double
__ieee754_lgamma_r(double x, int *signgamp)
{
double t,y,z,nadj,p,p1,p2,p3,q,r,w;
int i,hx,lx,ix;
EXTRACT_WORDS(hx,lx,x);
/* purge off +-inf, NaN, +-0, and negative arguments */
*signgamp = 1;
ix = hx&0x7fffffff;
if(__builtin_expect(ix>=0x7ff00000, 0)) return x*x;
if(__builtin_expect((ix|lx)==0, 0))
{
if (hx < 0)
*signgamp = -1;
return one/fabs(x);
}
if(__builtin_expect(ix<0x3b900000, 0)) {
/* |x|<2**-70, return -log(|x|) */
if(hx<0) {
*signgamp = -1;
return -__ieee754_log(-x);
} else return -__ieee754_log(x);
}
if(hx<0) {
if(__builtin_expect(ix>=0x43300000, 0))
/* |x|>=2**52, must be -integer */
return x/zero;
t = sin_pi(x);
if(t==zero) return one/fabsf(t); /* -integer */
nadj = __ieee754_log(pi/fabs(t*x));
if(t<zero) *signgamp = -1;
x = -x;
}
/* purge off 1 and 2 */
if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
/* for x < 2.0 */
else if(ix<0x40000000) {
if(ix<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */
r = -__ieee754_log(x);
if(ix>=0x3FE76944) {y = one-x; i= 0;}
else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
else {y = x; i=2;}
} else {
r = zero;
if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
else {y=x-one;i=2;}
}
switch(i) {
case 0:
z = y*y;
p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
p = y*p1+p2;
r += (p-0.5*y); break;
case 1:
z = y*y;
w = z*y;
p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */
p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
p = z*p1-(tt-w*(p2+y*p3));
r += (tf + p); break;
case 2:
p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
r += (-0.5*y + p1/p2);
}
}
else if(ix<0x40200000) { /* x < 8.0 */
i = (int)x;
t = zero;
y = x-(double)i;
p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
r = half*y+p/q;
z = one; /* lgamma(1+s) = log(s) + lgamma(s) */
switch(i) {
case 7: z *= (y+6.0); /* FALLTHRU */
case 6: z *= (y+5.0); /* FALLTHRU */
case 5: z *= (y+4.0); /* FALLTHRU */
case 4: z *= (y+3.0); /* FALLTHRU */
case 3: z *= (y+2.0); /* FALLTHRU */
r += __ieee754_log(z); break;
}
/* 8.0 <= x < 2**58 */
} else if (ix < 0x43900000) {
t = __ieee754_log(x);
z = one/x;
y = z*z;
w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
r = (x-half)*(t-one)+w;
} else
/* 2**58 <= x <= inf */
r = x*(__ieee754_log(x)-one);
/* NADJ is set for negative arguments but not otherwise,
resulting in warnings that it may be used uninitialized
although in the cases where it is used it has always been
set. */
DIAG_PUSH_NEEDS_COMMENT;
#if __GNUC_PREREQ (4, 7)
DIAG_IGNORE_NEEDS_COMMENT (4.9, "-Wmaybe-uninitialized");
#else
DIAG_IGNORE_NEEDS_COMMENT (4.9, "-Wuninitialized");
#endif
if(hx<0) r = nadj - r;
DIAG_POP_NEEDS_COMMENT;
return r;
}
double
lgamma_r(double x, int *signgamp)
{
double t,y,z,nadj,p,p1,p2,p3,q,r,w;
int i,hx,lx,ix;
nadj = 0;
EXTRACT_WORDS(hx,lx,x);
/* purge off +-inf, NaN, +-0, and negative arguments */
*signgamp = 1;
ix = hx&0x7fffffff;
if(ix>=0x7ff00000) return x*x;
if((ix|lx)==0) {
if(hx<0)
*signgamp = -1;
return one/zero;
}
if(ix<0x3b900000) { /* |x|<2**-70, return -log(|x|) */
if(hx<0) {
*signgamp = -1;
return - log(-x);
} else return - log(x);
}
if(hx<0) {
if(ix>=0x43300000) /* |x|>=2**52, must be -integer */
return one/zero;
t = sin_pi(x);
if(t==zero) return one/zero; /* -integer */
nadj = log(pi/fabs(t*x));
