1 | /* origin: FreeBSD /usr/src/lib/msun/src/e_j1.c */
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2 | /*
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3 | * ====================================================
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4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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5 | *
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6 | * Developed at SunSoft, a Sun Microsystems, Inc. business.
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7 | * Permission to use, copy, modify, and distribute this
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8 | * software is freely granted, provided that this notice
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9 | * is preserved.
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10 | * ====================================================
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11 | */
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12 | /* j1(x), y1(x)
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13 | * Bessel function of the first and second kinds of order zero.
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14 | * Method -- j1(x):
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15 | * 1. For tiny x, we use j1(x) = x/2 - x^3/16 + x^5/384 - ...
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16 | * 2. Reduce x to |x| since j1(x)=-j1(-x), and
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17 | * for x in (0,2)
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18 | * j1(x) = x/2 + x*z*R0/S0, where z = x*x;
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19 | * (precision: |j1/x - 1/2 - R0/S0 |<2**-61.51 )
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20 | * for x in (2,inf)
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21 | * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
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22 | * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
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23 | * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
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24 | * as follow:
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25 | * cos(x1) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
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26 | * = 1/sqrt(2) * (sin(x) - cos(x))
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27 | * sin(x1) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
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28 | * = -1/sqrt(2) * (sin(x) + cos(x))
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29 | * (To avoid cancellation, use
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30 | * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
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31 | * to compute the worse one.)
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32 | *
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33 | * 3 Special cases
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34 | * j1(nan)= nan
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35 | * j1(0) = 0
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36 | * j1(inf) = 0
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37 | *
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38 | * Method -- y1(x):
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39 | * 1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
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40 | * 2. For x<2.
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41 | * Since
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42 | * y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x^3-...)
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43 | * therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
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44 | * We use the following function to approximate y1,
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45 | * y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x^2
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46 | * where for x in [0,2] (abs err less than 2**-65.89)
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47 | * U(z) = U0[0] + U0[1]*z + ... + U0[4]*z^4
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48 | * V(z) = 1 + v0[0]*z + ... + v0[4]*z^5
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49 | * Note: For tiny x, 1/x dominate y1 and hence
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50 | * y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
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51 | * 3. For x>=2.
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52 | * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
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53 | * where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
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54 | * by method mentioned above.
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55 | */
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56 |
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57 | #include "libm.h"
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58 |
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59 | static double pone(double), qone(double);
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60 |
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61 | static const double
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62 | invsqrtpi = 5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
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63 | tpi = 6.36619772367581382433e-01; /* 0x3FE45F30, 0x6DC9C883 */
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64 |
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65 | static double common(uint32_t ix, double x, int y1, int sign)
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66 | {
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67 | double z,s,c,ss,cc;
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68 |
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69 | /*
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70 | * j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x-3pi/4)-q1(x)*sin(x-3pi/4))
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71 | * y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x-3pi/4)+q1(x)*cos(x-3pi/4))
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72 | *
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73 | * sin(x-3pi/4) = -(sin(x) + cos(x))/sqrt(2)
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74 | * cos(x-3pi/4) = (sin(x) - cos(x))/sqrt(2)
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75 | * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
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76 | */
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77 | s = sin(x);
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78 | if (y1)
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79 | s = -s;
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80 | c = cos(x);
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81 | cc = s-c;
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82 | if (ix < 0x7fe00000) {
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83 | /* avoid overflow in 2*x */
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84 | ss = -s-c;
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85 | z = cos(2*x);
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86 | if (s*c > 0)
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87 | cc = z/ss;
