text: /*******************************************************************************
   *
   * Copyright (C) 2014-2018 Wave Computing, Inc.
   *
   * Redistribution and use in source and binary forms, with or without
   * modification, are permitted provided that the following conditions are met:
   *
   * 1. Redistributions of source code must retain the above copyright notice,
   * this list of conditions and the following disclaimer.
   * 2. Redistributions in binary form must reproduce the above copyright notice,
   * this list of conditions and the following disclaimer in the documentation
   * and/or other materials provided with the distribution.
   * 3. Neither the name of the copyright holder nor the names of its
   * contributors may be used to endorse or promote products derived from this
   * software without specific prior written permission.
   *
   * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
   * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
   * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
   * ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
   * LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
   * CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
   * SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
   * INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
   * CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
   * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
   * POSSIBILITY OF SUCH DAMAGE.
   *
   ******************************************************************************/
  
  
  /* ef_j1.c -- float version of e_j1.c.
   * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
   */
  
  /*
   * ====================================================
   * 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 "low/_math.h"
  #include "low/_fpuinst.h"
  
  #ifdef __STDC__
  static float ponef(float), qonef(float);
  #else
  static float ponef(), qonef();
  #endif
  
  #ifdef __STDC__
  static const float 
  #else
  static float 
  #endif
  huge    = 1e30,
  one	= 1.0,
  invsqrtpi=  5.6418961287e-01, /* 0x3f106ebb */
  tpi      =  6.3661974669e-01, /* 0x3f22f983 */
  	/* R0/S0 on [0,2] */
  r00  = -6.2500000000e-02, /* 0xbd800000 */
  r01  =  1.4070566976e-03, /* 0x3ab86cfd */
  r02  = -1.5995563444e-05, /* 0xb7862e36 */
  r03  =  4.9672799207e-08, /* 0x335557d2 */
  s01  =  1.9153760746e-02, /* 0x3c9ce859 */
  s02  =  1.8594678841e-04, /* 0x3942fab6 */
  s03  =  1.1771846857e-06, /* 0x359dffc2 */
  s04  =  5.0463624390e-09, /* 0x31ad6446 */
  s05  =  1.2354227016e-11; /* 0x2d59567e */
  
  #ifdef __STDC__
  static const float zero    = 0.0;
  #else
  static float zero    = 0.0;
  #endif
  
  #ifdef __STDC__
  	float __ieee754_j1f(float x) 
  #else
  	float __ieee754_j1f(x) 
  	float x;
  #endif
  {
  	float z, s,c,ss,cc,r,u,v,y;
  	__int32_t hx,ix;
  
  	GET_FLOAT_WORD(hx,x);
  	ix = hx&0x7fffffff;
  #ifdef __MATH_NONFINITE__ 
    if(!FLT_UWORD_IS_FINITE(ix)) return one/(x*x);
  #endif /*__MATH_NONFINITE__ */
  
  #ifdef __MATH_SOFT_FLOAT__
    y=fabsf(x);
  #else /* __MATH_SOFT_FLOAT__ */
    __inst_abs_S(y,x);
  #endif /* __MATH_SOFT_FLOAT__ */
  
  	if(ix >= 0x40000000) 
    {	/* |x| >= 2.0 */
  		s = sinf(y);
  		c = cosf(y);
  		ss = -s-c;
  		cc = s-c;
  		if(ix<=FLT_UWORD_HALF_MAX) 
      {  /* make sure y+y not overflow */
  		  z = cosf(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>0x80000000) 
      {
  #ifdef __MATH_SOFT_FLOAT__
        z=__ieee754_sqrtf(y);
  #else /* __MATH_SOFT_FLOAT__ */
        __inst_sqrt_S(z,y);
  #endif /* __MATH_SOFT_FLOAT__ */
  
        z = (invsqrtpi*cc)/z;
      }
  		else
      {
  		  u = ponef(y); v = qonef(y);
  #ifdef __MATH_SOFT_FLOAT__
        z=__ieee754_sqrtf(y);
  #else /* __MATH_SOFT_FLOAT__ */
        __inst_sqrt_S(z,y);
  #endif /* __MATH_SOFT_FLOAT__ */
  
          
  		  z = invsqrtpi*(u*cc-v*ss)/z;
  		}
  		if(hx<0) return -z;
  		else  	 return  z;
  	}
  	if(ix<0x32000000) 
    {	/* |x|<2**-27 */
  	  if(huge+x>one) 
        return (float)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*(float)0.5+r/s);
  }
  
