github.com/afumu/libc@v0.0.6/musl/src/math/j0f.c (about)

     1  /* origin: FreeBSD /usr/src/lib/msun/src/e_j0f.c */
     2  /*
     3   * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
     4   */
     5  /*
     6   * ====================================================
     7   * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
     8   *
     9   * Developed at SunPro, a Sun Microsystems, Inc. business.
    10   * Permission to use, copy, modify, and distribute this
    11   * software is freely granted, provided that this notice
    12   * is preserved.
    13   * ====================================================
    14   */
    15  
    16  #define _GNU_SOURCE
    17  #include "libm.h"
    18  
    19  static float pzerof(float), qzerof(float);
    20  
    21  static const float
    22  invsqrtpi = 5.6418961287e-01, /* 0x3f106ebb */
    23  tpi       = 6.3661974669e-01; /* 0x3f22f983 */
    24  
    25  static float common(uint32_t ix, float x, int y0)
    26  {
    27  	float z,s,c,ss,cc;
    28  	/*
    29  	 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
    30  	 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
    31  	 */
    32  	s = sinf(x);
    33  	c = cosf(x);
    34  	if (y0)
    35  		c = -c;
    36  	cc = s+c;
    37  	if (ix < 0x7f000000) {
    38  		ss = s-c;
    39  		z = -cosf(2*x);
    40  		if (s*c < 0)
    41  			cc = z/ss;
    42  		else
    43  			ss = z/cc;
    44  		if (ix < 0x58800000) {
    45  			if (y0)
    46  				ss = -ss;
    47  			cc = pzerof(x)*cc-qzerof(x)*ss;
    48  		}
    49  	}
    50  	return invsqrtpi*cc/sqrtf(x);
    51  }
    52  
    53  /* R0/S0 on [0, 2.00] */
    54  static const float
    55  R02 =  1.5625000000e-02, /* 0x3c800000 */
    56  R03 = -1.8997929874e-04, /* 0xb947352e */
    57  R04 =  1.8295404516e-06, /* 0x35f58e88 */
    58  R05 = -4.6183270541e-09, /* 0xb19eaf3c */
    59  S01 =  1.5619102865e-02, /* 0x3c7fe744 */
    60  S02 =  1.1692678527e-04, /* 0x38f53697 */
    61  S03 =  5.1354652442e-07, /* 0x3509daa6 */
    62  S04 =  1.1661400734e-09; /* 0x30a045e8 */
    63  
    64  float j0f(float x)
    65  {
    66  	float z,r,s;
    67  	uint32_t ix;
    68  
    69  	GET_FLOAT_WORD(ix, x);
    70  	ix &= 0x7fffffff;
    71  	if (ix >= 0x7f800000)
    72  		return 1/(x*x);
    73  	x = fabsf(x);
    74  
    75  	if (ix >= 0x40000000) {  /* |x| >= 2 */
    76  		/* large ulp error near zeros */
    77  		return common(ix, x, 0);
    78  	}
    79  	if (ix >= 0x3a000000) {  /* |x| >= 2**-11 */
    80  		/* up to 4ulp error near 2 */
    81  		z = x*x;
    82  		r = z*(R02+z*(R03+z*(R04+z*R05)));
    83  		s = 1+z*(S01+z*(S02+z*(S03+z*S04)));
    84  		return (1+x/2)*(1-x/2) + z*(r/s);
    85  	}
    86  	if (ix >= 0x21800000)  /* |x| >= 2**-60 */
    87  		x = 0.25f*x*x;
    88  	return 1 - x;
    89  }
    90  
    91  static const float
    92  u00  = -7.3804296553e-02, /* 0xbd9726b5 */
    93  u01  =  1.7666645348e-01, /* 0x3e34e80d */
    94  u02  = -1.3818567619e-02, /* 0xbc626746 */
    95  u03  =  3.4745343146e-04, /* 0x39b62a69 */
    96  u04  = -3.8140706238e-06, /* 0xb67ff53c */
    97  u05  =  1.9559013964e-08, /* 0x32a802ba */
    98  u06  = -3.9820518410e-11, /* 0xae2f21eb */
    99  v01  =  1.2730483897e-02, /* 0x3c509385 */
   100  v02  =  7.6006865129e-05, /* 0x389f65e0 */
   101  v03  =  2.5915085189e-07, /* 0x348b216c */
   102  v04  =  4.4111031494e-10; /* 0x2ff280c2 */
   103  
   104  float y0f(float x)
   105  {
   106  	float z,u,v;
   107  	uint32_t ix;
   108  
   109  	GET_FLOAT_WORD(ix, x);
   110  	if ((ix & 0x7fffffff) == 0)
   111  		return -1/0.0f;
   112  	if (ix>>31)
   113  		return 0/0.0f;
   114  	if (ix >= 0x7f800000)
   115  		return 1/x;
   116  	if (ix >= 0x40000000) {  /* |x| >= 2.0 */
   117  		/* large ulp error near zeros */
   118  		return common(ix,x,1);
   119  	}
   120  	if (ix >= 0x39000000) {  /* x >= 2**-13 */
   121  		/* large ulp error at x ~= 0.89 */
   122  		z = x*x;
   123  		u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
   124  		v = 1+z*(v01+z*(v02+z*(v03+z*v04)));
   125  		return u/v + tpi*(j0f(x)*logf(x));
   126  	}
   127  	return u00 + tpi*logf(x);
   128  }
   129  
   130  /* The asymptotic expansions of pzero is
   131   *      1 - 9/128 s^2 + 11025/98304 s^4 - ...,  where s = 1/x.
