}
if (c != zero) c *= r;
if (s != zero) s *= r;
if (k != 0) {
c = scalbnl(c, k);
s = scalbnl(s, k);
}
LD_RE(ans) = c;
LD_IM(ans) = s;
}
return (ans);
}
long double
expm1l (long double x)
{
if (isnanl (x))
return x;
if (x >= (long double) LDBL_MAX_EXP * LOG2_PLUS_EPSILON)
/* x > LDBL_MAX_EXP * log(2)
hence exp(x) > 2^LDBL_MAX_EXP, overflows to Infinity. */
return HUGE_VALL;
if (x <= (long double) (- LDBL_MANT_DIG) * LOG2_PLUS_EPSILON)
/* x < (- LDBL_MANT_DIG) * log(2)
hence 0 < exp(x) < 2^-LDBL_MANT_DIG,
hence -1 < exp(x)-1 < -1 + 2^-LDBL_MANT_DIG
rounds to -1. */
return -1.0L;
if (x <= - LOG2_PLUS_EPSILON)
/* 0 < exp(x) < 1/2.
Just compute exp(x), then subtract 1. */
return expl (x) - 1.0L;
if (x == 0.0L)
/* Return a zero with the same sign as x. */
return x;
/* Decompose x into
x = n * log(2) + m * log(2)/256 + y
where
n is an integer, n >= -1,
m is an integer, -128 <= m <= 128,
y is a number, |y| <= log(2)/512 + epsilon = 0.00135...
Then
exp(x) = 2^n * exp(m * log(2)/256) * exp(y)
Compute each factor minus one, then combine them through the
formula (1+a)*(1+b) = 1 + (a+b*(1+a)),
that is (1+a)*(1+b) - 1 = a + b*(1+a).
The first factor is an ldexpl() call.
The second factor is a table lookup.
The third factor minus one is computed
- either as sinh(y) + sinh(y)^2 / (cosh(y) + 1)
where sinh(y) is computed through the power series:
sinh(y) = y + y^3/3! + y^5/5! + ...
and cosh(y) is computed as hypot(1, sinh(y)),
- or as exp(2*z) - 1 = 2 * tanh(z) / (1 - tanh(z))
where z = y/2
and tanh(z) is computed through its power series:
tanh(z) = z
- 1/3 * z^3
+ 2/15 * z^5
- 17/315 * z^7
+ 62/2835 * z^9
- 1382/155925 * z^11
+ 21844/6081075 * z^13
- 929569/638512875 * z^15
+ ...
Since |z| <= log(2)/1024 < 0.0007, the relative contribution of the
z^13 term is < 0.0007^12 < 2^-120 <= 2^-LDBL_MANT_DIG, therefore we
can truncate the series after the z^11 term.
Given the usual bounds LDBL_MAX_EXP <= 16384, LDBL_MANT_DIG <= 120, we
can estimate x: -84 <= x <= 11357.
This means, when dividing x by log(2), where we want x mod log(2)
to be precise to LDBL_MANT_DIG bits, we have to use an approximation
to log(2) that has 14+LDBL_MANT_DIG bits. */
{
long double nm = roundl (x * LOG2_BY_256_INVERSE); /* = 256 * n + m */
/* n has at most 15 bits, nm therefore has at most 23 bits, therefore
n * LOG2_HI_PART is computed exactly, and n * LOG2_LO_PART is computed
with an absolute error < 2^15 * 2e-10 * 2^-LDBL_MANT_DIG. */
long double y_tmp = x - nm * LOG2_BY_256_HI_PART;
long double y = y_tmp - nm * LOG2_BY_256_LO_PART;
long double z = 0.5L * y;
/* Coefficients of the power series for tanh(z). */
#define TANH_COEFF_1 1.0L
#define TANH_COEFF_3 -0.333333333333333333333333333333333333334L
#define TANH_COEFF_5 0.133333333333333333333333333333333333334L
#define TANH_COEFF_7 -0.053968253968253968253968253968253968254L
#define TANH_COEFF_9 0.0218694885361552028218694885361552028218L
#define TANH_COEFF_11 -0.00886323552990219656886323552990219656886L
#define TANH_COEFF_13 0.00359212803657248101692546136990581435026L
#define TANH_COEFF_15 -0.00145583438705131826824948518070211191904L
long double z2 = z * z;
long double tanh_z =
(((((TANH_COEFF_11
* z2 + TANH_COEFF_9)
* z2 + TANH_COEFF_7)
* z2 + TANH_COEFF_5)
* z2 + TANH_COEFF_3)
* z2 + TANH_COEFF_1)
* z;
long double exp_y_minus_1 = 2.0L * tanh_z / (1.0L - tanh_z);
int n = (int) roundl (nm * (1.0L / 256.0L));
int m = (int) nm - 256 * n;
/* expm1l_table[i] = exp((i - 128) * log(2)/256) - 1.
