2fe8fb192f
There is important information about booting non-ack images in docs/UPDATING. ack/aout-format images can't be built any more, and booting clang/ELF-format ones is a little different. Updating to the new boot monitor is recommended. Changes in this commit: . drop boot monitor -> allowing dropping ack support . facility to copy ELF boot files to /boot so that old boot monitor can still boot fairly easily, see UPDATING . no more ack-format libraries -> single-case libraries . some cleanup of OBJECT_FMT, COMPILER_TYPE, etc cases . drop several ack toolchain commands, but not all support commands (e.g. aal is gone but acksize is not yet). . a few libc files moved to netbsd libc dir . new /bin/date as minix date used code in libc/ . test compile fix . harmonize includes . /usr/lib is no longer special: without ack, /usr/lib plays no kind of special bootstrapping role any more and bootstrapping is done exclusively through packages, so releases depend even less on the state of the machine making them now. . rename nbsd_lib* to lib* . reduce mtree
274 lines
7.1 KiB
C
274 lines
7.1 KiB
C
/* @(#)e_jn.c 5.1 93/09/24 */
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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#include <sys/cdefs.h>
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#if defined(LIBM_SCCS) && !defined(lint)
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__RCSID("$NetBSD: e_jn.c,v 1.14 2010/11/29 15:10:06 drochner Exp $");
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#endif
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/*
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* __ieee754_jn(n, x), __ieee754_yn(n, x)
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* floating point Bessel's function of the 1st and 2nd kind
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* of order n
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*
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* Special cases:
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* y0(0)=y1(0)=yn(n,0) = -inf with division by zero signal;
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* y0(-ve)=y1(-ve)=yn(n,-ve) are NaN with invalid signal.
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* Note 2. About jn(n,x), yn(n,x)
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* For n=0, j0(x) is called,
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* for n=1, j1(x) is called,
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* for n<x, forward recursion us used starting
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* from values of j0(x) and j1(x).
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* for n>x, a continued fraction approximation to
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* j(n,x)/j(n-1,x) is evaluated and then backward
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* recursion is used starting from a supposed value
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* for j(n,x). The resulting value of j(0,x) is
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* compared with the actual value to correct the
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* supposed value of j(n,x).
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*
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* yn(n,x) is similar in all respects, except
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* that forward recursion is used for all
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* values of n>1.
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*
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*/
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#include "namespace.h"
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#include "math.h"
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#include "math_private.h"
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static const double
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invsqrtpi= 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
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two = 2.00000000000000000000e+00, /* 0x40000000, 0x00000000 */
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one = 1.00000000000000000000e+00; /* 0x3FF00000, 0x00000000 */
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static const double zero = 0.00000000000000000000e+00;
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double
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__ieee754_jn(int n, double x)
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{
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int32_t i,hx,ix,lx, sgn;
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double a, b, temp, di;
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double z, w;
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temp = 0;
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/* J(-n,x) = (-1)^n * J(n, x), J(n, -x) = (-1)^n * J(n, x)
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* Thus, J(-n,x) = J(n,-x)
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*/
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EXTRACT_WORDS(hx,lx,x);
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ix = 0x7fffffff&hx;
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/* if J(n,NaN) is NaN */
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if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
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if(n<0){
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n = -n;
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x = -x;
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hx ^= 0x80000000;
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}
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if(n==0) return(__ieee754_j0(x));
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if(n==1) return(__ieee754_j1(x));
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sgn = (n&1)&(hx>>31); /* even n -- 0, odd n -- sign(x) */
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x = fabs(x);
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if((ix|lx)==0||ix>=0x7ff00000) /* if x is 0 or inf */
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b = zero;
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else if((double)n<=x) {
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/* Safe to use J(n+1,x)=2n/x *J(n,x)-J(n-1,x) */
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if(ix>=0x52D00000) { /* x > 2**302 */
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/* (x >> n**2)
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* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Let s=sin(x), c=cos(x),
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* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
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*
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* n sin(xn)*sqt2 cos(xn)*sqt2
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* ----------------------------------
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* 0 s-c c+s
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* 1 -s-c -c+s
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* 2 -s+c -c-s
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* 3 s+c c-s
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*/
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switch(n&3) {
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case 0: temp = cos(x)+sin(x); break;
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case 1: temp = -cos(x)+sin(x); break;
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case 2: temp = -cos(x)-sin(x); break;
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case 3: temp = cos(x)-sin(x); break;
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}
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b = invsqrtpi*temp/sqrt(x);
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} else {
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a = __ieee754_j0(x);
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b = __ieee754_j1(x);
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for(i=1;i<n;i++){
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temp = b;
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b = b*((double)(i+i)/x) - a; /* avoid underflow */
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a = temp;
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}
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}
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} else {
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if(ix<0x3e100000) { /* x < 2**-29 */
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/* x is tiny, return the first Taylor expansion of J(n,x)
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* J(n,x) = 1/n!*(x/2)^n - ...
