double
__asinh (double x)
{
double w;
int32_t hx, ix;
GET_HIGH_WORD (hx, x);
ix = hx & 0x7fffffff;
if (__glibc_unlikely (ix < 0x3e300000)) /* |x|<2**-28 */
{
if (huge + x > one)
return x; /* return x inexact except 0 */
}
if (__glibc_unlikely (ix > 0x41b00000)) /* |x| > 2**28 */
{
if (ix >= 0x7ff00000)
return x + x; /* x is inf or NaN */
w = __ieee754_log (fabs (x)) + ln2;
}
else
{
double xa = fabs (x);
if (ix > 0x40000000) /* 2**28 > |x| > 2.0 */
{
w = __ieee754_log (2.0 * xa + one / (__ieee754_sqrt (xa * xa + one) +
xa));
}
else /* 2.0 > |x| > 2**-28 */
{
double t = xa * xa;
w = __log1p (xa + t / (one + __ieee754_sqrt (one + t)));
}
}
return __copysign (w, x);
}
double
__ieee754_acosh (double x)
{
int64_t hx;
EXTRACT_WORDS64 (hx, x);
if (hx > INT64_C (0x4000000000000000))
{
if (__builtin_expect (hx >= INT64_C (0x41b0000000000000), 0))
{
/* x > 2**28 */
if (hx >= INT64_C (0x7ff0000000000000))
/* x is inf of NaN */
return x + x;
else
return __ieee754_log (x) + ln2;/* acosh(huge)=log(2x) */
}
/* 2**28 > x > 2 */
double t = x * x;
return __ieee754_log (2.0 * x - one / (x + __ieee754_sqrt (t - one)));
}
else if (__builtin_expect (hx > INT64_C (0x3ff0000000000000), 1))
{
/* 1<x<2 */
double t = x - one;
return __log1p (t + __ieee754_sqrt (2.0 * t + t * t));
}
else if (__builtin_expect (hx == INT64_C (0x3ff0000000000000), 1))
return 0.0; /* acosh(1) = 0 */
else /* x < 1 */
return (x - x) / (x - x);
}
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(__builtin_expect(ix>=0x7ff00000, 0)) return one/(x+x*x);
if(__builtin_expect((ix|lx)==0, 0))
return -HUGE_VAL+x; /* -inf and overflow exception. */;
if(__builtin_expect(hx<0, 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(__builtin_expect(ix<=0x3c900000, 0)) { /* x < 2**-54 */
return(-tpi/x);
}
z = x*x;
#ifdef DO_NOT_USE_THIS
u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
#else
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;
#endif
return(x*(u/v) + tpi*(__ieee754_j1(x)*__ieee754_log(x)-one/x));
}
double
__ieee754_hypot (double x, double y)
{
x = fabs (x);
y = fabs (y);
TEST_INF_NAN (x, y);
if (y > x)
{
double t = x;
x = y;
y = t;
}
if (y == 0.0)
return x;
/* if y is higher enough, y * 2^60 might overflow. The tests if
y >= 1.7976931348623157e+308/2^60 (two60factor) and uses the
appropriate check to avoid the overflow exception generation. */
if (y > two60factor)
{
if ((x / y) > two60)
return x + y;
}
else
{
if (x > (y * two60))
return x + y;
}
if (x > two500)
{
x *= twoM600;
y *= twoM600;
return __ieee754_sqrt (x * x + y * y) / twoM600;
}
if (y < twoM500)
{
if (y <= pdnum)
{
x *= two1022;
y *= two1022;
double ret = __ieee754_sqrt (x * x + y * y) / two1022;
math_check_force_underflow_nonneg (ret);
return ret;
}
else
{
x *= two600;
y *= two600;
return __ieee754_sqrt (x * x + y * y) / two600;
}
}
return __ieee754_sqrt (x * x + y * y);
}
double
__ieee754_j0(double x)
{
double z, s,c,ss,cc,r,u,v,r1,r2,s1,s2,z2,z4;
int32_t hx,ix;
GET_HIGH_WORD(hx,x);
ix = hx&0x7fffffff;
if(ix>=0x7ff00000) return one/(x*x);
x = fabs(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;
}
/*
* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
*/
if(ix>0x48000000) z = (invsqrtpi*cc)/__ieee754_sqrt(x);
else {
u = pzero(x); v = qzero(x);
