* Note 2. About jn(n,x), yn(n,x)
* For n=0, j0(x) is called,
* for n=1, j1(x) is called,
- * for n<x, forward recursion us used starting
+ * for n<=x, forward recursion is used starting
* from values of j0(x) and j1(x).
* for n>x, a continued fraction approximation to
* j(n,x)/j(n-1,x) is evaluated and then backward
* yn(n,x) is similar in all respects, except
* that forward recursion is used for all
* values of n>1.
- *
*/
#include "libm.h"
double jn(int n, double x)
{
- int32_t i,hx,ix,lx,sgn;
- double a, b, temp, di;
- double z, w;
+ uint32_t ix, lx;
+ int nm1, i, sign;
+ double a, b, temp;
+
+ EXTRACT_WORDS(ix, lx, x);
+ sign = ix>>31;
+ ix &= 0x7fffffff;
+
+ if ((ix | (lx|-lx)>>31) > 0x7ff00000) /* nan */
+ return x;
/* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
* Thus, J(-n,x) = J(n,-x)
*/
- EXTRACT_WORDS(hx, lx, x);
- ix = 0x7fffffff & hx;
- /* if J(n,NaN) is NaN */
- if ((ix|((uint32_t)(lx|-lx))>>31) > 0x7ff00000)
- return x+x;
+ /* nm1 = |n|-1 is used instead of |n| to handle n==INT_MIN */
+ if (n == 0)
+ return j0(x);
if (n < 0) {
- n = -n;
+ nm1 = -(n+1);
x = -x;
- hx ^= 0x80000000;
- }
- if (n == 0) return j0(x);
- if (n == 1) return j1(x);
+ sign ^= 1;
+ } else
+ nm1 = n-1;
+ if (nm1 == 0)
+ return j1(x);
- sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
+ sign &= n; /* even n: 0, odd n: signbit(x) */
x = fabs(x);
- if ((ix|lx) == 0 || ix >= 0x7ff00000) /* if x is 0 or inf */
+ if ((ix|lx) == 0 || ix == 0x7ff00000) /* if x is 0 or inf */
b = 0.0;
- else if ((double)n <= x) {
+ else if (nm1 < x) {
/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
- if (ix >= 0x52D00000) { /* x > 2**302 */
+ if (ix >= 0x52d00000) { /* x > 2**302 */
/* (x >> n**2)
* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
* 2 -s+c -c-s
* 3 s+c c-s
*/
- switch(n&3) {
- case 0: temp = cos(x)+sin(x); break;
- case 1: temp = -cos(x)+sin(x); break;
- case 2: temp = -cos(x)-sin(x); break;
- case 3: temp = cos(x)-sin(x); break;
+ switch(nm1&3) {
+ case 0: temp = -cos(x)+sin(x); break;
+ case 1: temp = -cos(x)-sin(x); break;
+ case 2: temp = cos(x)-sin(x); break;
+ default:
+ case 3: temp = cos(x)+sin(x); break;
}
b = invsqrtpi*temp/sqrt(x);
} else {
a = j0(x);
b = j1(x);
- for (i=1; i<n; i++){
+ for (i=0; i<nm1; ) {
+ i++;
temp = b;
- b = b*((double)(i+i)/x) - a; /* avoid underflow */
+ b = b*(2.0*i/x) - a; /* avoid underflow */
a = temp;
}
}
/* x is tiny, return the first Taylor expansion of J(n,x)
* J(n,x) = 1/n!*(x/2)^n - ...
