static double pone(double), qone(double);
static const double
-huge = 1e300,
invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
-tpi = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
+tpi = 6.36619772367581382433e-01; /* 0x3FE45F30, 0x6DC9C883 */
+
+static double common(uint32_t ix, double x, int y1, int sign)
+{
+ double z,s,c,ss,cc;
+
+ /*
+ * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x-3pi/4)-q1(x)*sin(x-3pi/4))
+ * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x-3pi/4)+q1(x)*cos(x-3pi/4))
+ *
+ * sin(x-3pi/4) = -(sin(x) + cos(x))/sqrt(2)
+ * cos(x-3pi/4) = (sin(x) - cos(x))/sqrt(2)
+ * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
+ */
+ s = sin(x);
+ if (y1)
+ s = -s;
+ c = cos(x);
+ cc = s-c;
+ if (ix < 0x7fe00000) {
+ /* avoid overflow in 2*x */
+ ss = -s-c;
+ z = cos(2*x);
+ if (s*c > 0)
+ cc = z/ss;
+ else
+ ss = z/cc;
+ if (ix < 0x48000000) {
+ if (y1)
+ ss = -ss;
+ cc = pone(x)*cc-qone(x)*ss;
+ }
+ }
+ if (sign)
+ cc = -cc;
+ return invsqrtpi*cc/sqrt(x);
+}
+
/* R0/S0 on [0,2] */
+static const double
r00 = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
r01 = 1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */
r02 = -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */
double j1(double x)
{
- double z,s,c,ss,cc,r,u,v,y;
- int32_t hx,ix;
+ double z,r,s;
+ uint32_t ix;
+ int sign;
- GET_HIGH_WORD(hx, x);
- ix = hx & 0x7fffffff;
+ GET_HIGH_WORD(ix, x);
+ sign = ix>>31;
+ ix &= 0x7fffffff;
if (ix >= 0x7ff00000)
- return 1.0/x;
- y = fabs(x);
- if (ix >= 0x40000000) { /* |x| >= 2.0 */
- s = sin(y);
- c = cos(y);
- ss = -s-c;
- cc = s-c;
- if (ix < 0x7fe00000) { /* make sure y+y not overflow */
- z = cos(y+y);
- if (s*c > 0.0)
- 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)/sqrt(y);
- else {
- u = pone(y);
- v = qone(y);
- z = invsqrtpi*(u*cc-v*ss)/sqrt(y);
- }
- if (hx < 0)
- return -z;
- else
- return z;
- }
- if (ix < 0x3e400000) { /* |x| < 2**-27 */
- /* raise inexact if x!=0 */
- if (huge+x > 1.0)
- return 0.5*x;
- }
- z = x*x;
- r = z*(r00+z*(r01+z*(r02+z*r03)));
- s = 1.0+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
- r *= x;
- return x*0.5 + r/s;
+ return 1/(x*x);
+ if (ix >= 0x40000000) /* |x| >= 2 */
+ return common(ix, fabs(x), 0, sign);
+ if (ix >= 0x38000000) { /* |x| >= 2**-127 */
+ z = x*x;
+ r = z*(r00+z*(r01+z*(r02+z*r03)));
+ s = 1+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
+ z = r/s;
+ } else
+ /* avoid underflow, raise inexact if x!=0 */
+ z = x;
+ return (0.5 + z)*x;
}
static const double U0[5] = {
1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
};
-
double y1(double x)
{
- double z,s,c,ss,cc,u,v;
- int32_t hx,ix,lx;
+ double z,u,v;
+ uint32_t ix,lx;
- EXTRACT_WORDS(hx, lx, x);
- ix = 0x7fffffff & hx;
- /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
+ EXTRACT_WORDS(ix, lx, x);
+ /* y1(nan)=nan, y1(<0)=nan, y1(0)=-inf, y1(inf)=0 */
+ if ((ix<<1 | lx) == 0)
+ return -1/0.0;
+ if (ix>>31)
+ return 0/0.0;
if (ix >= 0x7ff00000)
- return 1.0/(x+x*x);
- if ((ix|lx) == 0)
- return -1.0/0.0;
- if (hx < 0)
- return 0.0/0.0;
- if (ix >= 0x40000000) { /* |x| >= 2.0 */
- s = sin(x);
- c = cos(x);
- ss = -s-c;
- cc = s-c;
- if (ix < 0x7fe00000) { /* make sure x+x not overflow */
- z = cos(x+x);
- if (s*c > 0.0)
- 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)/sqrt(x);
- else {
- u = pone(x);
- v = qone(x);
- z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
- }
- return z;
- }
- if (ix <= 0x3c900000) /* x < 2**-54 */
+ return 1/x;
+
+ if (ix >= 0x40000000) /* x >= 2 */
+ return common(ix, x, 1, 0);
+ if (ix < 0x3c900000) /* x < 2**-54 */
return -tpi/x;
z = x*x;
u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
- v = 1.0+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
- return x*(u/v) + tpi*(j1(x)*log(x)-1.0/x);
+ v = 1+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
+ return x*(u/v) + tpi*(j1(x)*log(x)-1/x);
}
/* For x >= 8, the asymptotic expansions of pone is
static double pone(double x)
{
const double *p,*q;
- double z,r,s;
- int32_t ix;
+ double_t z,r,s;
+ uint32_t ix;
GET_HIGH_WORD(ix, x);
ix &= 0x7fffffff;
if (ix >= 0x40200000){p = pr8; q = ps8;}
else if (ix >= 0x40122E8B){p = pr5; q = ps5;}
else if (ix >= 0x4006DB6D){p = pr3; q = ps3;}
- else if (ix >= 0x40000000){p = pr2; q = ps2;}
+ else /*ix >= 0x40000000*/ {p = pr2; q = ps2;}
z = 1.0/(x*x);
r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
static double qone(double x)
{
const double *p,*q;
- double s,r,z;
- int32_t ix;
+ double_t s,r,z;
+ uint32_t ix;
GET_HIGH_WORD(ix, x);
ix &= 0x7fffffff;
if (ix >= 0x40200000){p = qr8; q = qs8;}
else if (ix >= 0x40122E8B){p = qr5; q = qs5;}
else if (ix >= 0x4006DB6D){p = qr3; q = qs3;}
- else if (ix >= 0x40000000){p = qr2; q = qs2;}
+ else /*ix >= 0x40000000*/ {p = qr2; q = qs2;}
z = 1.0/(x*x);
r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
projects / musl / blobdiff