   132   * For x >= 2, We approximate pzero by
   133   *      pzero(x) = 1 + (R/S)
   134   * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
   135   *        S = 1 + pS0*s^2 + ... + pS4*s^10
   136   * and
   137   *      | pzero(x)-1-R/S | <= 2  ** ( -60.26)
   138   */
   139  static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
   140    0.0000000000e+00, /* 0x00000000 */
   141   -7.0312500000e-02, /* 0xbd900000 */
   142   -8.0816707611e+00, /* 0xc1014e86 */
   143   -2.5706311035e+02, /* 0xc3808814 */
   144   -2.4852163086e+03, /* 0xc51b5376 */
   145   -5.2530439453e+03, /* 0xc5a4285a */
   146  };
   147  static const float pS8[5] = {
   148    1.1653436279e+02, /* 0x42e91198 */
   149    3.8337448730e+03, /* 0x456f9beb */
   150    4.0597855469e+04, /* 0x471e95db */
   151    1.1675296875e+05, /* 0x47e4087c */
   152    4.7627726562e+04, /* 0x473a0bba */
   153  };
   154  static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
   155   -1.1412546255e-11, /* 0xad48c58a */
   156   -7.0312492549e-02, /* 0xbd8fffff */
   157   -4.1596107483e+00, /* 0xc0851b88 */
   158   -6.7674766541e+01, /* 0xc287597b */
   159   -3.3123129272e+02, /* 0xc3a59d9b */
   160   -3.4643338013e+02, /* 0xc3ad3779 */
   161  };
   162  static const float pS5[5] = {
   163    6.0753936768e+01, /* 0x42730408 */
   164    1.0512523193e+03, /* 0x44836813 */
   165    5.9789707031e+03, /* 0x45bad7c4 */
   166    9.6254453125e+03, /* 0x461665c8 */
   167    2.4060581055e+03, /* 0x451660ee */
   168  };
   169  
   170  static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
   171   -2.5470459075e-09, /* 0xb12f081b */
   172   -7.0311963558e-02, /* 0xbd8fffb8 */
   173   -2.4090321064e+00, /* 0xc01a2d95 */
   174   -2.1965976715e+01, /* 0xc1afba52 */
   175   -5.8079170227e+01, /* 0xc2685112 */
   176   -3.1447946548e+01, /* 0xc1fb9565 */
   177  };
   178  static const float pS3[5] = {
   179    3.5856033325e+01, /* 0x420f6c94 */
   180    3.6151397705e+02, /* 0x43b4c1ca */
   181    1.1936077881e+03, /* 0x44953373 */
   182    1.1279968262e+03, /* 0x448cffe6 */
   183    1.7358093262e+02, /* 0x432d94b8 */
   184  };
   185  
   186  static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
   187   -8.8753431271e-08, /* 0xb3be98b7 */
   188   -7.0303097367e-02, /* 0xbd8ffb12 */
   189   -1.4507384300e+00, /* 0xbfb9b1cc */
   190   -7.6356959343e+00, /* 0xc0f4579f */
   191   -1.1193166733e+01, /* 0xc1331736 */
   192   -3.2336456776e+00, /* 0xc04ef40d */
   193  };
   194  static const float pS2[5] = {
   195    2.2220300674e+01, /* 0x41b1c32d */
   196    1.3620678711e+02, /* 0x430834f0 */
   197    2.7047027588e+02, /* 0x43873c32 */
   198    1.5387539673e+02, /* 0x4319e01a */
   199    1.4657617569e+01, /* 0x416a859a */
   200  };
   201  
   202  static float pzerof(float x)
   203  {
   204  	const float *p,*q;
   205  	float_t z,r,s;
   206  	uint32_t ix;
   207  
   208  	GET_FLOAT_WORD(ix, x);
   209  	ix &= 0x7fffffff;
   210  	if      (ix >= 0x41000000){p = pR8; q = pS8;}
   211  	else if (ix >= 0x409173eb){p = pR5; q = pS5;}
   212  	else if (ix >= 0x4036d917){p = pR3; q = pS3;}
   213  	else /*ix >= 0x40000000*/ {p = pR2; q = pS2;}
   214  	z = 1.0f/(x*x);
   215  	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
   216  	s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
   217  	return 1.0f + r/s;
   218  }
   219  
   220  
   221  /* For x >= 8, the asymptotic expansions of qzero is
   222   *      -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
   223   * We approximate pzero by