*/
- if (n > 33) /* underflow */
+ if (nm1 > 32) /* underflow */
b = 0.0;
else {
temp = x*0.5;
b = temp;
- for (a=1.0,i=2; i<=n; i++) {
+ a = 1.0;
+ for (i=2; i<=nm1+1; i++) {
a *= (double)i; /* a = n! */
b *= temp; /* b = (x/2)^n */
}
* When Q(k) > 1e17 good for quadruple
*/
/* determine k */
- double t,v;
- double q0,q1,h,tmp;
- int32_t k,m;
+ double t,q0,q1,w,h,z,tmp,nf;
+ int k;
- w = (n+n)/(double)x; h = 2.0/(double)x;
- q0 = w;
+ nf = nm1 + 1.0;
+ w = 2*nf/x;
+ h = 2/x;
z = w+h;
+ q0 = w;
q1 = w*z - 1.0;
k = 1;
while (q1 < 1.0e9) {
q0 = q1;
q1 = tmp;
}
- m = n+n;
- for (t=0.0, i = 2*(n+k); i>=m; i -= 2)
- t = 1.0/(i/x-t);
+ for (t=0.0, i=k; i>=0; i--)
+ t = 1/(2*(i+nf)/x - t);
a = t;
b = 1.0;
/* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
* then recurrent value may overflow and the result is
* likely underflow to zero
*/
- tmp = n;
- v = 2.0/x;
- tmp = tmp*log(fabs(v*tmp));
+ tmp = nf*log(fabs(w));
if (tmp < 7.09782712893383973096e+02) {
- for (i=n-1,di=(double)(i+i); i>0; i--) {
+ for (i=nm1; i>0; i--) {
temp = b;
- b *= di;
- b = b/x - a;
+ b = b*(2.0*i)/x - a;
a = temp;
- di -= 2.0;
}
} else {
- for (i=n-1,di=(double)(i+i); i>0; i--) {
+ for (i=nm1; i>0; i--) {
temp = b;
- b *= di;
- b = b/x - a;
+ b = b*(2.0*i)/x - a;
a = temp;
- di -= 2.0;
/* scale b to avoid spurious overflow */
- if (b > 1e100) {
+ if (b > 0x1p500) {
a /= b;
t /= b;
b = 1.0;
b = t*w/a;
}
}
- if (sgn==1) return -b;
- return b;
+ return sign ? -b : b;
}
-
double yn(int n, double x)
{
- int32_t i,hx,ix,lx;
- int32_t sign;
+ uint32_t ix, lx, ib;
+ int nm1, sign, i;
double a, b, temp;
- EXTRACT_WORDS(hx, lx, x);
- ix = 0x7fffffff & hx;
- /* if Y(n,NaN) is NaN */
- if ((ix|((uint32_t)(lx|-lx))>>31) > 0x7ff00000)
- return x+x;
- if ((ix|lx) == 0)
- return -1.0/0.0;
- if (hx < 0)
- return 0.0/0.0;
- sign = 1;
- if (n < 0) {
- n = -n;
- sign = 1 - ((n&1)<<1);
- }
- if (n == 0)
- return y0(x);
- if (n == 1)
- return sign*y1(x);
+ EXTRACT_WORDS(ix, lx, x);
+ sign = ix>>31;
+ ix &= 0x7fffffff;
+
+ if ((ix | (lx|-lx)>>31) > 0x7ff00000) /* nan */
+ return x;
+ if (sign && (ix|lx)!=0) /* x < 0 */
+ return 0/0.0;
if (ix == 0x7ff00000)
return 0.0;
- if (ix >= 0x52D00000) { /* x > 2**302 */
+
+ if (n == 0)
+ return y0(x);
+ if (n < 0) {
+ nm1 = -(n+1);
+ sign = n&1;
+ } else {
+ nm1 = n-1;
+ sign = 0;
+ }
+ if (nm1 == 0)
+ return sign ? -y1(x) : y1(x);
+