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88 | else
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89 | ss = z/cc;
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90 | if (ix < 0x48000000) {
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91 | if (y1)
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92 | ss = -ss;
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93 | cc = pone(x)*cc-qone(x)*ss;
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94 | }
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95 | }
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96 | if (sign)
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97 | cc = -cc;
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98 | return invsqrtpi*cc/sqrt(x);
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99 | }
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100 |
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101 | /* R0/S0 on [0,2] */
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102 | static const double
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103 | r00 = -6.25000000000000000000e-02, /* 0xBFB00000, 0x00000000 */
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104 | r01 = 1.40705666955189706048e-03, /* 0x3F570D9F, 0x98472C61 */
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105 | r02 = -1.59955631084035597520e-05, /* 0xBEF0C5C6, 0xBA169668 */
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106 | r03 = 4.96727999609584448412e-08, /* 0x3E6AAAFA, 0x46CA0BD9 */
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107 | s01 = 1.91537599538363460805e-02, /* 0x3F939D0B, 0x12637E53 */
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108 | s02 = 1.85946785588630915560e-04, /* 0x3F285F56, 0xB9CDF664 */
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109 | s03 = 1.17718464042623683263e-06, /* 0x3EB3BFF8, 0x333F8498 */
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110 | s04 = 5.04636257076217042715e-09, /* 0x3E35AC88, 0xC97DFF2C */
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111 | s05 = 1.23542274426137913908e-11; /* 0x3DAB2ACF, 0xCFB97ED8 */
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112 |
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113 | double j1(double x)
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114 | {
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115 | double z,r,s;
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116 | uint32_t ix;
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117 | int sign;
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118 |
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119 | GET_HIGH_WORD(ix, x);
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120 | sign = ix>>31;
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121 | ix &= 0x7fffffff;
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122 | if (ix >= 0x7ff00000)
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123 | return 1/(x*x);
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124 | if (ix >= 0x40000000) /* |x| >= 2 */
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125 | return common(ix, fabs(x), 0, sign);
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126 | if (ix >= 0x38000000) { /* |x| >= 2**-127 */
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127 | z = x*x;
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128 | r = z*(r00+z*(r01+z*(r02+z*r03)));
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129 | s = 1+z*(s01+z*(s02+z*(s03+z*(s04+z*s05))));
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130 | z = r/s;
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131 | } else
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132 | /* avoid underflow, raise inexact if x!=0 */
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133 | z = x;
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134 | return (0.5 + z)*x;
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135 | }
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136 |
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137 | static const double U0[5] = {
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138 | -1.96057090646238940668e-01, /* 0xBFC91866, 0x143CBC8A */
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139 | 5.04438716639811282616e-02, /* 0x3FA9D3C7, 0x76292CD1 */
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140 | -1.91256895875763547298e-03, /* 0xBF5F55E5, 0x4844F50F */
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141 | 2.35252600561610495928e-05, /* 0x3EF8AB03, 0x8FA6B88E */
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142 | -9.19099158039878874504e-08, /* 0xBE78AC00, 0x569105B8 */
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143 | };
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144 | static const double V0[5] = {
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145 | 1.99167318236649903973e-02, /* 0x3F94650D, 0x3F4DA9F0 */
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146 | 2.02552581025135171496e-04, /* 0x3F2A8C89, 0x6C257764 */
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147 | 1.35608801097516229404e-06, /* 0x3EB6C05A, 0x894E8CA6 */
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148 | 6.22741452364621501295e-09, /* 0x3E3ABF1D, 0x5BA69A86 */
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149 | 1.66559246207992079114e-11, /* 0x3DB25039, 0xDACA772A */
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150 | };
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151 |
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152 | double y1(double x)
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153 | {
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154 | double z,u,v;
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155 | uint32_t ix,lx;
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156 |
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157 | EXTRACT_WORDS(ix, lx, x);
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158 | /* y1(nan)=nan, y1(<0)=nan, y1(0)=-inf, y1(inf)=0 */
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159 | if ((ix<<1 | lx) == 0)
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160 | return -1/0.0;
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161 | if (ix>>31)
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162 | return 0/0.0;
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163 | if (ix >= 0x7ff00000)
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164 | return 1/x;
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165 |
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166 | if (ix >= 0x40000000) /* x >= 2 */
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167 | return common(ix, x, 1, 0);
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168 | if (ix < 0x3c900000) /* x < 2**-54 */
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169 | return -tpi/x;
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170 | z = x*x;
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171 | u = U0[0]+z*(U0[1]+z*(U0[2]+z*(U0[3]+z*U0[4])));
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172 | v = 1+z*(V0[0]+z*(V0[1]+z*(V0[2]+z*(V0[3]+z*V0[4]))));
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173 | return x*(u/v) + tpi*(j1(x)*log(x)-1/x);
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174 | }
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175 |
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176 | /* For x >= 8, the asymptotic expansions of pone is
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177 | * 1 + 15/128 s^2 - 4725/2^15 s^4 - ..., where s = 1/x.
