diff options
Diffstat (limited to 'lib/mlibc/options/ansi/musl-generic-math/jn.c')
| -rw-r--r-- | lib/mlibc/options/ansi/musl-generic-math/jn.c | 280 |
1 files changed, 280 insertions, 0 deletions
diff --git a/lib/mlibc/options/ansi/musl-generic-math/jn.c b/lib/mlibc/options/ansi/musl-generic-math/jn.c new file mode 100644 index 0000000..4878a54 --- /dev/null +++ b/lib/mlibc/options/ansi/musl-generic-math/jn.c @@ -0,0 +1,280 @@ +/* origin: FreeBSD /usr/src/lib/msun/src/e_jn.c */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunSoft, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ +/* + * jn(n, x), yn(n, x) + * floating point Bessel's function of the 1st and 2nd kind + * of order n + * + * Special cases: + * y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal; + * y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal. + * 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 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 + * recursion is used starting from a supposed value + * for j(n,x). The resulting value of j(0,x) is + * compared with the actual value to correct the + * supposed value of j(n,x). + * + * yn(n,x) is similar in all respects, except + * that forward recursion is used for all + * values of n>1. + */ + +#include "libm.h" + +static const double invsqrtpi = 5.64189583547756279280e-01; /* 0x3FE20DD7, 0x50429B6D */ + +double jn(int n, double x) +{ + 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) + */ + /* nm1 = |n|-1 is used instead of |n| to handle n==INT_MIN */ + if (n == 0) + return j0(x); + if (n < 0) { + nm1 = -(n+1); + x = -x; + sign ^= 1; + } else + nm1 = n-1; + if (nm1 == 0) + return j1(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 */ + b = 0.0; + 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 */ + /* (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) + * Let s=sin(x), c=cos(x), + * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then + * + * n sin(xn)*sqt2 cos(xn)*sqt2 + * ---------------------------------- + * 0 s-c c+s + * 1 -s-c -c+s + * 2 -s+c -c-s + * 3 s+c c-s + */ + 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=0; i<nm1; ) { + i++; + temp = b; + b = b*(2.0*i/x) - a; /* avoid underflow */ + a = temp; + } + } + } else { + if (ix < 0x3e100000) { /* x < 2**-29 */ + /* x is tiny, return the first Taylor expansion of J(n,x) + * J(n,x) = 1/n!*(x/2)^n - ... + */ + if (nm1 > 32) /* underflow */ + b = 0.0; + else { + temp = x*0.5; + b = temp; + a = 1.0; + for (i=2; i<=nm1+1; i++) { + a *= (double)i; /* a = n! */ + b *= temp; /* b = (x/2)^n */ + } + b = b/a; + } + } else { + /* use backward recurrence */ + /* x x^2 x^2 + * J(n,x)/J(n-1,x) = ---- ------ ------ ..... + * 2n - 2(n+1) - 2(n+2) + * + * 1 1 1 + * (for large x) = ---- ------ ------ ..... + * 2n 2(n+1) 2(n+2) + * -- - ------ - ------ - + * x x x + * + * Let w = 2n/x and h=2/x, then the above quotient + * is equal to the continued fraction: + * 1 + * = ----------------------- + * 1 + * w - ----------------- + * 1 + * w+h - --------- + * w+2h - ... + * + * To determine how many terms needed, let + * Q(0) = w, Q(1) = w(w+h) - 1, + * Q(k) = (w+k*h)*Q(k-1) - Q(k-2), + * When Q(k) > 1e4 good for single + * When Q(k) > 1e9 good for double + * When Q(k) > 1e17 good for quadruple + */ + /* determine k */ + double t,q0,q1,w,h,z,tmp,nf; + int k; + + 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) { + k += 1; + z += h; + tmp = z*q1 - q0; + q0 = q1; + q1 = tmp; + } + 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) + * Hence, if n*(log(2n/x)) > ... + * single 8.8722839355e+01 + * double 7.09782712893383973096e+02 + * long double 1.1356523406294143949491931077970765006170e+04 + * then recurrent value may overflow and the result is + * likely underflow to zero + */ + tmp = nf*log(fabs(w)); + if (tmp < 7.09782712893383973096e+02) { + for (i=nm1; i>0; i--) { + temp = b; + b = b*(2.0*i)/x - a; + a = temp; + } + } else { + for (i=nm1; i>0; i--) { + temp = b; + b = b*(2.0*i)/x - a; + a = temp; + /* scale b to avoid spurious overflow */ + if (b > 0x1p500) { + a /= b; + t /= b; + b = 1.0; + } + } + } + z = j0(x); + w = j1(x); + if (fabs(z) >= fabs(w)) + b = t*z/b; + else + b = t*w/a; + } + } + return sign ? -b : b; +} + + +double yn(int n, double x) +{ + uint32_t ix, lx, ib; + int nm1, sign, i; + double a, b, temp; + + 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 (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) + * Let s=sin(x), c=cos(x), + * xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then + * + * n sin(xn)*sqt2 cos(xn)*sqt2 + * ---------------------------------- + * 0 s-c c+s + * 1 -s-c -c+s + * 2 -s+c -c-s + * 3 s+c c-s + */ + 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 { + a = y0(x); + b = y1(x); + /* quit if b is -inf */ + GET_HIGH_WORD(ib, b); + for (i=0; i<nm1 && ib!=0xfff00000; ){ + i++; + temp = b; + b = (2.0*i/x)*b - a; + GET_HIGH_WORD(ib, b); + a = temp; + } + } + return sign ? -b : b; +} |