Computed in GNU clisp through
(setf (long-float-digits) 128)
(setq a 0L0)
(setf (long-float-digits) 256)
(dotimes (i 257)
(format t " ~D,~%"
(float (- (exp (* (/ (- i 128) 256) (log 2L0))) 1) a))) */
static const long double expm1l_table[257] =
{
-0.292893218813452475599155637895150960716L,
-0.290976057839792401079436677742323809165L,
-0.289053698915417220095325702647879950038L,
-0.287126127947252846596498423285616993819L,
-0.285193330804014994382467110862430046956L,
-0.283255293316105578740250215722626632811L,
-0.281312001275508837198386957752147486471L,
-0.279363440435687168635744042695052413926L,
-0.277409596511476689981496879264164547161L,
-0.275450455178982509740597294512888729286L,
-0.273486002075473717576963754157712706214L,
-0.271516222799278089184548475181393238264L,
-0.269541102909676505674348554844689233423L,
-0.267560627926797086703335317887720824384L,
-0.265574783331509036569177486867109287348L,
-0.263583554565316202492529493866889713058L,
-0.261586927030250344306546259812975038038L,
-0.259584886088764114771170054844048746036L,
-0.257577417063623749727613604135596844722L,
-0.255564505237801467306336402685726757248L,
-0.253546135854367575399678234256663229163L,
-0.251522294116382286608175138287279137577L,
-0.2494929651867872398674385184702356751864L,
-0.247458134188296727960327722100283867508L,
-0.24541778620328863011699022448340323429L,
-0.243371906273695048903181511842366886387L,
-0.24132047940089265059510885341281062657L,
-0.239263490545592708236869372901757573532L,
-0.237200924627730846574373155241529522695L,
-0.23513276652635648805745654063657412692L,
-0.233059001079521999099699248246140670544L,
-0.230979613084171535783261520405692115669L,
-0.228894587296029588193854068954632579346L,
-0.226803908429489222568744221853864674729L,
-0.224707561157500020438486294646580877171L,
-0.222605530111455713940842831198332609562L,
-0.2204977998810815164831359552625710592544L,
-0.218384355014321147927034632426122058645L,
-0.2162651800172235534675441445217774245016L,
-0.214140259353829315375718509234297186439L,
-0.212009577446056756772364919909047495547L,
-0.209873118673587736597751517992039478005L,
-0.2077308673737531349400659265343210916196L,
-0.205582807841418027883101951185666435317L,
-0.2034289243288665510313756784404656320656L,
-0.201269201045686450868589852895683430425L,
-0.199103622158653323103076879204523186316L,
-0.196932171791614537151556053482436428417L,
-0.19475483402537284591023966632129970827L,
-0.192571592897569679960015418424270885733L,
-0.190382432402568125350119133273631796029L,
-0.188187336491335584102392022226559177731L,
-0.185986289071326116575890738992992661386L,
-0.183779274006362464829286135533230759947L,
-0.181566275116517756116147982921992768975L,
-0.17934727617799688564586793151548689933L,
-0.1771222609230175777406216376370887771665L,
-0.1748912130396911245164132617275148983224L,
-0.1726541161719028012138814282020908791644L,
-0.170410953919191957302175212789218768074L,
-0.168161709836631782476831771511804777363L,
-0.165906367434708746670203829291463807099L,
-0.1636449101792017131905953879307692887046L,
-0.161377321491060724103867675441291294819L,
-0.15910358474628545696887452376678510496L,
-0.15682368327580335203567701228614769857L,
-0.154537600365347409013071332406381692911L,
-0.152245319255333652509541396360635796882L,
-0.149946823140738265249318713251248832456L,
-0.147642095170974388162796469615281683674L,
-0.145331118449768586448102562484668501975L,
-0.143013876035036980698187522160833990549L,
-0.140690350938761042185327811771843747742L,
-0.138360526126863051392482883127641270248L,
-0.136024384519081218878475585385633792948L,
-0.133681908988844467561490046485836530346L,
-0.131333082363146875502898959063916619876L,
-0.128977887422421778270943284404535317759L,
-0.126616306900415529961291721709773157771L,
-0.1242483234840609219490048572320697039866L,
-0.121873919813350258443919690312343389353L,
-0.1194930784812080879189542126763637438278L,
-0.11710578203336358947830887503073906297L,
-0.1147120129682226132300120925687579825894L,
-0.1123117537367393737247203999003383961205L,
-0.1099049867422877955201404475637647649574L,
-0.1074916943405325099278897180135900838485L,
-0.1050718588392995019970556101123417014993L,
-0.102645462498446406786148378936109092823L,
-0.1002124875297324539725723033374854302454L,
-0.097772916096688059846161368344495155786L,
-0.0953267303144840657307406742107731280055L,
-0.092873912249800621875082699818829828767L,
-0.0904144439206957158520284361718212536293L,
-0.0879483072964733445019372468353990225585L,
-0.0854754842975513284540160873038416459095L,
-0.0829959567953287682564584052058555719614L,
-0.080509706612053141143695628825336081184L,
-0.078016715520687037466429613329061550362L,
-0.075516965244774535807472733052603963221L,
-0.073010437458307215803773464831151680239L,
-0.070497113785589807692349282254427317595L,
-0.067976975801105477595185454402763710658L,
-0.0654500050293807475554878955602008567352L,
-0.06291618294485004933500052502277673278L,
-0.0603754909717199109794126487955155117284L,
-0.0578279104838327751561896480162548451191L,
-0.055273422804530448266460732621318468453L,