if(t<zero) *signgamp = -1;
x = -x;
}
/* purge off 1 and 2 */
if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0;
/* for x < 2.0 */
else if(ix<0x40000000) {
if(ix<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */
r = - log(x);
if(ix>=0x3FE76944) {y = one-x; i= 0;}
else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;}
else {y = x; i=2;}
} else {
r = zero;
if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */
else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */
else {y=x-one;i=2;}
}
switch(i) {
case 0:
z = y*y;
p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
p = y*p1+p2;
r += (p-0.5*y); break;
case 1:
z = y*y;
w = z*y;
p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */
p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
p = z*p1-(tt-w*(p2+y*p3));
r += (tf + p); break;
case 2:
p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
r += (-0.5*y + p1/p2);
}
}
else if(ix<0x40200000) { /* x < 8.0 */
i = (int)x;
t = zero;
y = x-(double)i;
p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
r = half*y+p/q;
z = one; /* lgamma(1+s) = log(s) + lgamma(s) */
switch(i) {
case 7: z *= (y+6.0); /* FALLTHRU */
case 6: z *= (y+5.0); /* FALLTHRU */
case 5: z *= (y+4.0); /* FALLTHRU */
case 4: z *= (y+3.0); /* FALLTHRU */
case 3: z *= (y+2.0); /* FALLTHRU */
r += log(z); break;
}
/* 8.0 <= x < 2**58 */
} else if (ix < 0x43900000) {
t = log(x);
z = one/x;
y = z*z;
w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
r = (x-half)*(t-one)+w;
} else
/* 2**58 <= x <= inf */
r = x*(log(x)-one);
if(hx<0) r = nadj - r;
return r;
}
double complex csinh(double complex z)
{
double x, y, h;
int32_t hx, hy, ix, iy, lx, ly;
x = creal(z);
y = cimag(z);
EXTRACT_WORDS(hx, lx, x);
EXTRACT_WORDS(hy, ly, y);
ix = 0x7fffffff & hx;
iy = 0x7fffffff & hy;
/* Handle the nearly-non-exceptional cases where x and y are finite. */
if (ix < 0x7ff00000 && iy < 0x7ff00000) {
if ((iy | ly) == 0)
return CMPLX(sinh(x), y);
if (ix < 0x40360000) /* small x: normal case */
return CMPLX(sinh(x) * cos(y), cosh(x) * sin(y));
/* |x| >= 22, so cosh(x) ~= exp(|x|) */
if (ix < 0x40862e42) {
/* x < 710: exp(|x|) won't overflow */
h = exp(fabs(x)) * 0.5;
return CMPLX(copysign(h, x) * cos(y), h * sin(y));
} else if (ix < 0x4096bbaa) {
/* x < 1455: scale to avoid overflow */
z = __ldexp_cexp(CMPLX(fabs(x), y), -1);
return CMPLX(creal(z) * copysign(1, x), cimag(z));
} else {
/* x >= 1455: the result always overflows */
h = huge * x;
return CMPLX(h * cos(y), h * h * sin(y));
}
}
/*
* sinh(+-0 +- I Inf) = sign(d(+-0, dNaN))0 + I dNaN.
* The sign of 0 in the result is unspecified. Choice = normally
* the same as dNaN. Raise the invalid floating-point exception.
*
* sinh(+-0 +- I NaN) = sign(d(+-0, NaN))0 + I d(NaN).
* The sign of 0 in the result is unspecified. Choice = normally
* the same as d(NaN).
*/
if ((ix | lx) == 0 && iy >= 0x7ff00000)
return CMPLX(copysign(0, x * (y - y)), y - y);
/*
* sinh(+-Inf +- I 0) = +-Inf + I +-0.
*
* sinh(NaN +- I 0) = d(NaN) + I +-0.
*/
if ((iy | ly) == 0 && ix >= 0x7ff00000) {
if (((hx & 0xfffff) | lx) == 0)
return CMPLX(x, y);
return CMPLX(x, copysign(0, y));
}
/*
* sinh(x +- I Inf) = dNaN + I dNaN.
* Raise the invalid floating-point exception for finite nonzero x.
*
* sinh(x + I NaN) = d(NaN) + I d(NaN).