z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrt(x);
}
return z;
}
if(ix<0x3f200000) { /* |x| < 2**-13 */
math_force_eval(huge+x); /* raise inexact if x != 0 */
if(ix<0x3e400000) return one; /* |x|<2**-27 */
else return one - 0.25*x*x;
}
z = x*x;
#ifdef DO_NOT_USE_THIS
r = z*(R02+z*(R03+z*(R04+z*R05)));
s = one+z*(S01+z*(S02+z*(S03+z*S04)));
#else
r1 = z*R[2]; z2=z*z;
r2 = R[3]+z*R[4]; z4=z2*z2;
r = r1 + z2*r2 + z4*R[5];
s1 = one+z*S[1];
s2 = S[2]+z*S[3];
s = s1 + z2*s2 + z4*S[4];
#endif
if(ix < 0x3FF00000) { /* |x| < 1.00 */
return one + z*(-0.25+(r/s));
} else {
u = 0.5*x;
return((one+u)*(one-u)+z*(r/s));
}
}
double
__ieee754_j1(double x)
{
double z, s,c,ss,cc,r,u,v,y,r1,r2,s1,s2,s3,z2,z4;
int32_t hx,ix;
GET_HIGH_WORD(hx,x);
ix = hx&0x7fffffff;
if(__builtin_expect(ix>=0x7ff00000, 0)) return one/x;
y = fabs(x);
if(ix >= 0x40000000) { /* |x| >= 2.0 */
__sincos (y, &s, &c);
ss = -s-c;
cc = s-c;
if(ix<0x7fe00000) { /* make sure y+y not overflow */
z = __cos(y+y);
if ((s*c)>zero) cc = z/ss;
else ss = z/cc;
}
/*
* j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
* y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
*/
if(ix>0x48000000) z = (invsqrtpi*cc)/__ieee754_sqrt(y);
else {
u = pone(y); v = qone(y);
z = invsqrtpi*(u*cc-v*ss)/__ieee754_sqrt(y);
}
if(hx<0) return -z;
else return z;
}
if(__builtin_expect(ix<0x3e400000, 0)) { /* |x|<2**-27 */
if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */
}
z = x*x;
#ifdef DO_NOT_USE_THIS
r = z*(r00+z*(r01+z*(r02+z*r03)));
s = one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
r *= x;
#else
r1 = z*R[0]; z2=z*z;
r2 = R[1]+z*R[2]; z4=z2*z2;
r = r1 + z2*r2 + z4*R[3];
r *= x;
s1 = one+z*S[1];
s2 = S[2]+z*S[3];
s3 = S[4]+z*S[5];
s = s1 + z2*s2 + z4*s3;
#endif
return(x*0.5+r/s);
}
double
sqrt(double x) /* wrapper sqrt */
{
#ifdef _IEEE_LIBM
return __ieee754_sqrt(x);
#else
double z;
z = __ieee754_sqrt(x);
if(/* _LIB_VERSION == _IEEE_ || */ isnan(x)) return z;
if(x<0.0) {
return __kernel_standard(x,x,26); /* sqrt(negative) */
} else
return z;
#endif
}
float
__ieee754_hypotf(float x, float y)
{
double d_x, d_y;
int32_t ha, hb;
GET_FLOAT_WORD(ha,x);
ha &= 0x7fffffff;
GET_FLOAT_WORD(hb,y);
hb &= 0x7fffffff;
if (ha == 0x7f800000 && !issignaling (y))
return fabsf(x);
else if (hb == 0x7f800000 && !issignaling (x))
return fabsf(y);
else if (ha > 0x7f800000 || hb > 0x7f800000)
return fabsf(x) * fabsf(y);
else if (ha == 0)
return fabsf(y);
else if (hb == 0)
return fabsf(x);
d_x = (double) x;
d_y = (double) y;
return (float) __ieee754_sqrt(d_x * d_x + d_y * d_y);
}
float
__ieee754_hypotf (float x, float y)
{
TEST_INF_NAN (x, y);
return __ieee754_sqrt ((double) x * x + (double) y * y);
}
/* wrapper sqrt */
double
__sqrt (double x)
{
if (__builtin_expect (isless (x, 0.0), 0) && _LIB_VERSION != _IEEE_)
return __kernel_standard (x, x, 26); /* sqrt(negative) */
return __ieee754_sqrt (x);
}
void
Math_sqrt(void *fp)
{
F_Math_sqrt *f;
f = fp;
*f->ret = __ieee754_sqrt(f->x);
}
double
__ieee754_acos(double x)
{
double z,p,q,r,w,s,c,df;
int32_t hx,ix;
GET_HIGH_WORD(hx,x);
ix = hx&0x7fffffff;
if(ix>=0x3ff00000) { /* |x| >= 1 */
u_int32_t lx;
GET_LOW_WORD(lx,x);
if(((ix-0x3ff00000)|lx)==0) { /* |x|==1 */
if(hx>0) return 0.0; /* acos(1) = 0 */
else return pi+2.0*pio2_lo; /* acos(-1)= pi */
}