-0.0527120092065171793298906732865376926237L,
-0.0501436509117223676387482401930039000769L,
-0.0475683290911628981746625337821392744829L,
-0.044986024864805103778829470427200864833L,
-0.0423967193014263530636943648520845560749L,
-0.0398003934184762630513928111129293882558L,
-0.0371970281819375355214808849088086316225L,
-0.0345866045061864160477270517354652168038L,
-0.0319691032538527747009720477166542375817L,
-0.0293445052356798073922893825624102948152L,
-0.0267127912103833568278979766786970786276L,
-0.0240739418845108520444897665995250062307L,
-0.0214279379122998654908388741865642544049L,
-0.018774759895536286618755114942929674984L,
-0.016114388383412110943633198761985316073L,
-0.01344680387238284353202993186779328685225L,
-0.0107719868060245158708750409344163322253L,
-0.00808991757489031507008688867384418356197L,
-0.00540057651636682434752231377783368554176L,
-0.00270394391452987374234008615207739887604L,
0.0L,
0.00271127505020248543074558845036204047301L,
0.0054299011128028213513839559347998147001L,
0.00815589811841751578309489081720103927357L,
0.0108892860517004600204097905618605243881L,
0.01363008495148943884025892906393992959584L,
0.0163783149109530379404931137862940627635L,
0.0191339960777379496848780958207928793998L,
0.0218971486541166782344801347832994397821L,
0.0246677928971356451482890762708149276281L,
0.0274459491187636965388611939222137814994L,
0.0302316376860410128717079024539045670944L,
0.0330248790212284225001082839704609180866L,
0.0358256936019571200299832090180813718441L,
0.0386341019613787906124366979546397325796L,
0.0414501246883161412645460790118931264803L,
0.0442737824274138403219664787399290087847L,
0.0471050958792898661299072502271122405627L,
0.049944085800687266082038126515907909062L,
0.0527907730046263271198912029807463031904L,
0.05564517836055715880834132515293865216L,
0.0585073227945126901057721096837166450754L,
0.0613772272892620809505676780038837262945L,
0.0642549128844645497886112570015802206798L,
0.0671404006768236181695211209928091626068L,
0.070033711820241773542411936757623568504L,
0.0729348675259755513850354508738275853402L,
0.0758438890627910378032286484760570740623L,
0.0787607977571197937406800374384829584908L,
0.081685614993215201942115594422531125645L,
0.0846183622133092378161051719066143416095L,
0.0875590609177696653467978309440397078697L,
0.090507732665257659207010655760707978993L,
0.0934643990728858542282201462504471620805L,
0.096429081816376823386138295859248481766L,
0.099401802630221985463696968238829904039L,
0.1023825833078409435564142094256468575113L,
0.1053714457017412555882746962569503110404L,
0.1083684117236786380094236494266198501387L,
0.111373503344817603850149254228916637444L,
0.1143867425958925363088129569196030678004L,
0.1174081515673691990545799630857802666544L,
0.120437752409606684429003879866313012766L,
0.1234755673330198007337297397753214319548L,
0.1265216186082418997947986437870347776336L,
0.12957592856628814599726498884024982591L,
0.1326385195987192279870737236776230843835L,
0.135709414157805514240390330676117013429L,
0.1387886347566916537038302838415112547204L,
0.14187620396956162271229760828788093894L,
0.144972144431804219394413888222915895793L,
0.148076478840179006778799662697342680031L,
0.15118922995298270581775963520198253612L,
0.154310420590216039548221528724806960684L,
0.157440073633751029613085766293796821108L,
0.160578212027498746369459472576090986253L,
0.163724858777577513813573599092185312343L,
0.166880036952481570555516298414089287832L,
0.1700437696832501880802590357927385730016L,
0.1732160801636372475348043545132453888896L,
0.176396991650281276284645728483848641053L,
0.1795865274628759454861005667694405189764L,
0.182784710984341029924457204693850757963L,
0.185991565660993831371265649534215563735L,
0.189207115002721066717499970560475915293L,
0.192431382583151222142727558145431011481L,
0.1956643920398273745838370498654519757025L,
0.1989061670743804817703025579763002069494L,
0.202156731452703142096396957497765876L,
0.205416109005123825604211432558411335666L,
0.208684323626581577354792255889216998483L,
0.211961399276801194468168917732493045449L,
0.2152473599804688781165202513387984576236L,
0.218542229827408361758207148117394510722L,
0.221846032972757516903891841911570785834L,
0.225158793637145437709464594384845353705L,
0.2284805361068700056940089577927818403626L,
0.231811284734075935884556653212794816605L,
0.235151063936933305692912507415415760296L,
0.238499898199816567833368865859612431546L,
0.241857812073484048593677468726595605511L,
0.245224830175257932775204967486152674173L,
0.248600977189204736621766097302495545187L,
0.251986277866316270060206031789203597321L,
0.255380757024691089579390657442301194598L,
0.258784439549716443077860441815162618762L,
0.262197350394250708014010258518416459672L,
0.265619514578806324196273999873453036297L,
0.269050957191733222554419081032338004715L,
0.272491703389402751236692044184602176772L,
0.27594177839639210038120243475928938891L,
0.279401207505669226913587970027852545961L,
0.282870016078778280726669781021514051111L,
0.286348229546025533601482208069738348358L,
0.289835873406665812232747295491552189677L,
0.293332973229089436725559789048704304684L,
