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Revision 1.12, Mon Aug 20 16:01:38 2007 UTC (7 years ago) by drochner
Branch: MAIN
CVS Tags: yamt-pf42-baseX, yamt-pf42-base4, yamt-pf42-base3, yamt-pf42-base2, yamt-pf42-base, yamt-pf42, yamt-pagecache-tag8, yamt-pagecache-base9, yamt-pagecache-base8, yamt-pagecache-base7, yamt-pagecache-base6, yamt-pagecache-base5, yamt-pagecache-base4, yamt-pagecache-base3, yamt-pagecache-base2, yamt-pagecache-base, yamt-pagecache, wrstuden-revivesa-base-3, wrstuden-revivesa-base-2, wrstuden-revivesa-base-1, wrstuden-revivesa-base, wrstuden-revivesa, tls-maxphys-base, tls-maxphys, tls-earlyentropy-base, tls-earlyentropy, riastradh-xf86-video-intel-2-7-1-pre-2-21-15, riastradh-drm2-base3, riastradh-drm2-base2, riastradh-drm2-base1, riastradh-drm2-base, riastradh-drm2, netbsd-7-base, netbsd-7, netbsd-6-base, netbsd-6-1-RELEASE, netbsd-6-1-RC4, netbsd-6-1-RC3, netbsd-6-1-RC2, netbsd-6-1-RC1, netbsd-6-1-4-RELEASE, netbsd-6-1-3-RELEASE, netbsd-6-1-2-RELEASE, netbsd-6-1-1-RELEASE, netbsd-6-1, netbsd-6-0-RELEASE, netbsd-6-0-RC2, netbsd-6-0-RC1, netbsd-6-0-5-RELEASE, netbsd-6-0-4-RELEASE, netbsd-6-0-3-RELEASE, netbsd-6-0-2-RELEASE, netbsd-6-0-1-RELEASE, netbsd-6-0, netbsd-6, netbsd-5-base, netbsd-5-2-RELEASE, netbsd-5-2-RC1, netbsd-5-2-2-RELEASE, netbsd-5-2-1-RELEASE, netbsd-5-2, netbsd-5-1-RELEASE, netbsd-5-1-RC4, netbsd-5-1-RC3, netbsd-5-1-RC2, netbsd-5-1-RC1, netbsd-5-1-4-RELEASE, netbsd-5-1-3-RELEASE, netbsd-5-1-2-RELEASE, netbsd-5-1-1-RELEASE, netbsd-5-1, netbsd-5-0-RELEASE, netbsd-5-0-RC4, netbsd-5-0-RC3, netbsd-5-0-RC2, netbsd-5-0-RC1, netbsd-5-0-2-RELEASE, netbsd-5-0-1-RELEASE, netbsd-5-0, netbsd-5, mjf-devfs2-base, mjf-devfs2, matt-premerge-20091211, matt-nb6-plus-nbase, matt-nb6-plus-base, matt-nb6-plus, matt-nb5-pq3-base, matt-nb5-pq3, matt-nb5-mips64-u2-k2-k4-k7-k8-k9, matt-nb5-mips64-u1-k1-k5, matt-nb5-mips64-premerge-20101231, matt-nb5-mips64-premerge-20091211, matt-nb5-mips64-k15, matt-nb5-mips64, matt-nb4-mips64-k7-u2a-k9b, matt-mips64-premerge-20101231, matt-mips64-base2, matt-armv6-prevmlocking, matt-armv6-nbase, matt-armv6-base, matt-armv6, keiichi-mipv6-base, keiichi-mipv6, jym-xensuspend-nbase, jym-xensuspend-base, jym-xensuspend, hpcarm-cleanup-nbase, hpcarm-cleanup-base, cube-autoconf-base, cube-autoconf, cherry-xenmp-base, cherry-xenmp, bouyer-quota2-nbase, bouyer-quota2-base, bouyer-quota2, agc-symver-base, agc-symver, HEAD
Changes since 1.11: +2 -1 lines

Add C99 complex support, for double and float.
Most complex function implementations are from the "c9x-complex" library,
originating from the "cephes" math library, see
http://www.netlib.org/cephes/, from Stephen L. Moshier, incorporated and
redistributed with the NetBSD license by permission of the author.

Error behaviour and other boundary conditions (branch cuts)
need to be looked at.

For namespace sanity, I've done the rename/weak alias procedure to
most of the exported functions which are also used internally.
Didn't do so for sin/cos(f) yet because assembler implementations use
them directly, and renaming functions shared between the main libm
and the machine specific "overlay" might raise binary compatibility
issues.