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178 | * We approximate pone by
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179 | * pone(x) = 1 + (R/S)
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180 | * where R = pr0 + pr1*s^2 + pr2*s^4 + ... + pr5*s^10
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181 | * S = 1 + ps0*s^2 + ... + ps4*s^10
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182 | * and
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183 | * | pone(x)-1-R/S | <= 2 ** ( -60.06)
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184 | */
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185 |
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186 | static const double pr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
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187 | 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
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188 | 1.17187499999988647970e-01, /* 0x3FBDFFFF, 0xFFFFFCCE */
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189 | 1.32394806593073575129e+01, /* 0x402A7A9D, 0x357F7FCE */
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190 | 4.12051854307378562225e+02, /* 0x4079C0D4, 0x652EA590 */
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191 | 3.87474538913960532227e+03, /* 0x40AE457D, 0xA3A532CC */
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192 | 7.91447954031891731574e+03, /* 0x40BEEA7A, 0xC32782DD */
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193 | };
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194 | static const double ps8[5] = {
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195 | 1.14207370375678408436e+02, /* 0x405C8D45, 0x8E656CAC */
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196 | 3.65093083420853463394e+03, /* 0x40AC85DC, 0x964D274F */
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197 | 3.69562060269033463555e+04, /* 0x40E20B86, 0x97C5BB7F */
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198 | 9.76027935934950801311e+04, /* 0x40F7D42C, 0xB28F17BB */
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199 | 3.08042720627888811578e+04, /* 0x40DE1511, 0x697A0B2D */
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200 | };
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201 |
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202 | static const double pr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
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203 | 1.31990519556243522749e-11, /* 0x3DAD0667, 0xDAE1CA7D */
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204 | 1.17187493190614097638e-01, /* 0x3FBDFFFF, 0xE2C10043 */
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205 | 6.80275127868432871736e+00, /* 0x401B3604, 0x6E6315E3 */
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206 | 1.08308182990189109773e+02, /* 0x405B13B9, 0x452602ED */
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207 | 5.17636139533199752805e+02, /* 0x40802D16, 0xD052D649 */
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208 | 5.28715201363337541807e+02, /* 0x408085B8, 0xBB7E0CB7 */
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209 | };
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210 | static const double ps5[5] = {
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211 | 5.92805987221131331921e+01, /* 0x404DA3EA, 0xA8AF633D */
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212 | 9.91401418733614377743e+02, /* 0x408EFB36, 0x1B066701 */
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213 | 5.35326695291487976647e+03, /* 0x40B4E944, 0x5706B6FB */
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214 | 7.84469031749551231769e+03, /* 0x40BEA4B0, 0xB8A5BB15 */
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215 | 1.50404688810361062679e+03, /* 0x40978030, 0x036F5E51 */
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216 | };
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217 |
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218 | static const double pr3[6] = {
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219 | 3.02503916137373618024e-09, /* 0x3E29FC21, 0xA7AD9EDD */
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220 | 1.17186865567253592491e-01, /* 0x3FBDFFF5, 0x5B21D17B */
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221 | 3.93297750033315640650e+00, /* 0x400F76BC, 0xE85EAD8A */
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222 | 3.51194035591636932736e+01, /* 0x40418F48, 0x9DA6D129 */
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223 | 9.10550110750781271918e+01, /* 0x4056C385, 0x4D2C1837 */
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224 | 4.85590685197364919645e+01, /* 0x4048478F, 0x8EA83EE5 */
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225 | };
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226 | static const double ps3[5] = {
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227 | 3.47913095001251519989e+01, /* 0x40416549, 0xA134069C */
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228 | 3.36762458747825746741e+02, /* 0x40750C33, 0x07F1A75F */
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229 | 1.04687139975775130551e+03, /* 0x40905B7C, 0x5037D523 */
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230 | 8.90811346398256432622e+02, /* 0x408BD67D, 0xA32E31E9 */
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231 | 1.03787932439639277504e+02, /* 0x4059F26D, 0x7C2EED53 */
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232 | };
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233 |
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234 | static const double pr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
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235 | 1.07710830106873743082e-07, /* 0x3E7CE9D4, 0xF65544F4 */
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236 | 1.17176219462683348094e-01, /* 0x3FBDFF42, 0xBE760D83 */
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237 | 2.36851496667608785174e+00, /* 0x4002F2B7, 0xF98FAEC0 */
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238 | 1.22426109148261232917e+01, /* 0x40287C37, 0x7F71A964 */
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239 | 1.76939711271687727390e+01, /* 0x4031B1A8, 0x177F8EE2 */
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240 | 5.07352312588818499250e+00, /* 0x40144B49, 0xA574C1FE */
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241 | };
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242 | static const double ps2[5] = {
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243 | 2.14364859363821409488e+01, /* 0x40356FBD, 0x8AD5ECDC */
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244 | 1.25290227168402751090e+02, /* 0x405F5293, 0x14F92CD5 */
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245 | 2.32276469057162813669e+02, /* 0x406D08D8, 0xD5A2DBD9 */
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246 | 1.17679373287147100768e+02, /* 0x405D6B7A, 0xDA1884A9 */
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247 | 8.36463893371618283368e+00, /* 0x4020BAB1, 0xF44E5192 */
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248 | };
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249 |
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250 | static double pone(double x)
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251 | {
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252 | const double *p,*q;
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253 | double_t z,r,s;
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254 | uint32_t ix;
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255 |
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256 | GET_HIGH_WORD(ix, x);
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257 | ix &= 0x7fffffff;
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258 | if (ix >= 0x40200000){p = pr8; q = ps8;}
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259 | else if (ix >= 0x40122E8B){p = pr5; q = ps5;}
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260 | else if (ix >= 0x4006DB6D){p = pr3; q = ps3;}
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261 | else /*ix >= 0x40000000*/ {p = pr2; q = ps2;}
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262 | z = 1.0/(x*x);
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263 | r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
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264 | s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
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265 | return 1.0+ r/s;
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266 | }
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267 |
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268 | /* For x >= 8, the asymptotic expansions of qone is
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269 | * 3/8 s - 105/1024 s^3 - ..., where s = 1/x.