* Optionally raises the invalid floating-point exception for finite
* nonzero x. Choice = don't raise (except for signaling NaNs).
*/
if (ix < 0x7ff00000 && iy >= 0x7ff00000)
return CMPLX(y - y, x * (y - y));
/*
* sinh(+-Inf + I NaN) = +-Inf + I d(NaN).
* The sign of Inf in the result is unspecified. Choice = normally
* the same as d(NaN).
*
* sinh(+-Inf +- I Inf) = +Inf + I dNaN.
* The sign of Inf in the result is unspecified. Choice = always +.
* Raise the invalid floating-point exception.
*
* sinh(+-Inf + I y) = +-Inf cos(y) + I Inf sin(y)
*/
if (ix >= 0x7ff00000 && ((hx & 0xfffff) | lx) == 0) {
if (iy >= 0x7ff00000)
return CMPLX(x * x, x * (y - y));
return CMPLX(x * cos(y), INFINITY * sin(y));
}
/*
* sinh(NaN + I NaN) = d(NaN) + I d(NaN).
*
* sinh(NaN +- I Inf) = d(NaN) + I d(NaN).
* Optionally raises the invalid floating-point exception.
* Choice = raise.
*
* sinh(NaN + I y) = d(NaN) + I d(NaN).
* Optionally raises the invalid floating-point exception for finite
* nonzero y. Choice = don't raise (except for signaling NaNs).
*/
return CMPLX((x * x) * (y - y), (x + x) * (y - y));
}
double
__ieee754_log2(double x)
{
double f,hfsq,hi,lo,r,val_hi,val_lo,w,y;
int32_t i,k,hx;
u_int32_t lx;
EXTRACT_WORDS(hx,lx,x);
k=0;
if (hx < 0x00100000) { /* x < 2**-1022 */
if (((hx&0x7fffffff)|lx)==0)
return -two54/vzero; /* log(+-0)=-inf */
if (hx<0) return (x-x)/zero; /* log(-#) = NaN */
k -= 54; x *= two54; /* subnormal number, scale up x */
GET_HIGH_WORD(hx,x);
}
if (hx >= 0x7ff00000) return x+x;
if (hx == 0x3ff00000 && lx == 0)
return zero; /* log(1) = +0 */
k += (hx>>20)-1023;
hx &= 0x000fffff;
i = (hx+0x95f64)&0x100000;
SET_HIGH_WORD(x,hx|(i^0x3ff00000)); /* normalize x or x/2 */
k += (i>>20);
y = (double)k;
f = x - 1.0;
hfsq = 0.5*f*f;
r = k_log1p(f);
/*
* f-hfsq must (for args near 1) be evaluated in extra precision
* to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2).
* This is fairly efficient since f-hfsq only depends on f, so can
* be evaluated in parallel with R. Not combining hfsq with R also
* keeps R small (though not as small as a true `lo' term would be),
* so that extra precision is not needed for terms involving R.
*
* Compiler bugs involving extra precision used to break Dekker's
* theorem for spitting f-hfsq as hi+lo, unless double_t was used
* or the multi-precision calculations were avoided when double_t
* has extra precision. These problems are now automatically
* avoided as a side effect of the optimization of combining the
* Dekker splitting step with the clear-low-bits step.
*
* y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra
* precision to avoid a very large cancellation when x is very near
* these values. Unlike the above cancellations, this problem is
* specific to base 2. It is strange that adding +-1 is so much
* harder than adding +-ln2 or +-log10_2.
*
* This uses Dekker's theorem to normalize y+val_hi, so the
* compiler bugs are back in some configurations, sigh. And I
* don't want to used double_t to avoid them, since that gives a
* pessimization and the support for avoiding the pessimization
* is not yet available.
*
* The multi-precision calculations for the multiplications are
* routine.