  #ifdef __STDC__
  static const float U0[5] = {
  #else
  static float U0[5] = {
  #endif
   -1.9605709612e-01, /* 0xbe48c331 */
    5.0443872809e-02, /* 0x3d4e9e3c */
   -1.9125689287e-03, /* 0xbafaaf2a */
    2.3525259166e-05, /* 0x37c5581c */
   -9.1909917899e-08, /* 0xb3c56003 */
  };
  #ifdef __STDC__
  static const float V0[5] = {
  #else
  static float V0[5] = {
  #endif
    1.9916731864e-02, /* 0x3ca3286a */
    2.0255257550e-04, /* 0x3954644b */
    1.3560879779e-06, /* 0x35b602d4 */
    6.2274145840e-09, /* 0x31d5f8eb */
    1.6655924903e-11, /* 0x2d9281cf */
  };
  
  #ifdef __STDC__
  	float __ieee754_y1f(float x) 
  #else
  	float __ieee754_y1f(x) 
  	float x;
  #endif
  {
  	float z, s,c,ss,cc,u,v;
  	__int32_t hx,ix;
  
  	GET_FLOAT_WORD(hx,x);
    ix = 0x7fffffff&hx;
      /* if Y1(NaN) is NaN, Y1(-inf) is NaN, Y1(inf) is 0 */
  #ifdef __MATH_NONFINITE__ 
    if(!FLT_UWORD_IS_FINITE(ix)) return one/(x+x*x); 
  #endif /*__MATH_NONFINITE__ */
    if(FLT_UWORD_IS_ZERO(ix)) return -one/zero;
    if(hx<0) return zero/zero;
    if(ix >= 0x40000000) 
    {  /* |x| >= 2.0 */
      s = sinf(x);
      c = cosf(x);
      ss = -s-c;
      cc = s-c;
      if(ix<=FLT_UWORD_HALF_MAX) 
      {  /* make sure x+x not overflow */
        z = cosf(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)
      {
  #ifdef __MATH_SOFT_FLOAT__
        z=__ieee754_sqrtf(x);
  #else /* __MATH_SOFT_FLOAT__ */
        __inst_sqrt_S(z,x);
  #endif /* __MATH_SOFT_FLOAT__ */
  
        z = (invsqrtpi*ss)/z;
      }
      else 
      {
        u = ponef(x); v = qonef(x);
  #ifdef __MATH_SOFT_FLOAT__
        z=__ieee754_sqrtf(x);
  #else /* __MATH_SOFT_FLOAT__ */
        __inst_sqrt_S(z,x);
  #endif /* __MATH_SOFT_FLOAT__ */
        z = invsqrtpi*(u*ss+v*cc)/z;
      }
      return z;
    } 
    if(ix<=0x24800000) 
    {    /* 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_j1f(x)*__ieee754_logf(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)
   */
  
  #ifdef __STDC__
  static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  #else
  static float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  #endif
    0.0000000000e+00, /* 0x00000000 */
    1.1718750000e-01, /* 0x3df00000 */
    1.3239480972e+01, /* 0x4153d4ea */
    4.1205184937e+02, /* 0x43ce06a3 */
    3.8747453613e+03, /* 0x45722bed */
    7.9144794922e+03, /* 0x45f753d6 */
  };
  #ifdef __STDC__
  static const float ps8[5] = {
  #else
  static float ps8[5] = {
  #endif
    1.1420736694e+02, /* 0x42e46a2c */
    3.6509309082e+03, /* 0x45642ee5 */
    3.6956207031e+04, /* 0x47105c35 */
    9.7602796875e+04, /* 0x47bea166 */
    3.0804271484e+04, /* 0x46f0a88b */
  };
  