return (x-x)/(x-x); /* acos(|x|>1) is NaN */
}
if(ix<0x3fe00000) { /* |x| < 0.5 */
if(ix<=0x3c600000) return pio2_hi+pio2_lo;/*if|x|<2**-57*/
z = x*x;
p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
r = p/q;
return pio2_hi - (x - (pio2_lo-x*r));
} else if (hx<0) { /* x < -0.5 */
z = (one+x)*0.5;
p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
s = __ieee754_sqrt(z);
r = p/q;
w = r*s-pio2_lo;
return pi - 2.0*(s+w);
} else { /* x > 0.5 */
z = (one-x)*0.5;
s = __ieee754_sqrt(z);
df = s;
SET_LOW_WORD(df,0);
c = (z-df*df)/(s+df);
p = z*(pS0+z*(pS1+z*(pS2+z*(pS3+z*(pS4+z*pS5)))));
q = one+z*(qS1+z*(qS2+z*(qS3+z*qS4)));
r = p/q;
w = r*s+c;
return 2.0*(df+w);
}
}
//------------------------------------------------------------------------------
double Cmath::__ieee754_acosh( double x )
{
static const double one = 1.0;
static const double ln2 = 6.93147180559945286227e-01; // 0x3FE62E42, 0xFEFA39EF
double t;
Cmp_signed__int32 hx;
Cmp_unsigned__int32 lx;
extract_words( hx, lx, x );
if( hx < 0x3ff00000 )
{
// x < 1
return ( x - x ) / ( x - x );
}
else if( hx >= 0x41b00000 )
{
// x > 2**28
if( hx >= 0x7ff00000 )
{
// x is inf of NaN
return x + x;
}
else
{
return __ieee754_log( x ) + ln2; // acosh(huge)=log(2x)
}
}
else if( ( ( hx - 0x3ff00000 ) | lx ) == 0 )
{
return 0.0; // acosh(1) = 0
}
else if( hx > 0x40000000 )
{
// 2**28 > x > 2
t = x * x;
return __ieee754_log( 2.0 * x - one / ( x + __ieee754_sqrt( t - one ) ) );
}
else
{
// 1<x<2
t = x - one;
return log1p( t + __ieee754_sqrt( 2.0 * t + t * t ) );
}
}
double asinh(double x)
{
double t,w;
int32_t hx,ix;
GET_HIGH_WORD(hx,x);
ix = hx&0x7fffffff;
if(ix>=0x7ff00000) return x+x; /* x is inf or NaN */
if(ix< 0x3e300000) { /* |x|<2**-28 */
if(huge+x>one) return x; /* return x inexact except 0 */
}
if(ix>0x41b00000) { /* |x| > 2**28 */
w = __ieee754_log(fabs(x))+ln2;
} else if (ix>0x40000000) { /* 2**28 > |x| > 2.0 */
t = fabs(x);
w = __ieee754_log(2.0*t+one/(__ieee754_sqrt(x*x+one)+t));
} else { /* 2.0 > |x| > 2**-28 */
t = x*x;
w =log1p(fabs(x)+t/(one+__ieee754_sqrt(one+t)));
}
if(hx>0) return w; else return -w;
}
double
__ieee754_asin(double x)
{
double t,w,p,q,c,r,s;
int32_t hx,ix;
t = 0;
GET_HIGH_WORD(hx,x);
ix = hx&0x7fffffff;
if(ix>= 0x3ff00000) { /* |x|>= 1 */
u_int32_t lx;
GET_LOW_WORD(lx,x);
if(((ix-0x3ff00000)|lx)==0)
/* asin(1)=+-pi/2 with inexact */
return x*pio2_hi+x*pio2_lo;
return (x-x)/(x-x); /* asin(|x|>1) is NaN */
} else if (ix<0x3fe00000) { /* |x|<0.5 */
if(ix<0x3e400000) { /* if |x| < 2**-27 */
if(huge+x>one) return x;/* return x with inexact if x!=0*/
} else
t = x*x;
p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
w = p/q;
return x+x*w;
}
/* 1> |x|>= 0.5 */
w = one-fabs(x);
t = w*0.5;
p = t*(pS0+t*(pS1+t*(pS2+t*(pS3+t*(pS4+t*pS5)))));
q = one+t*(qS1+t*(qS2+t*(qS3+t*qS4)));
s = __ieee754_sqrt(t);
if(ix>=0x3FEF3333) { /* if |x| > 0.975 */
w = p/q;
t = pio2_hi-(2.0*(s+s*w)-pio2_lo);
} else {
w = s;
SET_LOW_WORD(w,0);
c = (t-w*w)/(s+w);
r = p/q;
p = 2.0*s*r-(pio2_lo-2.0*c);
q = pio4_hi-2.0*w;
t = pio4_hi-(p-q);
}
if(hx>0) return t; else return -t;
}
EXPORT(sqInt) primitiveSqrt(void) {
double rcvr;
double result;
rcvr = interpreterProxy->stackFloatValue(0);
if (interpreterProxy->failed()) {
return null;
}
if (rcvr < 0.0) {
return interpreterProxy->primitiveFail();