0.296839554651009665933754117792451159835L,
0.300355643379650651014140567070917791291L,
0.303881265191935898574523648951997368331L,
0.30741644593467724479715157747196172848L,
0.310961211524764341922991786330755849366L,
0.314515587949354658485983613383997794966L,
0.318079601266063994690185647066116617661L,
0.321653277603157514326511812330609226158L,
0.325236643159741294629537095498721674113L,
0.32882972420595439547865089632866510792L,
0.33243254708316144935164337949073577407L,
0.336045138204145773442627904371869759286L,
0.339667524053303005360030669724352576023L,
0.343299731186835263824217146181630875424L,
0.346941786232945835788173713229537282073L,
0.350593715892034391408522196060133960038L,
0.354255546936892728298014740140702804344L,
0.357927306212901046494536695671766697444L,
0.361609020638224755585535938831941474643L,
0.365300717204011815430698360337542855432L,
0.369002422974590611929601132982192832168L,
0.372714165087668369284997857144717215791L,
0.376435970754530100216322805518686960261L,
0.380167867260238095581945274358283464698L,
0.383909881963831954872659527265192818003L,
0.387662042298529159042861017950775988895L,
0.391424375771926187149835529566243446678L,
0.395196909966200178275574599249220994717L,
0.398979672538311140209528136715194969206L,
0.402772691220204706374713524333378817108L,
0.40657599381901544248361973255451684411L,
0.410389608217270704414375128268675481146L,
0.414213562373095048801688724209698078569L
};
long double t = expm1l_table[128 + m];
/* (1+t) * (1+exp_y_minus_1) - 1 = t + (1+t)*exp_y_minus_1 */
long double p_minus_1 = t + (1.0L + t) * exp_y_minus_1;
long double s = ldexpl (1.0L, n) - 1.0L;
/* (1+s) * (1+p_minus_1) - 1 = s + (1+s)*p_minus_1 */
return s + (1.0L + s) * p_minus_1;
}
}
long double
logl (long double x)
{
long double z, y, w;
long double t;
int k, e;
/* Check for IEEE special cases. */
/* log(NaN) = NaN. */
if (isnanl (x))
{
return x;
}
/* log(0) = -infinity. */
if (x == 0.0L)
{
return -0.5L / ZERO;
}
/* log ( x < 0 ) = NaN */
if (x < 0.0L)
{
return (x - x) / ZERO;
}
/* log (infinity) */
if (x + x == x)
{
return x + x;
}
/* Extract exponent and reduce domain to 0.703125 <= u < 1.40625 */
x = frexpl (x, &e);
if (x < 0.703125L)
{
x += x;
e--;
}
/* On this interval the table is not used due to cancellation error. */
if ((x <= 1.0078125L) && (x >= 0.9921875L))
{
z = x - 1.0L;
k = 64;
t = 1.0L;
}
else
{
k = floorl ((x - 0.5L) * 128.0L);
t = 0.5L + k / 128.0L;
z = (x - t) / t;
}
/* Series expansion of log(1+z). */
w = z * z;
y = ((((((((((((l15 * z
+ l14) * z
+ l13) * z
+ l12) * z
+ l11) * z
+ l10) * z
+ l9) * z
+ l8) * z
+ l7) * z
+ l6) * z
+ l5) * z
+ l4) * z
+ l3) * z * w;
y -= 0.5 * w;
y += e * ln2b; /* Base 2 exponent offset times ln(2). */
y += z;
y += logtbl[k-26]; /* log(t) - (t-1) */
y += (t - 1.0L);
y += e * ln2a;
return y;
}
int gl_isfinitel (long double x)
{
return !isnanl (x) && x - x == zerol;
}
long double
log1pl (long double x)
{
if (isnanl (x))
return x;
if (x <= -1.0L)
{
if (x == -1.0L)
/* Return -Infinity. */
return - HUGE_VALL;
else
{
/* Return NaN. */
#if defined _MSC_VER || (defined __sgi && !defined __GNUC__)
static long double zero;
return zero / zero;
#else
return 0.0L / 0.0L;
#endif
}
}
if (x < -0.5L || x > 1.0L)
return logl (1.0L + x);
/* Here -0.5 <= x <= 1.0. */
if (x == 0.0L)
/* Return a zero with the same sign as x. */
return x;
/* Decompose x into
1 + x = (1 + m/256) * (1 + y)
where
m is an integer, -128 <= m <= 256,
y is a number, |y| <= 1/256.
y is computed as
y = (256 * x - m) / (256 + m).
Then
log(1+x) = log(m/256) + log(1+y)
The first summand is a table lookup.
The second summand is computed
- either through the power series
log(1+y) = y
- 1/2 * y^2
+ 1/3 * y^3
- 1/4 * y^4
+ 1/5 * y^5
- 1/6 * y^6
+ 1/7 * y^7
- 1/8 * y^8
+ 1/9 * y^9
- 1/10 * y^10
+ 1/11 * y^11
- 1/12 * y^12
+ 1/13 * y^13
- 1/14 * y^14
+ 1/15 * y^15
- ...
- or as log(1+y) = log((1+z)/(1-z)) = 2 * atanh(z)
where z = y/(2+y)
and atanh(z) is computed through its power series:
atanh(z) = z
+ 1/3 * z^3
+ 1/5 * z^5
+ 1/7 * z^7
+ 1/9 * z^9
+ 1/11 * z^11
+ 1/13 * z^13
+ 1/15 * z^15
+ ...
Since |z| <= 1/511 < 0.002, the relative contribution of the z^15
term is < 1/15*0.002^14 < 2^-120 <= 2^-LDBL_MANT_DIG, therefore we
can truncate the series after the z^13 term. */
{
long double m = roundl (x * 256.0L);
long double y = ((x * 256.0L) - m) / (m + 256.0L);
long double z = y / (2.0L + y);
/* Coefficients of the power series for atanh(z). */
#define ATANH_COEFF_1 1.0L
#define ATANH_COEFF_3 0.333333333333333333333333333333333333334L
#define ATANH_COEFF_5 0.2L
#define ATANH_COEFF_7 0.142857142857142857142857142857142857143L
#define ATANH_COEFF_9 0.1111111111111111111111111111111111111113L
#define ATANH_COEFF_11 0.090909090909090909090909090909090909091L
#define ATANH_COEFF_13 0.076923076923076923076923076923076923077L
#define ATANH_COEFF_15 0.066666666666666666666666666666666666667L
long double z2 = z * z;
long double atanh_z =
((((((ATANH_COEFF_13
* z2 + ATANH_COEFF_11)
* z2 + ATANH_COEFF_9)
* z2 + ATANH_COEFF_7)
* z2 + ATANH_COEFF_5)
* z2 + ATANH_COEFF_3)
* z2 + ATANH_COEFF_1)
* z;
/* logl_table[i] = log((i + 128) / 256).