/* @(#)e_j1.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.
 * ====================================================
 */

#include <sys/cdefs.h>
#if defined(LIBM_SCCS) && !defined(lint)
__RCSID("$NetBSD: e_j1.c,v 1.12 2007/08/20 16:01:38 drochner Exp $");
#endif

/* __ieee754_j1(x), __ieee754_y1(x)
 * Bessel function of the first and second kinds of order zero.
 * Method -- j1(x):
 *	1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
 *	2. Reduce x to |x| since j1(x)=-j1(-x),  and
 *	   for x in (0,2)
 *		j1(x) = x/2 + x*z*R0/S0,  where z = x*x;
 *	   (precision:  |j1/x - 1/2 - R0/S0 |<2**-61.51 )
 *	   for x in (2,inf)
 * 		j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
 * 		y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
 * 	   where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
 *	   as follow:
 *		cos(x1) =  cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
 *			=  1/sqrt(2) * (sin(x) - cos(x))
 *		sin(x1) =  sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
 *			= -1/sqrt(2) * (sin(x) + cos(x))
 * 	   (To avoid cancellation, use
 *		sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 * 	    to compute the worse one.)
 *
 *	3 Special cases
 *		j1(nan)= nan
 *		j1(0) = 0
 *		j1(inf) = 0
 *
 * Method -- y1(x):
 *	1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
 *	2. For x<2.
 *	   Since
 *		y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
 *	   therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
 *	   We use the following function to approximate y1,
 *		y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
 *	   where for x in [0,2] (abs err less than 2**-65.89)
 *		U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
 *		V(z) = 1  + v0[0]*z + ... + v0[4]*z^5
 *	   Note: For tiny x, 1/x dominate y1 and hence
 *		y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
 *	3. For x>=2.
 * 		y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
 * 	   where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
 *	   by method mentioned above.
 */

#include "namespace.h"
#include "math.h"
#include "math_private.h"

static double pone(double), qone(double);

static const double
huge    = 1e300,
one	= 1.0,
invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
tpi      =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
	/* R0/S0 on [0,2] */
r00  = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
r01  =  1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */
r02  = -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */
r03  =  4.96727999609584448412e-08, /* 0x3E6AAAFA, 0x46CA0BD9 */
s01  =  1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */
s02  =  1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */
s03  =  1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */
s04  =  5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */
s05  =  1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */

static const double zero    = 0.0;

double
__ieee754_j1(double x)
{
	double z, s,c,ss,cc,r,u,v,y;
	int32_t hx,ix;

	GET_HIGH_WORD(hx,x);
	ix = hx&0x7fffffff;
	if(ix>=0x7ff00000) return one/x;
	y = fabs(x);
	if(ix >= 0x40000000) {	/* |x| >= 2.0 */
		s = sin(y);
		c = cos(y);
		ss = -s-c;
		cc = s-c;
		if(ix<0x7fe00000) {  /* make sure y+y not overflow */
		    z = cos(y+y);
		    if ((s*c)>zero) cc = z/ss;
		    else 	    ss = z/cc;
		}
	/*
	 * j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
	 * y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
	 */
		if(ix>0x48000000) z = (invsqrtpi*cc)/sqrt(y);
		else {
		    u = pone(y); v = qone(y);
		    z = invsqrtpi*(u*cc-v*ss)/sqrt(y);
		}
		if(hx<0) return -z;
		else  	 return  z;
	}
	if(ix<0x3e400000) {	/* |x|<2**-27 */
	    if(huge+x>one) return 0.5*x;/* inexact if x!=0 necessary */
	}
	z = x*x;
	r =  z*(r00+z*(r01+z*(r02+z*r03)));
	s =  one+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
	r *= x;
	return(x*0.5+r/s);
}

static const double U0[5] = {
 -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
  5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */
 -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
  2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */
 -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
};
static const double V0[5] = {
  1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
  2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
  1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
  6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
  1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
};

double
__ieee754_y1(double x)
{
	double z, s,c,ss,cc,u,v;
	int32_t hx,ix,lx;