}
result = __ieee754_sqrt(rcvr);
if (isnan(result)) {
return interpreterProxy->primitiveFail();
}
interpreterProxy->pop((interpreterProxy->methodArgumentCount()) + 1);
interpreterProxy->pushFloat(result);
}
primitiveSqrt(void)
{
// FloatMathPlugin>>#primitiveSqrt
double rcvr;
double result;
rcvr = stackFloatValue(0);
if (failed()) {
return null;
}
if (rcvr < 0.0) {
return primitiveFail();
}
result = __ieee754_sqrt(rcvr);
if (isnan(result)) {
return primitiveFail();
}
pop((methodArgumentCount()) + 1);
pushFloat(result);
}
double acosh(double x)
{
double t;
int32_t hx;
u_int32_t lx;
EXTRACT_WORDS(hx,lx,x);
if(hx<0x3ff00000) { /* x < 1 */
return (x-x)/(x-x);
} else if(hx >=0x41b00000) { /* x > 2**28 */
if(hx >=0x7ff00000) { /* x is inf of NaN */
return x+x;
} else
return __ieee754_log(x)+ln2; /* acosh(huge)=log(2x) */
} else if(((hx-0x3ff00000)|lx)==0) {
return 0.0; /* acosh(1) = 0 */
} else if (hx > 0x40000000) { /* 2**28 > x > 2 */
t=x*x;
return __ieee754_log(2.0*x-one/(x+__ieee754_sqrt(t-one)));
} else { /* 1<x<2 */
t = x-one;
return log1p(t+sqrt(2.0*t+t*t));
}
}
double
__ieee754_hypot (double x, double y)
{
double a, b, t1, t2, y1, y2, w;
int32_t j, k, ha, hb;
GET_HIGH_WORD (ha, x);
ha &= 0x7fffffff;
GET_HIGH_WORD (hb, y);
hb &= 0x7fffffff;
if (hb > ha)
{
a = y; b = x; j = ha; ha = hb; hb = j;
}
else
{
a = x; b = y;
}
SET_HIGH_WORD (a, ha); /* a <- |a| */
SET_HIGH_WORD (b, hb); /* b <- |b| */
if ((ha - hb) > 0x3c00000)
{
return a + b;
} /* x/y > 2**60 */
k = 0;
if (__glibc_unlikely (ha > 0x5f300000)) /* a>2**500 */
{
if (ha >= 0x7ff00000) /* Inf or NaN */
{
u_int32_t low;
w = a + b; /* for sNaN */
GET_LOW_WORD (low, a);
if (((ha & 0xfffff) | low) == 0)
w = a;
GET_LOW_WORD (low, b);
if (((hb ^ 0x7ff00000) | low) == 0)
w = b;
return w;
}
/* scale a and b by 2**-600 */
ha -= 0x25800000; hb -= 0x25800000; k += 600;
SET_HIGH_WORD (a, ha);
SET_HIGH_WORD (b, hb);
}
if (__builtin_expect (hb < 0x23d00000, 0)) /* b < 2**-450 */
{
if (hb <= 0x000fffff) /* subnormal b or 0 */
{
u_int32_t low;
GET_LOW_WORD (low, b);
if ((hb | low) == 0)
return a;
t1 = 0;
SET_HIGH_WORD (t1, 0x7fd00000); /* t1=2^1022 */
b *= t1;
a *= t1;
k -= 1022;
GET_HIGH_WORD (ha, a);
GET_HIGH_WORD (hb, b);
if (hb > ha)
{
t1 = a;
a = b;
b = t1;
j = ha;
ha = hb;
hb = j;
}
}
else /* scale a and b by 2^600 */
{
ha += 0x25800000; /* a *= 2^600 */
hb += 0x25800000; /* b *= 2^600 */
k -= 600;
SET_HIGH_WORD (a, ha);
SET_HIGH_WORD (b, hb);
}
}
/* medium size a and b */
w = a - b;
if (w > b)
{
t1 = 0;
SET_HIGH_WORD (t1, ha);
t2 = a - t1;
w = __ieee754_sqrt (t1 * t1 - (b * (-b) - t2 * (a + t1)));
}
else
{
a = a + a;
y1 = 0;
SET_HIGH_WORD (y1, hb);
y2 = b - y1;
t1 = 0;
SET_HIGH_WORD (t1, ha + 0x00100000);
t2 = a - t1;
w = __ieee754_sqrt (t1 * y1 - (w * (-w) - (t1 * y2 + t2 * b)));
}
if (k != 0)
{
u_int32_t high;
t1 = 1.0;
GET_HIGH_WORD (high, t1);
SET_HIGH_WORD (t1, high + (k << 20));
w *= t1;
math_check_force_underflow_nonneg (w);
return w;
}
else
return w;
}
// Special "jump island" routines, used to provide a bridge when
// a routine near one end of our oversized code segment has to call
// a routine near the other end. The need for these rest stops has
// been minimized by trying to arrange our code segment for good
// locality of reference.