+ if (ix >= 0x52d00000) { /* x > 2**302 */
/* (x >> n**2)
* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
* 2 -s+c -c-s
* 3 s+c c-s
*/
- switch(n&3) {
- case 0: temp = sin(x)-cos(x); break;
- case 1: temp = -sin(x)-cos(x); break;
- case 2: temp = -sin(x)+cos(x); break;
- case 3: temp = sin(x)+cos(x); break;
+ switch(nm1&3) {
+ case 0: temp = -sin(x)-cos(x); break;
+ case 1: temp = -sin(x)+cos(x); break;
+ case 2: temp = sin(x)+cos(x); break;
+ default:
+ case 3: temp = sin(x)-cos(x); break;
}
b = invsqrtpi*temp/sqrt(x);
} else {
- uint32_t high;
a = y0(x);
b = y1(x);
/* quit if b is -inf */
- GET_HIGH_WORD(high, b);
- for (i=1; i<n && high!=0xfff00000; i++){
+ GET_HIGH_WORD(ib, b);
+ for (i=0; i<nm1 && ib!=0xfff00000; ){
+ i++;
temp = b;
- b = ((double)(i+i)/x)*b - a;
- GET_HIGH_WORD(high, b);
+ b = (2.0*i/x)*b - a;
+ GET_HIGH_WORD(ib, b);
a = temp;
}
}
- if (sign > 0) return b;
- return -b;
+ return sign ? -b : b;
}
float jnf(int n, float x)
{
- int32_t i,hx,ix, sgn;
- float a, b, temp, di;
- float z, w;
+ uint32_t ix;
+ int nm1, sign, i;
+ float a, b, temp;
+
+ GET_FLOAT_WORD(ix, x);
+ sign = ix>>31;
+ ix &= 0x7fffffff;
+ if (ix > 0x7f800000) /* nan */
+ return x;
- /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
- * Thus, J(-n,x) = J(n,-x)
- */
- GET_FLOAT_WORD(hx, x);
- ix = 0x7fffffff & hx;
- /* if J(n,NaN) is NaN */
- if (ix > 0x7f800000)
- return x+x;
+ /* J(-n,x) = J(n,-x), use |n|-1 to avoid overflow in -n */
+ if (n == 0)
+ return j0f(x);
if (n < 0) {
- n = -n;
+ nm1 = -(n+1);
x = -x;
- hx ^= 0x80000000;
- }
- if (n == 0) return j0f(x);
- if (n == 1) return j1f(x);
+ sign ^= 1;
+ } else
+ nm1 = n-1;
+ if (nm1 == 0)
+ return j1f(x);
- sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
+ sign &= n; /* even n: 0, odd n: signbit(x) */
x = fabsf(x);
- if (ix == 0 || ix >= 0x7f800000) /* if x is 0 or inf */
+ if (ix == 0 || ix == 0x7f800000) /* if x is 0 or inf */
b = 0.0f;
- else if((float)n <= x) {
+ else if (nm1 < x) {
/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
a = j0f(x);
b = j1f(x);
- for (i=1; i<n; i++){
+ for (i=0; i<nm1; ){
+ i++;
temp = b;
- b = b*((float)(i+i)/x) - a; /* avoid underflow */
+ b = b*(2.0f*i/x) - a;
a = temp;
}
} else {
- if (ix < 0x30800000) { /* x < 2**-29 */
+ if (ix < 0x35800000) { /* x < 2**-20 */
/* x is tiny, return the first Taylor expansion of J(n,x)
* J(n,x) = 1/n!*(x/2)^n - ...
