1 | /* origin: FreeBSD /usr/src/lib/msun/src/e_j1f.c */
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2 | /*
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3 | * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
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4 | */
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5 | /*
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6 | * ====================================================
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7 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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8 | *
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9 | * Developed at SunPro, a Sun Microsystems, Inc. business.
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10 | * Permission to use, copy, modify, and distribute this
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11 | * software is freely granted, provided that this notice
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12 | * is preserved.
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13 | * ====================================================
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14 | */
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15 |
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16 | #define _GNU_SOURCE
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17 | #include "libm.h"
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18 |
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19 | static float ponef(float), qonef(float);
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20 |
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21 | static const float
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22 | invsqrtpi = 5.6418961287e-01, /* 0x3f106ebb */
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23 | tpi = 6.3661974669e-01; /* 0x3f22f983 */
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24 |
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25 | static float common(uint32_t ix, float x, int y1, int sign)
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26 | {
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27 | double z,s,c,ss,cc;
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28 |
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29 | s = sinf(x);
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30 | if (y1)
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31 | s = -s;
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32 | c = cosf(x);
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33 | cc = s-c;
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34 | if (ix < 0x7f000000) {
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35 | ss = -s-c;
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36 | z = cosf(2*x);
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37 | if (s*c > 0)
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38 | cc = z/ss;
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39 | else
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40 | ss = z/cc;
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41 | if (ix < 0x58800000) {
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42 | if (y1)
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43 | ss = -ss;
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44 | cc = ponef(x)*cc-qonef(x)*ss;
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45 | }
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46 | }
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47 | if (sign)
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48 | cc = -cc;
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49 | return invsqrtpi*cc/sqrtf(x);
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50 | }
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51 |
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52 | /* R0/S0 on [0,2] */
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53 | static const float
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54 | r00 = -6.2500000000e-02, /* 0xbd800000 */
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55 | r01 = 1.4070566976e-03, /* 0x3ab86cfd */
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56 | r02 = -1.5995563444e-05, /* 0xb7862e36 */
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57 | r03 = 4.9672799207e-08, /* 0x335557d2 */
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58 | s01 = 1.9153760746e-02, /* 0x3c9ce859 */
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59 | s02 = 1.8594678841e-04, /* 0x3942fab6 */
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60 | s03 = 1.1771846857e-06, /* 0x359dffc2 */
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61 | s04 = 5.0463624390e-09, /* 0x31ad6446 */
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62 | s05 = 1.2354227016e-11; /* 0x2d59567e */
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63 |
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64 | float j1f(float x)
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65 | {
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66 | float z,r,s;
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67 | uint32_t ix;
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68 | int sign;
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69 |
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70 | GET_FLOAT_WORD(ix, x);
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71 | sign = ix>>31;
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72 | ix &= 0x7fffffff;
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73 | if (ix >= 0x7f800000)
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74 | return 1/(x*x);
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75 | if (ix >= 0x40000000) /* |x| >= 2 */
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76 | return common(ix, fabsf(x), 0, sign);
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77 | if (ix >= 0x39000000) { /* |x| >= 2**-13 */
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78 | z = x*x;
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79 | r = z*(r00+z*(r01+z*(r02+z*r03)));
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80 | s = 1+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
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81 | z = 0.5f + r/s;
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82 | } else
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83 | z = 0.5f;
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84 | return z*x;
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85 | }
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86 |
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87 | static const float U0[5] = {
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88 | -1.9605709612e-01, /* 0xbe48c331 */
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89 | 5.0443872809e-02, /* 0x3d4e9e3c */
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90 | -1.9125689287e-03, /* 0xbafaaf2a */
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91 | 2.3525259166e-05, /* 0x37c5581c */
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92 | -9.1909917899e-08, /* 0xb3c56003 */
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93 | };
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94 | static const float V0[5] = {
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95 | 1.9916731864e-02, /* 0x3ca3286a */
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96 | 2.0255257550e-04, /* 0x3954644b */
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97 | 1.3560879779e-06, /* 0x35b602d4 */
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98 | 6.2274145840e-09, /* 0x31d5f8eb */
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99 | 1.6655924903e-11, /* 0x2d9281cf */
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100 | };
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101 |
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102 | float y1f(float x)
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103 | {
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104 | float z,u,v;
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105 | uint32_t ix;
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106 |
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107 | GET_FLOAT_WORD(ix, x);
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108 | if ((ix & 0x7fffffff) == 0)
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109 | return -1/0.0f;
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110 | if (ix>>31)
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111 | return 0/0.0f;