Computed in GNU clisp through
(setf (long-float-digits) 128)
(setq a 0L0)
(setf (long-float-digits) 256)
(dotimes (i 385)
(format t " ~D,~%"
(float (log (* (/ (+ i 128) 256) 1L0)) a))) */
static const long double logl_table[385] =
{
-0.693147180559945309417232121458176568075L,
-0.6853650401178903604697692213970398044L,
-0.677642994023980055266378075415729732197L,
-0.669980121278410931188432960495886651496L,
-0.662375521893191621046203913861404403985L,
-0.65482831625780871022347679633437927773L,
-0.647337644528651106250552853843513225963L,
-0.639902666041133026551361927671647791137L,
-0.632522558743510466836625989417756304788L,
-0.625196518651437560022666843685547154042L,
-0.617923759322357783718626781474514153438L,
-0.61070351134887071814907205278986876216L,
-0.60353502187025817679728065207969203929L,
-0.59641755410139419712166106497071313106L,
-0.58935038687830174459117031769420187977L,
-0.582332814219655195222425952134964639978L,
-0.575364144903561854878438011987654863008L,
-0.568443702058988073553825606077313299585L,
-0.561570822771226036828515992768693405624L,
-0.554744857700826173731906247856527380683L,
-0.547965170715447412135297057717612244552L,
-0.541231138534103334345428696561292056747L,
-0.534542150383306725323860946832334992828L,
-0.527897607664638146541620672180936254347L,
-0.52129692363328608707713317540302930314L,
-0.514739523087127012297831879947234599722L,
-0.50822484206593331675332852879892694707L,
-0.50175232756031585480793331389686769463L,
-0.495321437230025429054660050261215099L,
-0.488931639131254417913411735261937295862L,
-0.482582411452595671747679308725825054355L,
-0.476273242259330949798142595713829069596L,
-0.470003629245735553650937031148342064701L,
-0.463773079495099479425751396412036696525L,
-0.457581109247178400339643902517133157939L,
-0.451427243672800141272924605544662667972L,
-0.445311016655364052636629355711651820077L,
-0.43923197057898186527990882355156990061L,
-0.4331896561230192424451526269158655235L,
-0.427183632062807368078106194920633178807L,
-0.421213465076303550585562626925177406092L,
-0.415278729556489003230882088534775334993L,
-0.409379007429300711070330899107921801414L,
-0.403513887976902632538339065932507598071L,
-0.397682967666109433030550215403212372894L,
-0.391885849981783528404356583224421075418L,
-0.386122145265033447342107580922798666387L,
-0.380391470556048421030985561769857535915L,
-0.374693449441410693606984907867576972481L,
-0.369027711905733333326561361023189215893L,
-0.363393894187477327602809309537386757124L,
-0.357791638638807479160052541644010369001L,
-0.352220593589352099112142921677820359633L,
-0.346680413213736728498769933032403617363L,
-0.341170757402767124761784665198737642087L,
-0.33569129163814153519122263131727209364L,
-0.330241686870576856279407775480686721935L,
-0.324821619401237656369001967407777741178L,
-0.31943077076636122859621528874235306143L,
-0.314068827624975851026378775827156709194L,
-0.308735481649613269682442058976885699557L,
-0.303430429419920096046768517454655701024L,
-0.298153372319076331310838085093194799765L,
-0.292904016432932602487907019463045397996L,
-0.287682072451780927439219005993827431504L,
-0.282487255574676923482925918282353780414L,
-0.277319285416234343803903228503274262719L,
-0.272177885915815673288364959951380595626L,
-0.267062785249045246292687241862699949179L,
-0.261973715741573968558059642502581569596L,
-0.256910413785027239068190798397055267412L,
-0.251872619755070079927735679796875342712L,
-0.2468600779315257978846419408385075613265L,
-0.24187253642048672427253973837916408939L,
-0.2369097470783577150364265832942468196375L,
-0.2319714654377751430492321958603212094726L,
-0.2270574506353460848586128739534071682175L,
-0.222167465341154296870334265401817316702L,
-0.2173012756899813951520225351537951559L,
-0.212458651214193401740613666010165016867L,
-0.2076393647782445016154410442673876674964L,
-0.202843192514751471266885961812429707545L,
-0.1980699137620937948192675366153429027185L,
-0.193319311003495979595900706211132426563L,
-0.188591169807550022358923589720001638093L,
-0.183885278770137362613157202229852743197L,
-0.179201429457710992616226033183958974965L,
-0.174539416351899677264255125093377869519L,
-0.169899036795397472900424896523305726435L,
-0.165280090939102924303339903679875604517L,
-0.160682381690473465543308397998034325468L,
-0.156105714663061654850502877304344269052L,
-0.1515498981272009378406898175577424691056L,
-0.1470147429618096590348349122269674042104L,
-0.142500062607283030157283942253263107981L,
-0.1380056730194437167017517619422725179055L,
-0.1335313926245226231463436209313499745895L,
-0.129077042275142343345847831367985856258L,
-0.124642445207276597338493356591214304499L,
-0.1202274269981598003244753948319154994493L,
-0.115831815525121705099120059938680166568L,
-0.1114554409253228268966213677328042273655L,
-0.1070981355563671005131126851708522185606L,
-0.1027597339577689347753154133345778104976L,
-0.098440072813252519902888574928971234883L,
-0.094138990913861910035632096996525066015L,
-0.0898563291218610470766469347968659624282L,
-0.0855919303354035139161469686670511961825L,
-0.0813456394539524058873423550293617843895L,
-0.077117303344431289769666193261475917783L,
-0.072906770808087780565737488890929711303L,
-0.0687138925480518083746933774035034481663L,
-0.064538521137571171672923915683992928129L,
-0.0603805109889074798714456529545968095868L,
-0.0562397183228760777967376942769773768851L,
-0.0521160011390140183616307870527840213665L,
-0.0480092191863606077520036253234446621373L,
-0.0439192339348354905263921515528654458042L,
-0.0398459085471996706586162402473026835046L,
-0.0357891078515852792753420982122404025613L,
-0.0317486983145803011569962827485256299276L,