*/
- if (n > 33) /* underflow */
- b = 0.0f;
- else {
- temp = 0.5f * x;
- b = temp;
- for (a=1.0f,i=2; i<=n; i++) {
- a *= (float)i; /* a = n! */
- b *= temp; /* b = (x/2)^n */
- }
- b = b/a;
+ if (nm1 > 8) /* underflow */
+ nm1 = 8;
+ temp = 0.5f * x;
+ b = temp;
+ a = 1.0f;
+ for (i=2; i<=nm1+1; i++) {
+ a *= (float)i; /* a = n! */
+ b *= temp; /* b = (x/2)^n */
}
+ b = b/a;
} else {
/* use backward recurrence */
/* x x^2 x^2
* When Q(k) > 1e17 good for quadruple
*/
/* determine k */
- float t,v;
- float q0,q1,h,tmp;
- int32_t k,m;
+ float t,q0,q1,w,h,z,tmp,nf;
+ int k;
- w = (n+n)/x;
- h = 2.0f/x;
+ nf = nm1+1.0f;
+ w = 2*nf/x;
+ h = 2/x;
z = w+h;
q0 = w;
q1 = w*z - 1.0f;
k = 1;
- while (q1 < 1.0e9f) {
+ while (q1 < 1.0e4f) {
k += 1;
z += h;
tmp = z*q1 - q0;
q0 = q1;
q1 = tmp;
}
- m = n+n;
- for (t=0.0f, i = 2*(n+k); i>=m; i -= 2)
- t = 1.0f/(i/x-t);
+ for (t=0.0f, i=k; i>=0; i--)
+ t = 1.0f/(2*(i+nf)/x-t);
a = t;
b = 1.0f;
/* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
* then recurrent value may overflow and the result is
* likely underflow to zero
*/
- tmp = n;
- v = 2.0f/x;
- tmp = tmp*logf(fabsf(v*tmp));
+ tmp = nf*logf(fabsf(w));
if (tmp < 88.721679688f) {
- for (i=n-1,di=(float)(i+i); i>0; i--) {
+ for (i=nm1; i>0; i--) {
temp = b;
- b *= di;
- b = b/x - a;
+ b = 2.0f*i*b/x - a;
a = temp;
- di -= 2.0f;
}
} else {
- for (i=n-1,di=(float)(i+i); i>0; i--){
+ for (i=nm1; i>0; i--){
temp = b;
- b *= di;
- b = b/x - a;
+ b = 2.0f*i*b/x - a;
a = temp;
- di -= 2.0f;
/* scale b to avoid spurious overflow */
- if (b > 1e10f) {
+ if (b > 0x1p60f) {
a /= b;
t /= b;
b = 1.0f;
b = t*w/a;
}
}
- if (sgn == 1) return -b;
- return b;
+ return sign ? -b : b;
}
float ynf(int n, float x)
{
- int32_t i,hx,ix,ib;
- int32_t sign;
+ uint32_t ix, ib;
+ int nm1, sign, i;
float a, b, temp;
- GET_FLOAT_WORD(hx, x);
- ix = 0x7fffffff & hx;
- /* if Y(n,NaN) is NaN */
- if (ix > 0x7f800000)
- return x+x;
- if (ix == 0)
- return -1.0f/0.0f;
- if (hx < 0)
- return 0.0f/0.0f;
- sign = 1;
- if (n < 0) {
- n = -n;
- sign = 1 - ((n&1)<<1);
- }
- if (n == 0)
- return y0f(x);
- if (n == 1)
- return sign*y1f(x);
+ GET_FLOAT_WORD(ix, x);
+ sign = ix>>31;
+ ix &= 0x7fffffff;
+ if (ix > 0x7f800000) /* nan */
+ return x;
+ if (sign && ix != 0) /* x < 0 */
+ return 0/0.0f;
if (ix == 0x7f800000)
return 0.0f;
+ if (n == 0)
+ return y0f(x);
+ if (n < 0) {
+ nm1 = -(n+1);
+ sign = n&1;
+ } else {
+ nm1 = n-1;
+ sign = 0;
+ }
+ if (nm1 == 0)
+ return sign ? -y1f(x) : y1f(x);
+
a = y0f(x);
b = y1f(x);
/* quit if b is -inf */
GET_FLOAT_WORD(ib,b);
- for (i = 1; i < n && ib != 0xff800000; i++){
+ for (i = 0; i < nm1 && ib != 0xff800000; ) {
+ i++;
temp = b;
- b = ((float)(i+i)/x)*b - a;
+ b = (2.0f*i/x)*b - a;
GET_FLOAT_WORD(ib, b);
a = temp;
}
- if (sign > 0)
- return b;
- return -b;
+ return sign ? -b : b;
}
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