double jumpto__ieee754_sqrt(double x) {
return __ieee754_sqrt(x);
}
Err mathlib_sqrt(UInt16 refnum, double x, double *result) {
#pragma unused(refnum)
*result = __ieee754_sqrt(x);
return mlErrNone;
}
double
__ieee754_j1 (double x)
{
double z, s, c, ss, cc, r, u, v, y, r1, r2, s1, s2, s3, z2, z4;
int32_t hx, ix;
GET_HIGH_WORD (hx, x);
ix = hx & 0x7fffffff;
if (__glibc_unlikely (ix >= 0x7ff00000))
return one / x;
y = fabs (x);
if (ix >= 0x40000000) /* |x| >= 2.0 */
{
__sincos (y, &s, &c);
ss = -s - c;
cc = s - c;
if (ix < 0x7fe00000) /* make sure y+y not overflow */
{
z = __cos (y + y);
if ((s * c) > zero)
cc = z / ss;
else
ss = z / cc;
}
/*
* j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
* y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
*/
if (ix > 0x48000000)
z = (invsqrtpi * cc) / __ieee754_sqrt (y);
else
{
u = pone (y); v = qone (y);
z = invsqrtpi * (u * cc - v * ss) / __ieee754_sqrt (y);
}
if (hx < 0)
return -z;
else
return z;
}
if (__glibc_unlikely (ix < 0x3e400000)) /* |x|<2**-27 */
{
if (huge + x > one) /* inexact if x!=0 necessary */
{
double ret = 0.5 * x;
if (fabs (ret) < DBL_MIN)
{
double force_underflow = ret * ret;
math_force_eval (force_underflow);
}
return ret;
}
}
z = x * x;
r1 = z * R[0]; z2 = z * z;
r2 = R[1] + z * R[2]; z4 = z2 * z2;
r = r1 + z2 * r2 + z4 * R[3];
r *= x;
s1 = one + z * S[1];
s2 = S[2] + z * S[3];
s3 = S[4] + z * S[5];
s = s1 + z2 * s2 + z4 * s3;
return (x * 0.5 + r / s);
}
static double
gamma_positive (double x, int *exp2_adj)
{
int local_signgam;
if (x < 0.5)
{
*exp2_adj = 0;
return __ieee754_exp (__ieee754_lgamma_r (x + 1, &local_signgam)) / x;
}
else if (x <= 1.5)
{
*exp2_adj = 0;
return __ieee754_exp (__ieee754_lgamma_r (x, &local_signgam));
}
else if (x < 6.5)
{
/* Adjust into the range for using exp (lgamma). */
*exp2_adj = 0;
double n = __ceil (x - 1.5);
double x_adj = x - n;
double eps;
double prod = __gamma_product (x_adj, 0, n, &eps);
return (__ieee754_exp (__ieee754_lgamma_r (x_adj, &local_signgam))
* prod * (1.0 + eps));
}
else
{
double eps = 0;
double x_eps = 0;
double x_adj = x;
double prod = 1;
if (x < 12.0)
{
/* Adjust into the range for applying Stirling's
approximation. */
double n = __ceil (12.0 - x);
#if FLT_EVAL_METHOD != 0
volatile
#endif
double x_tmp = x + n;
x_adj = x_tmp;
x_eps = (x - (x_adj - n));
prod = __gamma_product (x_adj - n, x_eps, n, &eps);
}
/* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)).
Compute gamma (X_ADJ + X_EPS) using Stirling's approximation,
starting by computing pow (X_ADJ, X_ADJ) with a power of 2
factored out. */
double exp_adj = -eps;
double x_adj_int = __round (x_adj);
double x_adj_frac = x_adj - x_adj_int;
int x_adj_log2;
double x_adj_mant = __frexp (x_adj, &x_adj_log2);
if (x_adj_mant < M_SQRT1_2)
{
x_adj_log2--;
x_adj_mant *= 2.0;
}
*exp2_adj = x_adj_log2 * (int) x_adj_int;
double ret = (__ieee754_pow (x_adj_mant, x_adj)
* __ieee754_exp2 (x_adj_log2 * x_adj_frac)
* __ieee754_exp (-x_adj)
* __ieee754_sqrt (2 * M_PI / x_adj)
/ prod);
exp_adj += x_eps * __ieee754_log (x);
double bsum = gamma_coeff[NCOEFF - 1];
double x_adj2 = x_adj * x_adj;
for (size_t i = 1; i <= NCOEFF - 1; i++)
bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i];
exp_adj += bsum / x_adj;
return ret + ret * __expm1 (exp_adj);
}
}
double
__ieee754_y0(double x)
{
double z, s,c,ss,cc,u,v,z2,z4,z6,u1,u2,u3,v1,v2;
int32_t hx,ix,lx;
EXTRACT_WORDS(hx,lx,x);
ix = 0x7fffffff&hx;
/* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0, y0(0) is -inf. */
if(ix>=0x7ff00000) return one/(x+x*x);
if((ix|lx)==0) return -HUGE_VAL+x; /* -inf and overflow exception. */
if(hx<0) return zero/(zero*x);
if(ix >= 0x40000000) { /* |x| >= 2.0 */
/* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
* where x0 = x-pi/4
* Better formula:
* cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
* = 1/sqrt(2) * (sin(x) + cos(x))
* sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
* = 1/sqrt(2) * (sin(x) - cos(x))
* To avoid cancellation, use
* sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
* to compute the worse one.