-0.0277245480148548604671395114515163869272L,
-0.0237165266173160421183468505286730579517L,
-0.0197245053477785891192717326571593033246L,
-0.015748356968139168607549511460828269521L,
-0.0117879557520422404691605618900871263399L,
-0.0078431774610258928731840424909435816546L,
-0.00391389932113632909231778364357266484272L,
0.0L,
0.00389864041565732301393734309584290701073L,
0.00778214044205494894746290006113676367813L,
0.01165061721997527413559144280921434893315L,
0.0155041865359652541508540460424468358779L,
0.01934296284313093463590553454155047018545L,
0.0231670592815343782287991609622899165794L,
0.0269765876982020757480692925396595457815L,
0.0307716586667536883710282075967721640917L,
0.0345523815066597334073715005898328652816L,
0.038318864302136599193755325123797290346L,
0.042071213920687054375203805926962379448L,
0.045809536031294203166679267614663342114L,
0.049533935122276630882096208829824573267L,
0.0532445145188122828658701937865287769396L,
0.0569413764001384247590131015404494943015L,
0.0606246218164348425806061320404202632862L,
0.0642943507053972572162284502656114944857L,
0.0679506619085077493945652777726294140346L,
0.071593653187008817925605272752092034269L,
0.075223421237587525698605339983662414637L,
0.078840061707776024531540577859198294559L,
0.082443669211074591268160068668307805914L,
0.086034337341803153381797826721996075141L,
0.0896121586896871326199514693784845287854L,
0.093177224854183289768781353027759396216L,
0.096729626458551112295571056487463437015L,
0.1002694531636751493081301751297276601964L,
0.1037967936816435648260618037639746883066L,
0.1073117357890880506671750303711543368066L,
0.1108143663402901141948061693232119280986L,
0.1143047712800586336342591448151747734094L,
0.1177830356563834545387941094705217050686L,
0.1212492436328696851612122640808405265723L,
0.1247034785009572358634065153808632684918L,
0.128145822691930038174109886961074873852L,
0.1315763577887192725887161286894831624516L,
0.134995164537504830601983291147085645626L,
0.138402322859119135685325873601649187393L,
0.1417979118602573498789527352804727189846L,
0.1451820098444978972819350637405643235226L,
0.1485546943231371429098223170672938691604L,
0.151916042025841975071803424896884511328L,
0.1552661289111239515223833017101021786436L,
0.1586050301766385840933711746258415752456L,
0.161932820269313253240338285123614220592L,
0.165249572895307162875611449277240313729L,
0.1685553610298066669415865321701023169345L,
0.171850256926659222340098946055147264935L,
0.1751343321278491480142914649863898412374L,
0.1784076574728182971194002415109419683545L,
0.181670303107634678260605595617079739242L,
0.184922338494011992663903592659249621006L,
0.1881638324181829868259905803105539806714L,
0.191394852999629454609298807561308873447L,
0.194615467699671658858138593767269731516L,
0.1978257433299198803625720711969614690756L,
0.201025746060590741340908337591797808969L,
0.204215541428690891503820386196239272214L,
0.2073951943460705871587455788490062338536L,
0.210564769107349637669552812732351513721L,
0.2137243293977181388619051976331987647734L,
0.216873938300614359619089525744347498479L,
0.220013658305282095907358638661628360712L,
0.2231435513142097557662950903098345033745L,
0.226263678650453389361787082280390161607L,
0.229374101064845829991480725046139871551L,
0.232474878743094064920705078095567528222L,
0.235566071312766909077588218941043410137L,
0.2386477378501750099171491363522813392526L,
0.241719936887145168144307515913513900104L,
0.244782726417690916434704717466314811104L,
0.247836163904581256780602765746524747999L,
0.25088030628580941658844644154994089393L,
0.253915209980963444137323297906606667466L,
0.256940930897500425446759867911224262093L,
0.259957524436926066972079494542311044577L,
0.26296504550088135182072917321108602859L,
0.265963548497137941339125926537543389269L,
0.268953087345503958932974357924497845489L,
0.271933715483641758831669494532999161983L,
0.274905485872799249167009582983018668293L,
0.277868451003456306186350032923401233082L,
0.280822662900887784639519758873134832073L,
0.28376817313064459834690122235025476666L,
0.286705032803954314653250930842073965668L,
0.289633292583042676878893055525668970004L,
0.292553002686377439978201258664126644308L,
0.295464212893835876386681906054964195182L,
0.298366972551797281464900430293496918012L,
0.301261330578161781012875538233755492657L,
0.304147335467296717015819874720446989991L,
0.30702503529491186207512454053537790169L,
0.309894477722864687861624550833227164546L,
0.31275571000389688838624655968831903216L,
0.315608778986303334901366180667483174144L,
0.318453731118534615810247213590599595595L,
0.321290612453734292057863145522557457887L,
0.324119468654211976090670760434987352183L,
0.326940344995853320592356894073809191681L,
0.329753286372467981814422811920789810952L,
0.332558337300076601412275626573419425269L,
0.335355541921137830257179579814166199074L,
0.338144944008716397710235913939267433111L,
0.340926586970593210305089199780356208443L,
0.34370051385331844468019789211029452987L,
0.346466767346208580918462188425772950712L,
0.349225389785288304181275421187371759687L,
0.35197642315717818465544745625943892599L,
0.354719909102929028355011218999317665826L,
0.357455888921803774226009490140904474434L,
0.360184403575007796281574967493016620926L,
0.362905493689368453137824345977489846141L,
0.365619199560964711319396875217046453067L,
0.368325561158707653048230154050398826898L,
0.371024618127872663911964910806824955394L,
0.373716409793584080821016832715823506644L,
0.376400975164253065997877633436251593315L,
0.379078352934969458390853345631019858882L,
0.38174858149084833985966626493567607862L,
0.384411698910332039734790062481290868519L,
0.387067742968448287898902502261817665695L,
0.38971675114002521337046360400352086705L,