	EXTRACT_WORDS(hx,lx,x);
        ix = 0x7fffffff&hx;
    /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
	if(ix>=0x7ff00000) return  one/(x+x*x);
        if((ix|lx)==0) return -one/zero;
        if(hx<0) return zero/zero;
        if(ix >= 0x40000000) {  /* |x| >= 2.0 */
                s = sin(x);
                c = cos(x);
                ss = -s-c;
                cc = s-c;
                if(ix<0x7fe00000) {  /* make sure x+x not overflow */
                    z = cos(x+x);
                    if ((s*c)>zero) cc = z/ss;
                    else            ss = z/cc;
                }
        /* y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
         * where x0 = x-3pi/4
         *      Better formula:
         *              cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
         *                      =  1/sqrt(2) * (sin(x) - cos(x))
         *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
         *                      = -1/sqrt(2) * (cos(x) + sin(x))
         * To avoid cancellation, use
         *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
         * to compute the worse one.
         */
                if(ix>0x48000000) z = (invsqrtpi*ss)/sqrt(x);
                else {
                    u = pone(x); v = qone(x);
                    z = invsqrtpi*(u*ss+v*cc)/sqrt(x);
                }
                return z;
        }
        if(ix<=0x3c900000) {    /* x < 2**-54 */
            return(-tpi/x);
        }
        z = x*x;
        u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
        v = one+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
        return(x*(u/v) + tpi*(__ieee754_j1(x)*__ieee754_log(x)-one/x));
}

/* For x >= 8, the asymptotic expansions of pone is
 *	1 + 15/128 s^2 - 4725/2^15 s^4 - ...,	where s = 1/x.
 * We approximate pone by
 * 	pone(x) = 1 + (R/S)
 * where  R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
 * 	  S = 1 + ps0*s^2 + ... + ps4*s^10
 * and
 *	| pone(x)-1-R/S | <= 2  ** ( -60.06)
 */

static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
  1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
  1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
  4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
  3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
  7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
};
static const double ps8[5] = {
  1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
  3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
  3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
  9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
  3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
};

static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
  1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
  6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
  1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
  5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
  5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
};
static const double ps5[5] = {
  5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
  9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
  5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
  7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
  1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
};

static const double pr3[6] = {
  3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
  1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
  3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
  3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
  9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
  4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
};
static const double ps3[5] = {
  3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
  3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
  1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
  8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
  1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
};

static const double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
  1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
  2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
  1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
  1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
  5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
};
static const double ps2[5] = {
  2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
  1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
  2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
  1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
  8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
};

static double
pone(double x)
{
	const double *p,*q;
	double z,r,s;
        int32_t ix;

	p = q = 0;
	GET_HIGH_WORD(ix,x);
	ix &= 0x7fffffff;
        if(ix>=0x40200000)     {p = pr8; q= ps8;}
        else if(ix>=0x40122E8B){p = pr5; q= ps5;}
        else if(ix>=0x4006DB6D){p = pr3; q= ps3;}
        else if(ix>=0x40000000){p = pr2; q= ps2;}
        z = one/(x*x);
        r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
        s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
        return one+ r/s;
}


/* For x >= 8, the asymptotic expansions of qone is
 *	3/8 s - 105/1024 s^3 - ..., where s = 1/x.
 * We approximate pone by
 * 	qone(x) = s*(0.375 + (R/S))
 * where  R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
 * 	  S = 1 + qs1*s^2 + ... + qs6*s^12
 * and
 *	| qone(x)/s -0.375-R/S | <= 2  ** ( -61.13)
 */

static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
 -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
 -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
 -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
 -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
 -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
};
static const double qs8[6] = {
  1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */
  7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */
  1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */
  7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */
  6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */
 -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
};

static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
 -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
 -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
 -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
 -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
 -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
};
static const double qs5[6] = {
  8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */
  1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */
  1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */
  4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */
  2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */
 -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
};

static const double qr3[6] = {
 -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
 -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
 -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
 -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
 -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
 -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
};
static const double qs3[6] = {
  4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */
  6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */
  3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */
  5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */
  1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */
 -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
};

static const double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
 -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
 -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
 -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
 -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
 -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
};
static const double qs2[6] = {
  2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */
  2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */
  7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */
  7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */
  1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */
 -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
};

static double
qone(double x)
{
	const double *p,*q;
	double  s,r,z;
	int32_t ix;

	p = q = 0;
	GET_HIGH_WORD(ix,x);
	ix &= 0x7fffffff;
	if(ix>=0x40200000)     {p = qr8; q= qs8;}
	else if(ix>=0x40122E8B){p = qr5; q= qs5;}
	else if(ix>=0x4006DB6D){p = qr3; q= qs3;}
	else if(ix>=0x40000000){p = qr2; q= qs2;}
	z = one/(x*x);
	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
	s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
	return (.375 + r/s)/x;
}