*/
hi = f - hfsq;
SET_LOW_WORD(hi,0);
lo = (f - hi) - hfsq + r;
val_hi = hi*ivln2hi;
val_lo = (lo+hi)*ivln2lo + lo*ivln2hi;
/* spadd(val_hi, val_lo, y), except for not using double_t: */
w = y + val_hi;
val_lo += (y - w) + val_hi;
val_hi = w;
return val_lo + val_hi;
}
double
__ieee754_y1 (double x)
{
double z, s, c, ss, cc, u, v, u1, u2, v1, v2, v3, z2, z4;
int32_t hx, ix, lx;
EXTRACT_WORDS (hx, lx, x);
ix = 0x7fffffff & hx;
/* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
if (__glibc_unlikely (ix >= 0x7ff00000))
return one / (x + x * x);
if (__glibc_unlikely ((ix | lx) == 0))
return -1 / zero; /* -inf and divide by zero exception. */
/* -inf and overflow exception. */;
if (__glibc_unlikely (hx < 0))
return zero / (zero * x);
if (ix >= 0x40000000) /* |x| >= 2.0 */
{
__sincos (x, &s, &c);
ss = -s - c;
cc = s - c;
if (ix < 0x7fe00000) /* make sure x+x not overflow */
{
z = __cos (x + x);
if ((s * c) > zero)
cc = z / ss;
else
ss = z / cc;
}
/* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
* where x0 = x-3pi/4
* Better formula:
* cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
* = 1/sqrt(2) * (sin(x) - cos(x))
* sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
* = -1/sqrt(2) * (cos(x) + sin(x))
* To avoid cancellation, use
* sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
* to compute the worse one.
*/
if (ix > 0x48000000)
z = (invsqrtpi * ss) / __ieee754_sqrt (x);
else
{
u = pone (x); v = qone (x);
z = invsqrtpi * (u * ss + v * cc) / __ieee754_sqrt (x);
}
return z;
}
if (__glibc_unlikely (ix <= 0x3c900000)) /* x < 2**-54 */
{
z = -tpi / x;
if (isinf (z))
__set_errno (ERANGE);
return z;
}
z = x * x;
u1 = U0[0] + z * U0[1]; z2 = z * z;
u2 = U0[2] + z * U0[3]; z4 = z2 * z2;
u = u1 + z2 * u2 + z4 * U0[4];
v1 = one + z * V0[0];
v2 = V0[1] + z * V0[2];
v3 = V0[3] + z * V0[4];
v = v1 + z2 * v2 + z4 * v3;
return (x * (u / v) + tpi * (__ieee754_j1 (x) * __ieee754_log (x) - one / x));
}
double complex
ctanh(double complex z)
{
double x, y;
double t, beta, s, rho, denom;
uint32_t hx, ix, lx;
x = creal(z);
y = cimag(z);
EXTRACT_WORDS(hx, lx, x);
ix = hx & 0x7fffffff;
/*
* ctanh(NaN + i 0) = NaN + i 0
*
* ctanh(NaN + i y) = NaN + i NaN for y != 0
*
* The imaginary part has the sign of x*sin(2*y), but there's no
* special effort to get this right.
*
* ctanh(+-Inf +- i Inf) = +-1 +- 0
*
* ctanh(+-Inf + i y) = +-1 + 0 sin(2y) for y finite
*
* The imaginary part of the sign is unspecified. This special
* case is only needed to avoid a spurious invalid exception when
* y is infinite.
*/
if (ix >= 0x7ff00000) {
if ((ix & 0xfffff) | lx) /* x is NaN */
return (CMPLX(x, (y == 0 ? y : x * y)));
SET_HIGH_WORD(x, hx - 0x40000000); /* x = copysign(1, x) */
return (CMPLX(x, copysign(0, isinf(y) ? y : sin(y) * cos(y))));
}
/*
* ctanh(x + i NAN) = NaN + i NaN
* ctanh(x +- i Inf) = NaN + i NaN
*/
if (!isfinite(y))
return (CMPLX(y - y, y - y));
/*
* ctanh(+-huge + i +-y) ~= +-1 +- i 2sin(2y)/exp(2x), using the
* approximation sinh^2(huge) ~= exp(2*huge) / 4.
* We use a modified formula to avoid spurious overflow.
*/
if (ix >= 0x40360000) { /* x >= 22 */
double exp_mx = exp(-fabs(x));
return (CMPLX(copysign(1, x),
4 * sin(y) * cos(y) * exp_mx * exp_mx));
}
/* Kahan's algorithm */
t = tan(y);
beta = 1.0 + t * t; /* = 1 / cos^2(y) */
s = sinh(x);
rho = sqrt(1 + s * s); /* = cosh(x) */
denom = 1 + beta * s * s;
return (CMPLX((beta * rho * s) / denom, t / denom));
}