0.392358760602863872479379611988215363485L,
0.39499380824086897810639403636498176831L,
0.397621930647138489104829072973405554918L,
0.40024316412701270692932510199513117008L,
0.402857544701083514655197565487057707577L,
0.405465108108164381978013115464349136572L,
0.408065889808221748430198682969084124381L,
0.410659924985268385934306203175822787661L,
0.41324724855021933092547601552548590025L,
0.415827895143710965613328892954902305356L,
0.418401899138883817510763261966760106515L,
0.42096929464412963612886716150679597245L,
0.423530115505803295718430478017910109426L,
0.426084395310900063124544879595476618897L,
0.428632167389698760206812276426639053152L,
0.43117346481837134085917247895559499848L,
0.433708320421559393435847903042186017095L,
0.436236766774918070349041323061121300663L,
0.438758836207627937745575058511446738878L,
0.441274560804875229489496441661301225362L,
0.443783972410300981171768440588146426918L,
0.446287102628419511532590180619669006749L,
0.448783982827006710512822115683937186274L,
0.451274644139458585144692383079012478686L,
0.453759117467120506644794794442263270651L,
0.456237433481587594380805538163929748437L,
0.458709622626976664843883309250877913511L,
0.461175715122170166367999925597855358603L,
0.463635740963032513092182277331163919118L,
0.466089729924599224558619247504769399859L,
0.468537711563239270375665237462973542708L,
0.470979715218791012546897856056359251373L,
0.473415770016672131372578393236978550606L,
0.475845904869963914265209586304381412175L,
0.478270148481470280383546145497464809096L,
0.480688529345751907676618455448011551209L,
0.48310107575113582273837458485214554795L,
0.485507815781700807801791077190788900579L,
0.487908777319238973246173184132656942487L,
0.490303988045193838150346159645746860531L,
0.492693475442575255695076950020077845328L,
0.495077266797851514597964584842833665358L,
0.497455389202818942250859256731684928918L,
0.499827869556449329821331415247044141512L,
0.502194734566715494273584171951812573586L,
0.504556010752395287058308531738174929982L,
0.506911724444854354113196312660089270034L,
0.509261901789807946804074919228323824878L,
0.51160656874906207851888487520338193135L,
0.51394575110223431680100608827421759311L,
0.51627947444845449617281928478756106467L,
0.518607764208045632152976996364798698556L,
0.520930645624185312409809834659637709188L,
0.52324814376454783651680722493487084164L,
0.525560283522927371382427602307131424923L,
0.527867089620842385113892217778300963557L,
0.530168586609121617841419630845212405063L,
0.532464798869471843873923723460142242606L,
0.534755750616027675477923292032637111077L,
0.537041465896883654566729244153832299024L,
0.539321968595608874655355158077341155752L,
0.54159728243274437157654230390043409897L,
0.543867430967283517663338989065998323965L,
0.546132437598135650382397209231209163864L,
0.548392325565573162748150286179863158565L,
0.550647117952662279259948179204913460093L,
0.552896837686677737580717902230624314327L,
0.55514150754050159271548035951590405017L,
0.557381150134006357049816540361233647898L,
0.559615787935422686270888500526826593487L,
0.561845443262691817915664819160697456814L,
0.564070138284802966071384290090190711817L,
0.566289895023115872590849979337124343595L,
0.568504735352668712078738764866962263577L,
0.5707146810034715448536245647415894503L,
0.572919753561785509092756726626261068625L,
0.575119974471387940421742546569273429365L,
0.577315365034823604318112061519496401506L,
0.579505946414642223855274409488070989814L,
0.58169173963462248252061075372537234071L,
0.583872765580982679097413356975291104927L,
0.586049045003578208904119436287324349516L,
0.588220598517086043034868221609113995052L,
0.590387446602176374641916708123598757576L,
0.59254960960667159874199020959329739696L,
0.594707107746692789514343546529205333192L,
0.59685996110779383658731192302565801002L,
0.59900818964608339938160002446165150206L,
0.601151813189334836191674317068856441547L,
0.603290851438084262340585186661310605647L,
0.6054253239667168894375677681414899356L,
0.607555250224541795501085152791125371894L,
0.609680649536855273481833501660588408785L,
0.611801541105992903529889766428814783686L,
0.613917944012370492196929119645563790777L,
0.616029877215514019647565928196700650293L,
0.618137359555078733872689126674816271683L,
0.620240409751857528851494632567246856773L,
0.62233904640877874159710264120869663505L,
0.62443328801189350104253874405467311991L,
0.626523152931352759778820859734204069282L,
0.628608659422374137744308205774183639946L,
0.6306898256261987050837261409313532241L,
0.63276666957103782954578646850357975849L,
0.634839209173010211969493840510489008123L,
0.63690746223706923162049442718119919119L,
0.63897144645792072137962398326473680873L,
0.64103117942093129105560133440539254671L,
0.643086678603027315392053859585132960477L,
0.645137961373584701665228496134731905937L,
0.647185044995309550122320631377863036675L,
0.64922794662510981889083996990531112227L,
0.651266683314958103396333353349672108398L,
0.653301272012745638758615881210873884572L,
0.65533172956312763209494967856962559648L,
0.657358072708360030141890023245936165513L,
0.659380318089127826115336413370955804038L,
0.661398482245365008260235838709650938148L,
0.66341258161706625109695030429080128179L,
0.665422632545090448950092610006660181147L,
0.667428651271956189947234166318980478403L,
0.669430653942629267298885270929503510123L,
0.67142865660530232331713904200189252584L,
0.67342267521216672029796038880101726475L,
0.67541272562017673108090414397019748722L,
0.677398823591806140809682609997348298556L,
0.67938098479579735014710062847376425181L,
0.681359224807903068948071559568089441735L,
0.683333559111620688164363148387750369654L,
0.68530400309891941654404807896723298642L,