  #ifdef __STDC__
  static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  #else
  static float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  #endif
    1.3199052094e-11, /* 0x2d68333f */
    1.1718749255e-01, /* 0x3defffff */
    6.8027510643e+00, /* 0x40d9b023 */
    1.0830818176e+02, /* 0x42d89dca */
    5.1763616943e+02, /* 0x440168b7 */
    5.2871520996e+02, /* 0x44042dc6 */
  };
  #ifdef __STDC__
  static const float ps5[5] = {
  #else
  static float ps5[5] = {
  #endif
    5.9280597687e+01, /* 0x426d1f55 */
    9.9140142822e+02, /* 0x4477d9b1 */
    5.3532670898e+03, /* 0x45a74a23 */
    7.8446904297e+03, /* 0x45f52586 */
    1.5040468750e+03, /* 0x44bc0180 */
  };
  
  #ifdef __STDC__
  static const float pr3[6] = {
  #else
  static float pr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
  #endif
    3.0250391081e-09, /* 0x314fe10d */
    1.1718686670e-01, /* 0x3defffab */
    3.9329774380e+00, /* 0x407bb5e7 */
    3.5119403839e+01, /* 0x420c7a45 */
    9.1055007935e+01, /* 0x42b61c2a */
    4.8559066772e+01, /* 0x42423c7c */
  };
  #ifdef __STDC__
  static const float ps3[5] = {
  #else
  static float ps3[5] = {
  #endif
    3.4791309357e+01, /* 0x420b2a4d */
    3.3676245117e+02, /* 0x43a86198 */
    1.0468714600e+03, /* 0x4482dbe3 */
    8.9081134033e+02, /* 0x445eb3ed */
    1.0378793335e+02, /* 0x42cf936c */
  };
  
  #ifdef __STDC__
  static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  #else
  static float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  #endif
    1.0771083225e-07, /* 0x33e74ea8 */
    1.1717621982e-01, /* 0x3deffa16 */
    2.3685150146e+00, /* 0x401795c0 */
    1.2242610931e+01, /* 0x4143e1bc */
    1.7693971634e+01, /* 0x418d8d41 */
    5.0735230446e+00, /* 0x40a25a4d */
  };
  #ifdef __STDC__
  static const float ps2[5] = {
  #else
  static float ps2[5] = {
  #endif
    2.1436485291e+01, /* 0x41ab7dec */
    1.2529022980e+02, /* 0x42fa9499 */
    2.3227647400e+02, /* 0x436846c7 */
    1.1767937469e+02, /* 0x42eb5bd7 */
    8.3646392822e+00, /* 0x4105d590 */
  };
  
  #ifdef __STDC__
  	static float ponef(float x)
  #else
  	static float ponef(x)
  	float x;
  #endif
  {
  #ifdef __STDC__
  	const float *p,*q;
  #else
  	float *p,*q;
  #endif
  	float z,r,s;
    __int32_t ix;
  	GET_FLOAT_WORD(ix,x);
  	ix &= 0x7fffffff;
    if(ix>=0x41000000)     
    {
      p = pr8; q= ps8;
    }
    else if(ix>=0x40f71c58)
    {
      p = pr5; q= ps5;
    }
    else if(ix>=0x4036db68)
    {
      p = pr3; q= ps3;
    }
    else 
    {
      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 qone 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)
   */
  
  #ifdef __STDC__
  static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  #else
  static float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
  #endif
    0.0000000000e+00, /* 0x00000000 */
   -1.0253906250e-01, /* 0xbdd20000 */
   -1.6271753311e+01, /* 0xc1822c8d */
   -7.5960174561e+02, /* 0xc43de683 */
   -1.1849806641e+04, /* 0xc639273a */
   -4.8438511719e+04, /* 0xc73d3683 */
  };
  #ifdef __STDC__
  static const float qs8[6] = {
  #else
  static float qs8[6] = {
  #endif
    1.6139537048e+02, /* 0x43216537 */
    7.8253862305e+03, /* 0x45f48b17 */
    1.3387534375e+05, /* 0x4802bcd6 */
    7.1965775000e+05, /* 0x492fb29c */
    6.6660125000e+05, /* 0x4922be94 */
   -2.9449025000e+05, /* 0xc88fcb48 */
  };
  
