| /* |
| * ==================================================== |
| * 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. |
| * ==================================================== |
| */ |
| |
| /* Modifications for 128-bit long double are |
| Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov> |
| and are incorporated herein by permission of the author. The author |
| reserves the right to distribute this material elsewhere under different |
| copying permissions. These modifications are distributed here under |
| the following terms: |
| |
| This library is free software; you can redistribute it and/or |
| modify it under the terms of the GNU Lesser General Public |
| License as published by the Free Software Foundation; either |
| version 2.1 of the License, or (at your option) any later version. |
| |
| This library is distributed in the hope that it will be useful, |
| but WITHOUT ANY WARRANTY; without even the implied warranty of |
| MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
| Lesser General Public License for more details. |
| |
| You should have received a copy of the GNU Lesser General Public |
| License along with this library; if not, see |
| <http://www.gnu.org/licenses/>. */ |
| |
| /* |
| * __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 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 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 "quadmath-imp.h" |
| |
| static const __float128 |
| invsqrtpi = 5.6418958354775628694807945156077258584405E-1Q, |
| two = 2, |
| one = 1, |
| zero = 0; |
| |
| |
| __float128 |
| jnq (int n, __float128 x) |
| { |
| uint32_t se; |
| int32_t i, ix, sgn; |
| __float128 a, b, temp, di, ret; |
| __float128 z, w; |
| ieee854_float128 u; |
| |
| |
| /* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x) |
| * Thus, J(-n,x) = J(n,-x) |
| */ |
| |
| u.value = x; |
| se = u.words32.w0; |
| ix = se & 0x7fffffff; |
| |
| /* if J(n,NaN) is NaN */ |
| if (ix >= 0x7fff0000) |
| { |
| if ((u.words32.w0 & 0xffff) | u.words32.w1 | u.words32.w2 | u.words32.w3) |
| return x + x; |
| } |
| |
| if (n < 0) |
| { |
| n = -n; |
| x = -x; |
| se ^= 0x80000000; |
| } |
| if (n == 0) |
| return (j0q (x)); |
| if (n == 1) |
| return (j1q (x)); |
| sgn = (n & 1) & (se >> 31); /* even n -- 0, odd n -- sign(x) */ |
| x = fabsq (x); |
| |
| { |
| SET_RESTORE_ROUNDF128 (FE_TONEAREST); |
| if (x == 0 || ix >= 0x7fff0000) /* if x is 0 or inf */ |
| return sgn == 1 ? -zero : zero; |
| else if ((__float128) n <= x) |
| { |
| /* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */ |
| if (ix >= 0x412D0000) |
| { /* x > 2**302 */ |
| |
| /* ??? Could use an expansion for large x here. */ |
| |
| /* (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 |
| */ |
| __float128 s; |
| __float128 c; |
| sincosq (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 / sqrtq (x); |
| } |
| else |
| { |
| a = j0q (x); |
| b = j1q (x); |
| for (i = 1; i < n; i++) |
| { |
| temp = b; |
| b = b * ((__float128) (i + i) / x) - a; /* avoid underflow */ |
| a = temp; |
| } |
| } |
| } |
| else |
| { |
| if (ix < 0x3fc60000) |
| { /* x < 2**-57 */ |
| /* x is tiny, return the first Taylor expansion of J(n,x) |
| * J(n,x) = 1/n!*(x/2)^n - ... |
| */ |
| if (n >= 400) /* underflow, result < 10^-4952 */ |
| b = zero; |
| else |
| { |
| temp = x * 0.5; |
| b = temp; |
| for (a = one, i = 2; i <= n; i++) |
| { |
| a *= (__float128) 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 */ |
| __float128 t, v; |
| __float128 q0, q1, h, tmp; |
| int32_t k, m; |
| w = (n + n) / (__float128) x; |
| h = 2 / (__float128) x; |
| q0 = w; |
| z = w + h; |
| q1 = w * z - 1; |
| k = 1; |
| while (q1 < 1.0e17Q) |
| { |
| 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 * logq (fabsq (v * tmp)); |
| |
| if (tmp < 1.1356523406294143949491931077970765006170e+04Q) |
| { |
| for (i = n - 1, di = (__float128) (i + i); i > 0; i--) |
| { |
| temp = b; |
| b *= di; |
| b = b / x - a; |
| a = temp; |
| di -= two; |
| } |
| } |
| else |
| { |
| for (i = n - 1, di = (__float128) (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 > 1e100Q) |
| { |
| 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 = j0q (x); |
| w = j1q (x); |
| if (fabsq (z) >= fabsq (w)) |
| b = (t * z / b); |
| else |
| b = (t * w / a); |
| } |
| } |
| if (sgn == 1) |
| ret = -b; |
| else |
| ret = b; |
| } |
| if (ret == 0) |
| { |
| ret = copysignq (FLT128_MIN, ret) * FLT128_MIN; |
| errno = ERANGE; |
| } |
| else |
| math_check_force_underflow (ret); |
| return ret; |
| } |
| |
| |
| __float128 |
| ynq (int n, __float128 x) |
| { |
| uint32_t se; |
| int32_t i, ix; |
| int32_t sign; |
| __float128 a, b, temp, ret; |
| ieee854_float128 u; |
| |
| u.value = x; |
| se = u.words32.w0; |
| ix = se & 0x7fffffff; |
| |
| /* if Y(n,NaN) is NaN */ |
| if (ix >= 0x7fff0000) |
| { |
| if ((u.words32.w0 & 0xffff) | u.words32.w1 | u.words32.w2 | u.words32.w3) |
| return x + x; |
| } |
| if (x <= 0) |
| { |
| if (x == 0) |
| return ((n < 0 && (n & 1) != 0) ? 1 : -1) / 0.0Q; |
| if (se & 0x80000000) |
| return zero / (zero * x); |
| } |
| sign = 1; |
| if (n < 0) |
| { |
| n = -n; |
| sign = 1 - ((n & 1) << 1); |
| } |
| if (n == 0) |
| return (y0q (x)); |
| { |
| SET_RESTORE_ROUNDF128 (FE_TONEAREST); |
| if (n == 1) |
| { |
| ret = sign * y1q (x); |
| goto out; |
| } |
| if (ix >= 0x7fff0000) |
| return zero; |
| if (ix >= 0x412D0000) |
| { /* x > 2**302 */ |
| |
| /* ??? See comment above on the possible futility of this. */ |
| |
| /* (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 |
| */ |
| __float128 s; |
| __float128 c; |
| sincosq (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 / sqrtq (x); |
| } |
| else |
| { |
| a = y0q (x); |
| b = y1q (x); |
| /* quit if b is -inf */ |
| u.value = b; |
| se = u.words32.w0 & 0xffff0000; |
| for (i = 1; i < n && se != 0xffff0000; i++) |
| { |
| temp = b; |
| b = ((__float128) (i + i) / x) * b - a; |
| u.value = b; |
| se = u.words32.w0 & 0xffff0000; |
| a = temp; |
| } |
| } |
| /* If B is +-Inf, set up errno accordingly. */ |
| if (! finiteq (b)) |
| errno = ERANGE; |
| if (sign > 0) |
| ret = b; |
| else |
| ret = -b; |
| } |
| out: |
| if (isinfq (ret)) |
| ret = copysignq (FLT128_MAX, ret) * FLT128_MAX; |
| return ret; |
| } |