*/
__sincos (x, &s, &c);
ss = s-c;
cc = s+c;
/*
* j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
* y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
*/
if(ix<0x7fe00000) { /* make sure x+x not overflow */
z = -__cos(x+x);
if ((s*c)<zero) cc = z/ss;
else ss = z/cc;
}
if(ix>0x48000000) z = (invsqrtpi*ss)/__ieee754_sqrt(x);
else {
u = pzero(x); v = qzero(x);
z = invsqrtpi*(u*ss+v*cc)/__ieee754_sqrt(x);
}
return z;
}
if(ix<=0x3e400000) { /* x < 2**-27 */
return(U[0] + tpi*__ieee754_log(x));
}
z = x*x;
#ifdef DO_NOT_USE_THIS
u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
v = one+z*(v01+z*(v02+z*(v03+z*v04)));
#else
u1 = U[0]+z*U[1]; z2=z*z;
u2 = U[2]+z*U[3]; z4=z2*z2;
u3 = U[4]+z*U[5]; z6=z4*z2;
u = u1 + z2*u2 + z4*u3 + z6*U[6];
v1 = one+z*V[0];
v2 = V[1]+z*V[2];
v = v1 + z2*v2 + z4*V[3];
#endif
return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x)));
}
double attribute_hidden __ieee754_hypot(double x, double y)
{
double a=x,b=y,t1,t2,y1,y2,w;
int32_t j,k,ha,hb;
GET_HIGH_WORD(ha,x);
ha &= 0x7fffffff;
GET_HIGH_WORD(hb,y);
hb &= 0x7fffffff;
if(hb > ha) {
a=y;
b=x;
j=ha;
ha=hb;
hb=j;
}
else {
a=x;
b=y;
}
SET_HIGH_WORD(a,ha); /* a <- |a| */
SET_HIGH_WORD(b,hb); /* b <- |b| */
if((ha-hb)>0x3c00000) {
return a+b; /* x/y > 2**60 */
}
k=0;
if(ha > 0x5f300000) { /* a>2**500 */
if(ha >= 0x7ff00000) { /* Inf or NaN */
u_int32_t low;
w = a+b; /* for sNaN */
GET_LOW_WORD(low,a);
if(((ha&0xfffff)|low)==0) w = a;
GET_LOW_WORD(low,b);
if(((hb^0x7ff00000)|low)==0) w = b;
return w;
}
/* scale a and b by 2**-600 */
ha -= 0x25800000;
hb -= 0x25800000;
k += 600;
SET_HIGH_WORD(a,ha);
SET_HIGH_WORD(b,hb);
}
if(hb < 0x20b00000) { /* b < 2**-500 */
if(hb <= 0x000fffff) { /* subnormal b or 0 */
u_int32_t low;
GET_LOW_WORD(low,b);
if((hb|low)==0) return a;
t1=0;
SET_HIGH_WORD(t1,0x7fd00000); /* t1=2^1022 */
b *= t1;
a *= t1;
k -= 1022;
} else { /* scale a and b by 2^600 */
ha += 0x25800000; /* a *= 2^600 */
hb += 0x25800000; /* b *= 2^600 */
k -= 600;
SET_HIGH_WORD(a,ha);
SET_HIGH_WORD(b,hb);
}
}
/* medium size a and b */
w = a-b;
if (w>b) {
t1 = 0;
SET_HIGH_WORD(t1,ha);
t2 = a-t1;
w = __ieee754_sqrt(t1*t1-(b*(-b)-t2*(a+t1)));
} else {
a = a+a;
y1 = 0;
SET_HIGH_WORD(y1,hb);
y2 = b - y1;
t1 = 0;
SET_HIGH_WORD(t1,ha+0x00100000);
t2 = a - t1;
w = __ieee754_sqrt(t1*y1-(w*(-w)-(t1*y2+t2*b)));
}
if(k!=0) {
u_int32_t high;
t1 = 1.0;
GET_HIGH_WORD(high,t1);
SET_HIGH_WORD(t1,high+(k<<20));
return t1*w;
} else return w;
}
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 Cmath::__ieee754_hypot( double x, double y )
{
double a = x, b = y, t1, t2, y1, y2, w;
Cmp_signed__int32 j, k, ha, hb;
get_high_word( ha, x );
ha &= 0x7fffffff;
get_high_word( hb, y );
hb &= 0x7fffffff;
if( hb > ha )
{
a = y;
b = x;
j = ha;
ha = hb;
hb = j;
}
else
{
a = x;
b = y;
}
set_high_word( a, ha ); // a <- |a|
set_high_word( b, hb ); // b <- |b|
if( ( ha - hb ) > 0x3c00000 )
{
return a + b;
} // x/y > 2**60
k = 0;
if( ha > 0x5f300000 )
{
// a>2**500
if( ha >= 0x7ff00000 )
{
// Inf or NaN
Cmp_unsigned__int32 low;
w = a + b; // for sNaN
get_low_word( low, a );
if( ( ( ha & 0xfffff ) | low ) == 0 )
{
w = a;
}
get_low_word( low, b );
if( ( ( hb ^ 0x7ff00000 ) | low ) == 0 )
{
w = b;
}
return w;
}
// scale a and b by 2**-600
ha -= 0x25800000; hb -= 0x25800000; k += 600;
set_high_word( a, ha );
set_high_word( b, hb );
}
if( hb < 0x20b00000 )
{
// b < 2**-500
if( hb <= 0x000fffff )
{
// subnormal b or 0
Cmp_unsigned__int32 low;
get_low_word( low, b );
if( ( hb | low ) == 0 )
{
return a;
}
t1 = 0;
set_high_word( t1, 0x7fd00000 ); // t1=2^1022
b *= t1;
a *= t1;