   224   *      qzero(x) = s*(-1.25 + (R/S))
   225   * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
   226   *        S = 1 + qS0*s^2 + ... + qS5*s^12
   227   * and
   228   *      | qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
   229   */
   230  static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
   231    0.0000000000e+00, /* 0x00000000 */
   232    7.3242187500e-02, /* 0x3d960000 */
   233    1.1768206596e+01, /* 0x413c4a93 */
   234    5.5767340088e+02, /* 0x440b6b19 */
   235    8.8591972656e+03, /* 0x460a6cca */
   236    3.7014625000e+04, /* 0x471096a0 */
   237  };
   238  static const float qS8[6] = {
   239    1.6377603149e+02, /* 0x4323c6aa */
   240    8.0983447266e+03, /* 0x45fd12c2 */
   241    1.4253829688e+05, /* 0x480b3293 */
   242    8.0330925000e+05, /* 0x49441ed4 */
   243    8.4050156250e+05, /* 0x494d3359 */
   244   -3.4389928125e+05, /* 0xc8a7eb69 */
   245  };
   246  
   247  static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
   248    1.8408595828e-11, /* 0x2da1ec79 */
   249    7.3242180049e-02, /* 0x3d95ffff */
   250    5.8356351852e+00, /* 0x40babd86 */
   251    1.3511157227e+02, /* 0x43071c90 */
   252    1.0272437744e+03, /* 0x448067cd */
   253    1.9899779053e+03, /* 0x44f8bf4b */
   254  };
   255  static const float qS5[6] = {
   256    8.2776611328e+01, /* 0x42a58da0 */
   257    2.0778142090e+03, /* 0x4501dd07 */
   258    1.8847289062e+04, /* 0x46933e94 */
   259    5.6751113281e+04, /* 0x475daf1d */
   260    3.5976753906e+04, /* 0x470c88c1 */
   261   -5.3543427734e+03, /* 0xc5a752be */
   262  };
   263  
   264  static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
   265    4.3774099900e-09, /* 0x3196681b */
   266    7.3241114616e-02, /* 0x3d95ff70 */
   267    3.3442313671e+00, /* 0x405607e3 */
   268    4.2621845245e+01, /* 0x422a7cc5 */
   269    1.7080809021e+02, /* 0x432acedf */
   270    1.6673394775e+02, /* 0x4326bbe4 */
   271  };
   272  static const float qS3[6] = {
   273    4.8758872986e+01, /* 0x42430916 */
   274    7.0968920898e+02, /* 0x44316c1c */
   275    3.7041481934e+03, /* 0x4567825f */
   276    6.4604252930e+03, /* 0x45c9e367 */
   277    2.5163337402e+03, /* 0x451d4557 */
   278   -1.4924745178e+02, /* 0xc3153f59 */
   279  };
   280  
   281  static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
   282    1.5044444979e-07, /* 0x342189db */
   283    7.3223426938e-02, /* 0x3d95f62a */
   284    1.9981917143e+00, /* 0x3fffc4bf */
   285    1.4495602608e+01, /* 0x4167edfd */
   286    3.1666231155e+01, /* 0x41fd5471 */
   287    1.6252708435e+01, /* 0x4182058c */
   288  };
   289  static const float qS2[6] = {
   290    3.0365585327e+01, /* 0x41f2ecb8 */
   291    2.6934811401e+02, /* 0x4386ac8f */
   292    8.4478375244e+02, /* 0x44533229 */
   293    8.8293585205e+02, /* 0x445cbbe5 */
   294    2.1266638184e+02, /* 0x4354aa98 */
   295   -5.3109550476e+00, /* 0xc0a9f358 */
   296  };
   297  
   298  static float qzerof(float x)
   299  {
   300  	const float *p,*q;
   301  	float_t s,r,z;
   302  	uint32_t ix;
   303  
   304  	GET_FLOAT_WORD(ix, x);
   305  	ix &= 0x7fffffff;
   306  	if      (ix >= 0x41000000){p = qR8; q = qS8;}
   307  	else if (ix >= 0x409173eb){p = qR5; q = qS5;}
   308  	else if (ix >= 0x4036d917){p = qR3; q = qS3;}
   309  	else /*ix >= 0x40000000*/ {p = qR2; q = qS2;}
   310  	z = 1.0f/(x*x);
   311  	r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
   312  	s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
   313  	return (-.125f + r/s)/x;
   314  }