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authorSzabolcs Nagy <nsz@port70.net>2013-01-01 22:20:45 +0100
committerSzabolcs Nagy <nsz@port70.net>2013-01-01 22:20:45 +0100
commit5652d70054daf3c2c9b6d475fdf9d24a940e51aa (patch)
treebe9e8823682335c7fda681166948d20e61031b23 /src/math/jn.c
parent5bb6b24952e3f95ede42b60ac64a33ac34b8e272 (diff)
downloadmusl-5652d70054daf3c2c9b6d475fdf9d24a940e51aa.tar.gz
math: bessel cleanup (jn.c and jnf.c)
both jn and yn functions had integer overflow issues for large and small n to handle these issues nm1 (== |n|-1) is used instead of n and -n in the code and some loops are changed to make sure the iteration counter does not overflow (another solution could be to use larger integer type or even double but that has more size and runtime cost, on x87 loading int64_t or even uint32_t into an fpu register is more than two times slower than loading int32_t, and using double for n slows down iteration logic) yn(-1,0) now returns inf posix2008 specifies that on overflow and at +-0 all y0,y1,yn functions return -inf, this is not consistent with math when n<0 odd integer in yn (eg. when x->0, yn(-1,x)->inf, but historically yn(-1,0) seems to be special cased and returned -inf) some threshold values in jnf and ynf were fixed that seems to be incorrectly copy-pasted from the double version
Diffstat (limited to 'src/math/jn.c')
-rw-r--r--src/math/jn.c171
1 files changed, 87 insertions, 84 deletions
diff --git a/src/math/jn.c b/src/math/jn.c
index d95af15d..4878a54f 100644
--- a/src/math/jn.c
+++ b/src/math/jn.c
@@ -20,7 +20,7 @@
* 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
@@ -32,7 +32,6 @@
* yn(n,x) is similar in all respects, except
* that forward recursion is used for all
* values of n>1.
- *
*/
#include "libm.h"
@@ -41,33 +40,39 @@ static const double invsqrtpi = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x504
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)
@@ -81,19 +86,21 @@ double jn(int n, double x)
* 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;
}
}
@@ -102,12 +109,13 @@ double jn(int n, double x)
/* 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 */
}
@@ -143,13 +151,14 @@ double jn(int n, double x)
* 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) {
@@ -159,9 +168,8 @@ double jn(int n, double x)
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)
@@ -172,26 +180,20 @@ double jn(int n, double x)
* 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;
@@ -206,39 +208,40 @@ double jn(int n, double x)
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)
@@ -252,26 +255,26 @@ double yn(int n, double x)
* 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;
}