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112 | if (ix >= 0x7f800000)
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113 | return 1/x;
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114 | if (ix >= 0x40000000) /* |x| >= 2.0 */
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115 | return common(ix,x,1,0);
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116 | if (ix < 0x33000000) /* x < 2**-25 */
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117 | return -tpi/x;
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118 | z = x*x;
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119 | u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
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120 | v = 1.0f+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
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121 | return x*(u/v) + tpi*(j1f(x)*logf(x)-1.0f/x);
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122 | }
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123 |
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124 | /* For x >= 8, the asymptotic expansions of pone is
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125 | * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
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126 | * We approximate pone by
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127 | * pone(x) = 1 + (R/S)
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128 | * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
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129 | * S = 1 + ps0*s^2 + ... + ps4*s^10
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130 | * and
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131 | * | pone(x)-1-R/S | <= 2 ** ( -60.06)
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132 | */
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133 |
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134 | static const float pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
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135 | 0.0000000000e+00, /* 0x00000000 */
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136 | 1.1718750000e-01, /* 0x3df00000 */
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137 | 1.3239480972e+01, /* 0x4153d4ea */
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138 | 4.1205184937e+02, /* 0x43ce06a3 */
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139 | 3.8747453613e+03, /* 0x45722bed */
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140 | 7.9144794922e+03, /* 0x45f753d6 */
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141 | };
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142 | static const float ps8[5] = {
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143 | 1.1420736694e+02, /* 0x42e46a2c */
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144 | 3.6509309082e+03, /* 0x45642ee5 */
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145 | 3.6956207031e+04, /* 0x47105c35 */
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146 | 9.7602796875e+04, /* 0x47bea166 */
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147 | 3.0804271484e+04, /* 0x46f0a88b */
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148 | };
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149 |
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150 | static const float pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
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151 | 1.3199052094e-11, /* 0x2d68333f */
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152 | 1.1718749255e-01, /* 0x3defffff */
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153 | 6.8027510643e+00, /* 0x40d9b023 */
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154 | 1.0830818176e+02, /* 0x42d89dca */
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155 | 5.1763616943e+02, /* 0x440168b7 */
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156 | 5.2871520996e+02, /* 0x44042dc6 */
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157 | };
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158 | static const float ps5[5] = {
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159 | 5.9280597687e+01, /* 0x426d1f55 */
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160 | 9.9140142822e+02, /* 0x4477d9b1 */
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161 | 5.3532670898e+03, /* 0x45a74a23 */
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162 | 7.8446904297e+03, /* 0x45f52586 */
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163 | 1.5040468750e+03, /* 0x44bc0180 */
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164 | };
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165 |
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166 | static const float pr3[6] = {
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167 | 3.0250391081e-09, /* 0x314fe10d */
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168 | 1.1718686670e-01, /* 0x3defffab */
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169 | 3.9329774380e+00, /* 0x407bb5e7 */
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170 | 3.5119403839e+01, /* 0x420c7a45 */
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171 | 9.1055007935e+01, /* 0x42b61c2a */
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172 | 4.8559066772e+01, /* 0x42423c7c */
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173 | };
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174 | static const float ps3[5] = {
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175 | 3.4791309357e+01, /* 0x420b2a4d */
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176 | 3.3676245117e+02, /* 0x43a86198 */
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177 | 1.0468714600e+03, /* 0x4482dbe3 */
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178 | 8.9081134033e+02, /* 0x445eb3ed */
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179 | 1.0378793335e+02, /* 0x42cf936c */
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180 | };
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181 |
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182 | static const float pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
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183 | 1.0771083225e-07, /* 0x33e74ea8 */
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184 | 1.1717621982e-01, /* 0x3deffa16 */
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185 | 2.3685150146e+00, /* 0x401795c0 */
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186 | 1.2242610931e+01, /* 0x4143e1bc */
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187 | 1.7693971634e+01, /* 0x418d8d41 */
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188 | 5.0735230446e+00, /* 0x40a25a4d */
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189 | };
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190 | static const float ps2[5] = {
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191 | 2.1436485291e+01, /* 0x41ab7dec */
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192 | 1.2529022980e+02, /* 0x42fa9499 */
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193 | 2.3227647400e+02, /* 0x436846c7 */
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194 | 1.1767937469e+02, /* 0x42eb5bd7 */
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195 | 8.3646392822e+00, /* 0x4105d590 */
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196 | };
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197 |
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198 | static float ponef(float x)
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199 | {
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200 | const float *p,*q;
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201 | float_t z,r,s;
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202 | uint32_t ix;
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203 |
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204 | GET_FLOAT_WORD(ix, x);
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205 | ix &= 0x7fffffff;
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206 | if (ix >= 0x41000000){p = pr8; q = ps8;}
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207 | else if (ix >= 0x409173eb){p = pr5; q = ps5;}
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208 | else if (ix >= 0x4036d917){p = pr3; q = ps3;}
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209 | else /*ix >= 0x40000000*/ {p = pr2; q = ps2;}
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210 | z = 1.0f/(x*x);
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211 | r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