k -= 1022;
}
else
{
// scale a and b by 2^600
ha += 0x25800000; // a *= 2^600
hb += 0x25800000; // b *= 2^600
k -= 600;
set_high_word( a, ha );
set_high_word( b, hb );
}
}
// medium size a and b
w = a - b;
if( w > b )
{
t1 = 0;
set_high_word( t1, ha );
t2 = a - t1;
w = __ieee754_sqrt( t1 * t1 - ( b * ( -b ) - t2 * ( a + t1 ) ) );
}
else
{
a = a + a;
y1 = 0;
set_high_word( y1, hb );
y2 = b - y1;
t1 = 0;
set_high_word( t1, ha + 0x00100000 );
t2 = a - t1;
w = __ieee754_sqrt( t1 * y1 - ( w * (-w) - ( t1 * y2 + t2 * b ) ) );
}
if( k != 0 )
{
Cmp_unsigned__int32 high;
t1 = 1.0;
get_high_word( high, t1 );
set_high_word( t1, high + ( k << 20 ) );
return t1 * w;
}
else
{
return w;
}
}
__complex__ double
__kernel_casinh (__complex__ double x, int adj)
{
__complex__ double res;
double rx, ix;
__complex__ double y;
/* Avoid cancellation by reducing to the first quadrant. */
rx = fabs (__real__ x);
ix = fabs (__imag__ x);
if (rx >= 1.0 / DBL_EPSILON || ix >= 1.0 / DBL_EPSILON)
{
/* For large x in the first quadrant, x + csqrt (1 + x * x)
is sufficiently close to 2 * x to make no significant
difference to the result; avoid possible overflow from
the squaring and addition. */
__real__ y = rx;
__imag__ y = ix;
if (adj)
{
double t = __real__ y;
__real__ y = __copysign (__imag__ y, __imag__ x);
__imag__ y = t;
}
res = __clog (y);
__real__ res += M_LN2;
}
else if (rx >= 0.5 && ix < DBL_EPSILON / 8.0)
{
double s = __ieee754_hypot (1.0, rx);
__real__ res = __ieee754_log (rx + s);
if (adj)
__imag__ res = __ieee754_atan2 (s, __imag__ x);
else
__imag__ res = __ieee754_atan2 (ix, s);
}
else if (rx < DBL_EPSILON / 8.0 && ix >= 1.5)
{
double s = __ieee754_sqrt ((ix + 1.0) * (ix - 1.0));
__real__ res = __ieee754_log (ix + s);
if (adj)
__imag__ res = __ieee754_atan2 (rx, __copysign (s, __imag__ x));
else
__imag__ res = __ieee754_atan2 (s, rx);
}
else if (ix > 1.0 && ix < 1.5 && rx < 0.5)
{
if (rx < DBL_EPSILON * DBL_EPSILON)
{
double ix2m1 = (ix + 1.0) * (ix - 1.0);
double s = __ieee754_sqrt (ix2m1);
__real__ res = __log1p (2.0 * (ix2m1 + ix * s)) / 2.0;
if (adj)
__imag__ res = __ieee754_atan2 (rx, __copysign (s, __imag__ x));
else
__imag__ res = __ieee754_atan2 (s, rx);
}
else
{
double ix2m1 = (ix + 1.0) * (ix - 1.0);
double rx2 = rx * rx;
double f = rx2 * (2.0 + rx2 + 2.0 * ix * ix);
double d = __ieee754_sqrt (ix2m1 * ix2m1 + f);
double dp = d + ix2m1;
double dm = f / dp;
double r1 = __ieee754_sqrt ((dm + rx2) / 2.0);
double r2 = rx * ix / r1;
__real__ res = __log1p (rx2 + dp + 2.0 * (rx * r1 + ix * r2)) / 2.0;
if (adj)
__imag__ res = __ieee754_atan2 (rx + r1, __copysign (ix + r2,
__imag__ x));
else
__imag__ res = __ieee754_atan2 (ix + r2, rx + r1);
}
}
else if (ix == 1.0 && rx < 0.5)
{
if (rx < DBL_EPSILON / 8.0)
{
__real__ res = __log1p (2.0 * (rx + __ieee754_sqrt (rx))) / 2.0;
if (adj)
__imag__ res = __ieee754_atan2 (__ieee754_sqrt (rx),
__copysign (1.0, __imag__ x));
else
__imag__ res = __ieee754_atan2 (1.0, __ieee754_sqrt (rx));
}
else
{
double d = rx * __ieee754_sqrt (4.0 + rx * rx);
double s1 = __ieee754_sqrt ((d + rx * rx) / 2.0);
double s2 = __ieee754_sqrt ((d - rx * rx) / 2.0);
__real__ res = __log1p (rx * rx + d + 2.0 * (rx * s1 + s2)) / 2.0;
if (adj)
__imag__ res = __ieee754_atan2 (rx + s1, __copysign (1.0 + s2,
__imag__ x));
else
__imag__ res = __ieee754_atan2 (1.0 + s2, rx + s1);
}
}
else if (ix < 1.0 && rx < 0.5)
{
if (ix >= DBL_EPSILON)
{
if (rx < DBL_EPSILON * DBL_EPSILON)
{
double onemix2 = (1.0 + ix) * (1.0 - ix);
double s = __ieee754_sqrt (onemix2);
__real__ res = __log1p (2.0 * rx / s) / 2.0;
if (adj)