  #ifdef __STDC__
  static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  #else
  static float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
  #endif
   -2.0897993405e-11, /* 0xadb7d219 */
   -1.0253904760e-01, /* 0xbdd1fffe */
   -8.0564479828e+00, /* 0xc100e736 */
   -1.8366960144e+02, /* 0xc337ab6b */
   -1.3731937256e+03, /* 0xc4aba633 */
   -2.6124443359e+03, /* 0xc523471c */
  };
  #ifdef __STDC__
  static const float qs5[6] = {
  #else
  static float qs5[6] = {
  #endif
    8.1276550293e+01, /* 0x42a28d98 */
    1.9917987061e+03, /* 0x44f8f98f */
    1.7468484375e+04, /* 0x468878f8 */
    4.9851425781e+04, /* 0x4742bb6d */
    2.7948074219e+04, /* 0x46da5826 */
   -4.7191835938e+03, /* 0xc5937978 */
  };
  
  #ifdef __STDC__
  static const float qr3[6] = {
  #else
  static float qr3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
  #endif
   -5.0783124372e-09, /* 0xb1ae7d4f */
   -1.0253783315e-01, /* 0xbdd1ff5b */
   -4.6101160049e+00, /* 0xc0938612 */
   -5.7847221375e+01, /* 0xc267638e */
   -2.2824453735e+02, /* 0xc3643e9a */
   -2.1921012878e+02, /* 0xc35b35cb */
  };
  #ifdef __STDC__
  static const float qs3[6] = {
  #else
  static float qs3[6] = {
  #endif
    4.7665153503e+01, /* 0x423ea91e */
    6.7386511230e+02, /* 0x4428775e */
    3.3801528320e+03, /* 0x45534272 */
    5.5477290039e+03, /* 0x45ad5dd5 */
    1.9031191406e+03, /* 0x44ede3d0 */
   -1.3520118713e+02, /* 0xc3073381 */
  };
  
  #ifdef __STDC__
  static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  #else
  static float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
  #endif
   -1.7838172539e-07, /* 0xb43f8932 */
   -1.0251704603e-01, /* 0xbdd1f475 */
   -2.7522056103e+00, /* 0xc0302423 */
   -1.9663616180e+01, /* 0xc19d4f16 */
   -4.2325313568e+01, /* 0xc2294d1f */
   -2.1371921539e+01, /* 0xc1aaf9b2 */
  };
  #ifdef __STDC__
  static const float qs2[6] = {
  #else
  static float qs2[6] = {
  #endif
    2.9533363342e+01, /* 0x41ec4454 */
    2.5298155212e+02, /* 0x437cfb47 */
    7.5750280762e+02, /* 0x443d602e */
    7.3939318848e+02, /* 0x4438d92a */
    1.5594900513e+02, /* 0x431bf2f2 */
   -4.9594988823e+00, /* 0xc09eb437 */
  };
  
  #ifdef __STDC__
  	static float qonef(float x)
  #else
  	static float qonef(x)
  	float x;
  #endif
  {
  #ifdef __STDC__
  	const float *p,*q;
  #else
  	float *p,*q;
  #endif
  	float  s,r,z;
  	__int32_t ix;
  	GET_FLOAT_WORD(ix,x);
  	ix &= 0x7fffffff;
  	if(ix>=0x40200000)     
    {
      p = qr8; q= qs8;
    }
  	else if(ix>=0x40f71c58)
    {
      p = qr5; q= qs5;
    }
  	else if(ix>=0x4036db68)
    {
      p = qr3; q= qs3;
    }
    else 
    {
      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 ((float).375 + r/s)/x;
  }
  