0.687270572070960267497006884394346103924L,
0.689233281238808980324914337814603903233L,
0.691192145724141958859604629216309755938L,
0.693147180559945309417232121458176568075L
};
return logl_table[128 + (int)m] + 2.0L * atanh_z;
}
}
long double __tgammal_r(long double x, int* sgngaml)
{
long double p, q, z;
int i;
*sgngaml = 1;
#ifdef NANS
if (isnanl(x))
return (NANL);
#endif
#ifdef INFINITIES
#ifdef NANS
if (x == INFINITYL)
return (x);
if (x == -INFINITYL)
return (NANL);
#else
if (!isfinite(x))
return (x);
#endif
#endif
q = fabsl(x);
if (q > 13.0L)
{
if (q > MAXGAML)
goto goverf;
if (x < 0.0L)
{
p = floorl(q);
if (p == q)
{
gsing:
_SET_ERRNO(EDOM);
mtherr("tgammal", SING);
#ifdef INFINITIES
return (INFINITYL);
#else
return (*sgngaml * MAXNUML);
#endif
}
i = p;
if ((i & 1) == 0)
*sgngaml = -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/MAXNUML)
{
goverf:
_SET_ERRNO(ERANGE);
mtherr("tgammal", OVERFLOW);
#ifdef INFINITIES
return(*sgngaml * INFINITYL);
#else
return(*sgngaml * MAXNUML);
#endif
}
z = PIL/z;
}
else
{
z = stirf(x);
}
return (*sgngaml * 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 );
return (z * p / q);
Small:
if (x == 0.0L)
{
goto gsing;
}
else
{
if (x < 0.0L)
{
x = -x;
q = z / (x * polevll(x, SN, 8));
}
else
q = z / (x * polevll(x, S, 8));
}
return q;
}
static int
print_float (struct printf_info *pinfo, char *startp, char *endp, int *signp, snv_long_double n)
{
int prec, fmtch;
char *p, *t;
snv_long_double fract;
int expcnt, gformat = 0;
snv_long_double integer, tmp;
char expbuf[10];
prec = pinfo->prec;
fmtch = pinfo->spec;
t = startp;
*signp = 0;
/* Do the special cases: nans, infinities, zero, and negative numbers. */
if (isnanl (n))
{
/* Not-a-numbers are printed as a simple string. */
*t++ = fmtch < 'a' ? 'N' : 'n';
*t++ = fmtch < 'a' ? 'A' : 'a';
*t++ = fmtch < 'a' ? 'N' : 'n';
return t - startp;
}
/* Zero and infinity also can have a sign in front of them. */
if (copysignl (1.0, n) < 0.0)
{
n = -1.0 * n;
*signp = '-';
}
if (isinfl (n))
{
/* Infinities are printed as a simple string. */
*t++ = fmtch < 'a' ? 'I' : 'i';
*t++ = fmtch < 'a' ? 'N' : 'n';
*t++ = fmtch < 'a' ? 'F' : 'f';
goto set_signp;
}
expcnt = 0;
fract = modfl (n, &integer);
/* get an extra slot for rounding. */
*t++ = '0';
/* get integer portion of number; put into the end of the buffer; the
.01 is added for modfl (356.0 / 10, &integer) returning .59999999... */
for (p = endp - 1; p >= startp && integer; ++expcnt)
{
tmp = modfl (integer / 10, &integer);
*p-- = '0' + ((int) ((tmp + .01L) * 10));
}
switch (fmtch)
{
case 'g':
case 'G':
gformat = 1;
/* a precision of 0 is treated as a precision of 1. */
if (!prec)
pinfo->prec = ++prec;
/* ``The style used depends on the value converted; style e
will be used only if the exponent resulting from the
conversion is less than -4 or greater than the precision.''
-- ANSI X3J11 */
if (expcnt > prec || (!expcnt && fract && fract < .0001L))
{
/* g/G format counts "significant digits, not digits of
precision; for the e/E format, this just causes an
off-by-one problem, i.e. g/G considers the digit
before the decimal point significant and e/E doesn't
count it as precision. */
--prec;
fmtch -= 2; /* G->E, g->e */
goto eformat;
}
else
{
/* Decrement precision */
if (n != 0.0L)
prec -= (endp - p) - 1;
else
prec--;
goto fformat;
}
case 'f':
case 'F':
fformat:
/* reverse integer into beginning of buffer */
if (expcnt)
for (; ++p < endp; *t++ = *p);
else
*t++ = '0';
/* If precision required or alternate flag set, add in a
decimal point. */
if (pinfo->prec || pinfo->alt)
*t++ = '.';
/* if requires more precision and some fraction left */
if (fract)
{
if (prec)
{
/* For %g, if no integer part, don't count initial
zeros as significant digits. */
do
{
fract = modfl (fract * 10, &tmp);
*t++ = '0' + ((int) tmp);
}
while (!tmp && !expcnt && gformat);
while (--prec && fract)
{
fract = modfl (fract * 10, &tmp);
*t++ = '0' + ((int) tmp);
}
}
if (fract)
startp =
print_float_round (fract, (int *) NULL, startp, t - 1, (char) 0, signp);
}
break;
case 'e':
case 'E':
eformat:
if (expcnt)
{
*t++ = *++p;
if (pinfo->prec || pinfo->alt)
*t++ = '.';
/* if requires more precision and some integer left */
for (; prec && ++p < endp; --prec)
*t++ = *p;
/* if done precision and more of the integer component,
round using it; adjust fract so we don't re-round
later. */
if (!prec && ++p < endp)
{
fract = 0;
startp = print_float_round ((snv_long_double) 0, &expcnt, startp, t - 1, *p, signp);
}
/* adjust expcnt for digit in front of decimal */
--expcnt;
}
/* until first fractional digit, decrement exponent */
else if (fract)
{
/* adjust expcnt for digit in front of decimal */
for (expcnt = -1;; --expcnt)
{
fract = modfl (fract * 10, &tmp);
if (tmp)
break;
}
*t++ = '0' + ((int) tmp);
if (pinfo->prec || pinfo->alt)
*t++ = '.';
}
else
{
*t++ = '0';
if (pinfo->prec || pinfo->alt)
*t++ = '.';
}
/* if requires more precision and some fraction left */
if (fract)
{
if (prec)
do
{
fract = modfl (fract * 10, &tmp);
*t++ = '0' + ((int) tmp);
}
while (--prec && fract);
if (fract)
startp = print_float_round (fract, &expcnt, startp, t - 1, (char) 0, signp);
}
break;
default:
abort ();
}
/* %e/%f/%#g add 0's for precision, others trim 0's */
if (gformat && !pinfo->alt)
{
while (t > startp && *--t == '0');
if (*t != '.')
++t;
}
else
for (; prec--; *t++ = '0');
if (fmtch == 'e' || fmtch == 'E')
{
*t++ = fmtch;
if (expcnt < 0)
{
expcnt = -expcnt;
*t++ = '-';
}
else
*t++ = '+';
p = expbuf;
do
*p++ = '0' + (expcnt % 10);
while ((expcnt /= 10) > 9);
*p++ = '0' + expcnt;
while (p > expbuf)
*t++ = *--p;
}
set_signp:
if (!*signp)
{
if (pinfo->showsign)
*signp = '+';
else if (pinfo->space)
*signp = ' ';
}
return (t - startp);
}