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*/
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if(n>33) /* underflow */
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b = zero;
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else {
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temp = x*0.5; b = temp;
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for (a=one,i=2;i<=n;i++) {
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a *= (double)i; /* a = n! */
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b *= temp; /* b = (x/2)^n */
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}
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b = b/a;
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}
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} else {
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/* use backward recurrence */
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/* x x^2 x^2
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* J(n,x)/J(n-1,x) = ---- ------ ------ .....
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* 2n - 2(n+1) - 2(n+2)
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*
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* 1 1 1
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* (for large x) = ---- ------ ------ .....
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* 2n 2(n+1) 2(n+2)
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* -- - ------ - ------ -
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* x x x
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*
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* Let w = 2n/x and h=2/x, then the above quotient
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* is equal to the continued fraction:
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* 1
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* = -----------------------
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* 1
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* w - -----------------
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* 1
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* w+h - ---------
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* w+2h - ...
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*
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* To determine how many terms needed, let
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* Q(0) = w, Q(1) = w(w+h) - 1,
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* Q(k) = (w+k*h)*Q(k-1) - Q(k-2),
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* When Q(k) > 1e4 good for single
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* When Q(k) > 1e9 good for double
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* When Q(k) > 1e17 good for quadruple
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*/
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/* determine k */
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double t,v;
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double q0,q1,h,tmp; int32_t k,m;
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w = (n+n)/(double)x; h = 2.0/(double)x;
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q0 = w; z = w+h; q1 = w*z - 1.0; k=1;
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while(q1<1.0e9) {
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k += 1; z += h;
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tmp = z*q1 - q0;
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q0 = q1;
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q1 = tmp;
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}
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m = n+n;
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for(t=zero, i = 2*(n+k); i>=m; i -= 2) t = one/(i/x-t);
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a = t;
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b = one;
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/* estimate log((2/x)^n*n!) = n*log(2/x)+n*ln(n)
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* Hence, if n*(log(2n/x)) > ...
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* single 8.8722839355e+01
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* double 7.09782712893383973096e+02
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* long double 1.1356523406294143949491931077970765006170e+04
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* then recurrent value may overflow and the result is
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* likely underflow to zero
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*/
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tmp = n;
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v = two/x;
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tmp = tmp*__ieee754_log(fabs(v*tmp));
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if(tmp<7.09782712893383973096e+02) {
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for(i=n-1,di=(double)(i+i);i>0;i--){
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temp = b;
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b *= di;
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b = b/x - a;
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a = temp;
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di -= two;
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}
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} else {
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for(i=n-1,di=(double)(i+i);i>0;i--){
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temp = b;
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b *= di;
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b = b/x - a;
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a = temp;
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di -= two;
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/* scale b to avoid spurious overflow */
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if(b>1e100) {
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a /= b;
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t /= b;
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b = one;
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}
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}
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}
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z = __ieee754_j0(x);
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w = __ieee754_j1(x);
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if (fabs(z) >= fabs(w))
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b = (t*z/b);
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else
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b = (t*w/a);
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}
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}
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if(sgn==1) return -b; else return b;
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}
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double
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__ieee754_yn(int n, double x)
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{
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int32_t i,hx,ix,lx;
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int32_t sign;
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double a, b, temp;
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temp = 0;
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EXTRACT_WORDS(hx,lx,x);
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ix = 0x7fffffff&hx;
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/* if Y(n,NaN) is NaN */
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if((ix|((u_int32_t)(lx|-lx))>>31)>0x7ff00000) return x+x;
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if((ix|lx)==0) return -one/zero;
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if(hx<0) return zero/zero;
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sign = 1;
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if(n<0){
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n = -n;
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sign = 1 - ((n&1)<<1);
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}
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if(n==0) return(__ieee754_y0(x));
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if(n==1) return(sign*__ieee754_y1(x));
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if(ix==0x7ff00000) return zero;
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if(ix>=0x52D00000) { /* x > 2**302 */
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/* (x >> n**2)
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* Jn(x) = cos(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Yn(x) = sin(x-(2n+1)*pi/4)*sqrt(2/x*pi)
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* Let s=sin(x), c=cos(x),
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* xn=x-(2n+1)*pi/4, sqt2 = sqrt(2),then
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*
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* n sin(xn)*sqt2 cos(xn)*sqt2
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* ----------------------------------
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* 0 s-c c+s
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* 1 -s-c -c+s
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* 2 -s+c -c-s
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* 3 s+c c-s
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*/
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switch(n&3) {
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case 0: temp = sin(x)-cos(x); break;
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case 1: temp = -sin(x)-cos(x); break;
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case 2: temp = -sin(x)+cos(x); break;
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case 3: temp = sin(x)+cos(x); break;
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}
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b = invsqrtpi*temp/sqrt(x);
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} else {
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u_int32_t high;
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a = __ieee754_y0(x);
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b = __ieee754_y1(x);
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/* quit if b is -inf */
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GET_HIGH_WORD(high,b);
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for(i=1;i<n&&high!=0xfff00000;i++){
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temp = b;
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b = ((double)(i+i)/x)*b - a;
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GET_HIGH_WORD(high,b);
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a = temp;
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}
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}
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if(sign>0) return b; else return -b;
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}
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