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212 | s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
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213 | return 1.0f + r/s;
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214 | }
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215 |
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216 | /* For x >= 8, the asymptotic expansions of qone is
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217 | * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
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218 | * We approximate pone by
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219 | * qone(x) = s*(0.375 + (R/S))
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220 | * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
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221 | * S = 1 + qs1*s^2 + ... + qs6*s^12
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222 | * and
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223 | * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
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224 | */
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225 |
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226 | static const float qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
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227 | 0.0000000000e+00, /* 0x00000000 */
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228 | -1.0253906250e-01, /* 0xbdd20000 */
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229 | -1.6271753311e+01, /* 0xc1822c8d */
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230 | -7.5960174561e+02, /* 0xc43de683 */
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231 | -1.1849806641e+04, /* 0xc639273a */
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232 | -4.8438511719e+04, /* 0xc73d3683 */
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233 | };
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234 | static const float qs8[6] = {
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235 | 1.6139537048e+02, /* 0x43216537 */
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236 | 7.8253862305e+03, /* 0x45f48b17 */
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237 | 1.3387534375e+05, /* 0x4802bcd6 */
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238 | 7.1965775000e+05, /* 0x492fb29c */
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239 | 6.6660125000e+05, /* 0x4922be94 */
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240 | -2.9449025000e+05, /* 0xc88fcb48 */
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241 | };
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242 |
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243 | static const float qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
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244 | -2.0897993405e-11, /* 0xadb7d219 */
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245 | -1.0253904760e-01, /* 0xbdd1fffe */
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246 | -8.0564479828e+00, /* 0xc100e736 */
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247 | -1.8366960144e+02, /* 0xc337ab6b */
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248 | -1.3731937256e+03, /* 0xc4aba633 */
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249 | -2.6124443359e+03, /* 0xc523471c */
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250 | };
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251 | static const float qs5[6] = {
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252 | 8.1276550293e+01, /* 0x42a28d98 */
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253 | 1.9917987061e+03, /* 0x44f8f98f */
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254 | 1.7468484375e+04, /* 0x468878f8 */
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255 | 4.9851425781e+04, /* 0x4742bb6d */
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256 | 2.7948074219e+04, /* 0x46da5826 */
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257 | -4.7191835938e+03, /* 0xc5937978 */
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258 | };
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259 |
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260 | static const float qr3[6] = {
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261 | -5.0783124372e-09, /* 0xb1ae7d4f */
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262 | -1.0253783315e-01, /* 0xbdd1ff5b */
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263 | -4.6101160049e+00, /* 0xc0938612 */
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264 | -5.7847221375e+01, /* 0xc267638e */
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265 | -2.2824453735e+02, /* 0xc3643e9a */
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266 | -2.1921012878e+02, /* 0xc35b35cb */
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267 | };
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268 | static const float qs3[6] = {
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269 | 4.7665153503e+01, /* 0x423ea91e */
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270 | 6.7386511230e+02, /* 0x4428775e */
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271 | 3.3801528320e+03, /* 0x45534272 */
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272 | 5.5477290039e+03, /* 0x45ad5dd5 */
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273 | 1.9031191406e+03, /* 0x44ede3d0 */
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274 | -1.3520118713e+02, /* 0xc3073381 */
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275 | };
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276 |
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277 | static const float qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
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278 | -1.7838172539e-07, /* 0xb43f8932 */
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279 | -1.0251704603e-01, /* 0xbdd1f475 */
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280 | -2.7522056103e+00, /* 0xc0302423 */
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281 | -1.9663616180e+01, /* 0xc19d4f16 */
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282 | -4.2325313568e+01, /* 0xc2294d1f */
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283 | -2.1371921539e+01, /* 0xc1aaf9b2 */
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284 | };
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285 | static const float qs2[6] = {
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286 | 2.9533363342e+01, /* 0x41ec4454 */
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287 | 2.5298155212e+02, /* 0x437cfb47 */
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288 | 7.5750280762e+02, /* 0x443d602e */
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289 | 7.3939318848e+02, /* 0x4438d92a */
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290 | 1.5594900513e+02, /* 0x431bf2f2 */
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291 | -4.9594988823e+00, /* 0xc09eb437 */
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292 | };
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293 |
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294 | static float qonef(float x)
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295 | {
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296 | const float *p,*q;
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297 | float_t s,r,z;
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298 | uint32_t ix;
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299 |
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300 | GET_FLOAT_WORD(ix, x);
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301 | ix &= 0x7fffffff;
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302 | if (ix >= 0x41000000){p = qr8; q = qs8;}
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303 | else if (ix >= 0x409173eb){p = qr5; q = qs5;}
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304 | else if (ix >= 0x4036d917){p = qr3; q = qs3;}
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305 | else /*ix >= 0x40000000*/ {p = qr2; q = qs2;}
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306 | z = 1.0f/(x*x);
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307 | r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
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308 | s = 1.0f+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
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309 | return (.375f + r/s)/x;
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310 | }
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