__imag__ res = __ieee754_atan2 (s, __imag__ x);
else
__imag__ res = __ieee754_atan2 (ix, s);
}
else
{
double onemix2 = (1.0 + ix) * (1.0 - ix);
double rx2 = rx * rx;
double f = rx2 * (2.0 + rx2 + 2.0 * ix * ix);
double d = __ieee754_sqrt (onemix2 * onemix2 + f);
double dp = d + onemix2;
double dm = f / dp;
double r1 = __ieee754_sqrt ((dp + rx2) / 2.0);
double r2 = rx * ix / r1;
__real__ res
= __log1p (rx2 + dm + 2.0 * (rx * r1 + ix * r2)) / 2.0;
if (adj)
__imag__ res = __ieee754_atan2 (rx + r1,
__copysign (ix + r2,
__imag__ x));
else
__imag__ res = __ieee754_atan2 (ix + r2, rx + r1);
}
}
else
{
double s = __ieee754_hypot (1.0, rx);
__real__ res = __log1p (2.0 * rx * (rx + s)) / 2.0;
if (adj)
__imag__ res = __ieee754_atan2 (s, __imag__ x);
else
__imag__ res = __ieee754_atan2 (ix, s);
}
math_check_force_underflow_nonneg (__real__ res);
}
else
{
__real__ y = (rx - ix) * (rx + ix) + 1.0;
__imag__ y = 2.0 * rx * ix;
y = __csqrt (y);
__real__ y += rx;
__imag__ y += ix;
if (adj)
{
double t = __real__ y;
__real__ y = __copysign (__imag__ y, __imag__ x);
__imag__ y = t;
}
res = __clog (y);
}
/* Give results the correct sign for the original argument. */
__real__ res = __copysign (__real__ res, __real__ x);
__imag__ res = __copysign (__imag__ res, (adj ? 1.0 : __imag__ x));
return res;
}
double sqrt(double x) { return __ieee754_sqrt(x); }
__complex__ double
__csqrt (__complex__ double x)
{
__complex__ double res;
int rcls = fpclassify (__real__ x);
int icls = fpclassify (__imag__ x);
if (__builtin_expect (rcls <= FP_INFINITE || icls <= FP_INFINITE, 0))
{
if (icls == FP_INFINITE)
{
__real__ res = HUGE_VAL;
__imag__ res = __imag__ x;
}
else if (rcls == FP_INFINITE)
{
if (__real__ x < 0.0)
{
__real__ res = icls == FP_NAN ? __nan ("") : 0;
__imag__ res = __copysign (HUGE_VAL, __imag__ x);
}
else
{
__real__ res = __real__ x;
__imag__ res = (icls == FP_NAN
? __nan ("") : __copysign (0.0, __imag__ x));
}
}
else
{
__real__ res = __nan ("");
__imag__ res = __nan ("");
}
}
else
{
if (__builtin_expect (icls == FP_ZERO, 0))
{
if (__real__ x < 0.0)
{
__real__ res = 0.0;
__imag__ res = __copysign (__ieee754_sqrt (-__real__ x),
__imag__ x);
}
else
{
__real__ res = fabs (__ieee754_sqrt (__real__ x));
__imag__ res = __copysign (0.0, __imag__ x);
}
}
else if (__builtin_expect (rcls == FP_ZERO, 0))
{
double r;
if (fabs (__imag__ x) >= 2.0 * DBL_MIN)
r = __ieee754_sqrt (0.5 * fabs (__imag__ x));
else
r = 0.5 * __ieee754_sqrt (2.0 * fabs (__imag__ x));
__real__ res = r;
__imag__ res = __copysign (r, __imag__ x);
}
else
{
double d, r, s;
int scale = 0;
if (fabs (__real__ x) > DBL_MAX / 4.0)
{
scale = 1;
__real__ x = __scalbn (__real__ x, -2 * scale);
__imag__ x = __scalbn (__imag__ x, -2 * scale);
}
else if (fabs (__imag__ x) > DBL_MAX / 4.0)
{
scale = 1;
if (fabs (__real__ x) >= 4.0 * DBL_MIN)
__real__ x = __scalbn (__real__ x, -2 * scale);
else
__real__ x = 0.0;
__imag__ x = __scalbn (__imag__ x, -2 * scale);
}
else if (fabs (__real__ x) < DBL_MIN
&& fabs (__imag__ x) < DBL_MIN)
{
scale = -(DBL_MANT_DIG / 2);
__real__ x = __scalbn (__real__ x, -2 * scale);
__imag__ x = __scalbn (__imag__ x, -2 * scale);
}
d = __ieee754_hypot (__real__ x, __imag__ x);
/* Use the identity 2 Re res Im res = Im x
to avoid cancellation error in d +/- Re x. */
if (__real__ x > 0)
{
r = __ieee754_sqrt (0.5 * (d + __real__ x));
s = 0.5 * (__imag__ x / r);
}
else
{
s = __ieee754_sqrt (0.5 * (d - __real__ x));
r = fabs (0.5 * (__imag__ x / s));
}
if (scale)
{
r = __scalbn (r, scale);
s = __scalbn (s, scale);
}
__real__ res = r;
__imag__ res = __copysign (s, __imag__ x);
}
}
return res;
}
