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Diffstat (limited to 'REORG.TODO/sysdeps/ieee754/dbl-64/e_jn.c')
| -rw-r--r-- | REORG.TODO/sysdeps/ieee754/dbl-64/e_jn.c | 347 |
1 files changed, 347 insertions, 0 deletions
diff --git a/REORG.TODO/sysdeps/ieee754/dbl-64/e_jn.c b/REORG.TODO/sysdeps/ieee754/dbl-64/e_jn.c new file mode 100644 index 0000000000..3fecf82f10 --- /dev/null +++ b/REORG.TODO/sysdeps/ieee754/dbl-64/e_jn.c @@ -0,0 +1,347 @@ +/* @(#)e_jn.c 5.1 93/09/24 */ +/* + * ==================================================== + * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. + * + * Developed at SunPro, a Sun Microsystems, Inc. business. + * Permission to use, copy, modify, and distribute this + * software is freely granted, provided that this notice + * is preserved. + * ==================================================== + */ + +/* + * __ieee754_jn(n, x), __ieee754_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 overflow 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 us 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 <errno.h> +#include <float.h> +#include <math.h> +#include <math_private.h> + +static const double + invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */ + two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */ + one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */ + +static const double zero = 0.00000000000000000000e+00; + +double +__ieee754_jn (int n, double x) +{ + int32_t i, hx, ix, lx, sgn; + double a, b, temp, di, ret; + double z, w; + + /* 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 (__glibc_unlikely ((ix | ((u_int32_t) (lx | -lx)) >> 31) > 0x7ff00000)) + return x + x; + if (n < 0) + { + n = -n; + x = -x; + hx ^= 0x80000000; + } + if (n == 0) + return (__ieee754_j0 (x)); + if (n == 1) + return (__ieee754_j1 (x)); + sgn = (n & 1) & (hx >> 31); /* even n -- 0, odd n -- sign(x) */ + x = fabs (x); + { + SET_RESTORE_ROUND (FE_TONEAREST); + if (__glibc_unlikely ((ix | lx) == 0 || ix >= 0x7ff00000)) + /* if x is 0 or inf */ + return sgn == 1 ? -zero : zero; + else if ((double) n <= 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 + */ + double s; + double c; + __sincos (x, &s, &c); + switch (n & 3) + { + case 0: temp = c + s; break; + case 1: temp = -c + s; break; + case 2: temp = -c - s; break; + case 3: temp = c - s; break; + } + b = invsqrtpi * temp / __ieee754_sqrt (x); + } + else + { + a = __ieee754_j0 (x); + b = __ieee754_j1 (x); + for (i = 1; i < n; i++) + { + temp = b; + b = b * ((double) (i + 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 (n > 33) /* underflow */ + b = zero; + else + { + temp = x * 0.5; b = temp; + for (a = one, i = 2; i <= n; 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, v; + double q0, q1, h, tmp; int32_t k, m; + w = (n + n) / (double) x; h = 2.0 / (double) x; + q0 = w; z = w + h; q1 = w * z - 1.0; k = 1; + while (q1 < 1.0e9) + { + k += 1; z += h; + tmp = z * q1 - q0; + q0 = q1; + q1 = tmp; + } + m = n + n; + for (t = zero, i = 2 * (n + k); i >= m; i -= 2) + t = one / (i / x - t); + a = t; + b = one; + /* 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 = n; + v = two / x; + tmp = tmp * __ieee754_log (fabs (v * tmp)); + if (tmp < 7.09782712893383973096e+02) + { + for (i = n - 1, di = (double) (i + i); i > 0; i--) + { + temp = b; + b *= di; + b = b / x - a; + a = temp; + di -= two; + } + } + else + { + for (i = n - 1, di = (double) (i + i); i > 0; i--) + { + temp = b; + b *= di; + b = b / x - a; + a = temp; + di -= two; + /* scale b to avoid spurious overflow */ + if (b > 1e100) + { + a /= b; + t /= b; + b = one; + } + } + } + /* j0() and j1() suffer enormous loss of precision at and + * near zero; however, we know that their zero points never + * coincide, so just choose the one further away from zero. + */ + z = __ieee754_j0 (x); + w = __ieee754_j1 (x); + if (fabs (z) >= fabs (w)) + b = (t * z / b); + else + b = (t * w / a); + } + } + if (sgn == 1) + ret = -b; + else + ret = b; + ret = math_narrow_eval (ret); + } + if (ret == 0) + { + ret = math_narrow_eval (__copysign (DBL_MIN, ret) * DBL_MIN); + __set_errno (ERANGE); + } + else + math_check_force_underflow (ret); + return ret; +} +strong_alias (__ieee754_jn, __jn_finite) + +double +__ieee754_yn (int n, double x) +{ + int32_t i, hx, ix, lx; + int32_t sign; + double a, b, temp, ret; + + EXTRACT_WORDS (hx, lx, x); + ix = 0x7fffffff & hx; + /* if Y(n,NaN) is NaN */ + if (__glibc_unlikely ((ix | ((u_int32_t) (lx | -lx)) >> 31) > 0x7ff00000)) + return x + x; + if (__glibc_unlikely ((ix | lx) == 0)) + return -HUGE_VAL + x; + /* -inf and overflow exception. */; + if (__glibc_unlikely (hx < 0)) + return zero / (zero * x); + sign = 1; + if (n < 0) + { + n = -n; + sign = 1 - ((n & 1) << 1); + } + if (n == 0) + return (__ieee754_y0 (x)); + { + SET_RESTORE_ROUND (FE_TONEAREST); + if (n == 1) + { + ret = sign * __ieee754_y1 (x); + goto out; + } + if (__glibc_unlikely (ix == 0x7ff00000)) + return zero; + 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 + */ + double c; + double s; + __sincos (x, &s, &c); + switch (n & 3) + { + case 0: temp = s - c; break; + case 1: temp = -s - c; break; + case 2: temp = -s + c; break; + case 3: temp = s + c; break; + } + b = invsqrtpi * temp / __ieee754_sqrt (x); + } + else + { + u_int32_t high; + a = __ieee754_y0 (x); + b = __ieee754_y1 (x); + /* quit if b is -inf */ + GET_HIGH_WORD (high, b); + for (i = 1; i < n && high != 0xfff00000; i++) + { + temp = b; + b = ((double) (i + i) / x) * b - a; + GET_HIGH_WORD (high, b); + a = temp; + } + /* If B is +-Inf, set up errno accordingly. */ + if (!isfinite (b)) + __set_errno (ERANGE); + } + if (sign > 0) + ret = b; + else + ret = -b; + } + out: + if (isinf (ret)) + ret = __copysign (DBL_MAX, ret) * DBL_MAX; + return ret; +} +strong_alias (__ieee754_yn, __yn_finite) |