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270 | * We approximate pone by
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271 | * qone(x) = s*(0.375 + (R/S))
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272 | * where R = qr1*s^2 + qr2*s^4 + ... + qr5*s^10
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273 | * S = 1 + qs1*s^2 + ... + qs6*s^12
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274 | * and
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275 | * | qone(x)/s -0.375-R/S | <= 2 ** ( -61.13)
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276 | */
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277 |
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278 | static const double qr8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
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279 | 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
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280 | -1.02539062499992714161e-01, /* 0xBFBA3FFF, 0xFFFFFDF3 */
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281 | -1.62717534544589987888e+01, /* 0xC0304591, 0xA26779F7 */
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282 | -7.59601722513950107896e+02, /* 0xC087BCD0, 0x53E4B576 */
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283 | -1.18498066702429587167e+04, /* 0xC0C724E7, 0x40F87415 */
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284 | -4.84385124285750353010e+04, /* 0xC0E7A6D0, 0x65D09C6A */
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285 | };
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286 | static const double qs8[6] = {
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287 | 1.61395369700722909556e+02, /* 0x40642CA6, 0xDE5BCDE5 */
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288 | 7.82538599923348465381e+03, /* 0x40BE9162, 0xD0D88419 */
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289 | 1.33875336287249578163e+05, /* 0x4100579A, 0xB0B75E98 */
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290 | 7.19657723683240939863e+05, /* 0x4125F653, 0x72869C19 */
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291 | 6.66601232617776375264e+05, /* 0x412457D2, 0x7719AD5C */
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292 | -2.94490264303834643215e+05, /* 0xC111F969, 0x0EA5AA18 */
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293 | };
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294 |
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295 | static const double qr5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
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296 | -2.08979931141764104297e-11, /* 0xBDB6FA43, 0x1AA1A098 */
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297 | -1.02539050241375426231e-01, /* 0xBFBA3FFF, 0xCB597FEF */
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298 | -8.05644828123936029840e+00, /* 0xC0201CE6, 0xCA03AD4B */
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299 | -1.83669607474888380239e+02, /* 0xC066F56D, 0x6CA7B9B0 */
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300 | -1.37319376065508163265e+03, /* 0xC09574C6, 0x6931734F */
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301 | -2.61244440453215656817e+03, /* 0xC0A468E3, 0x88FDA79D */
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302 | };
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303 | static const double qs5[6] = {
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304 | 8.12765501384335777857e+01, /* 0x405451B2, 0xFF5A11B2 */
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305 | 1.99179873460485964642e+03, /* 0x409F1F31, 0xE77BF839 */
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306 | 1.74684851924908907677e+04, /* 0x40D10F1F, 0x0D64CE29 */
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307 | 4.98514270910352279316e+04, /* 0x40E8576D, 0xAABAD197 */
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308 | 2.79480751638918118260e+04, /* 0x40DB4B04, 0xCF7C364B */
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309 | -4.71918354795128470869e+03, /* 0xC0B26F2E, 0xFCFFA004 */
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310 | };
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311 |
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312 | static const double qr3[6] = {
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313 | -5.07831226461766561369e-09, /* 0xBE35CFA9, 0xD38FC84F */
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314 | -1.02537829820837089745e-01, /* 0xBFBA3FEB, 0x51AEED54 */
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315 | -4.61011581139473403113e+00, /* 0xC01270C2, 0x3302D9FF */
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316 | -5.78472216562783643212e+01, /* 0xC04CEC71, 0xC25D16DA */
|
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317 | -2.28244540737631695038e+02, /* 0xC06C87D3, 0x4718D55F */
|
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318 | -2.19210128478909325622e+02, /* 0xC06B66B9, 0x5F5C1BF6 */
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319 | };
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320 | static const double qs3[6] = {
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321 | 4.76651550323729509273e+01, /* 0x4047D523, 0xCCD367E4 */
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322 | 6.73865112676699709482e+02, /* 0x40850EEB, 0xC031EE3E */
|
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323 | 3.38015286679526343505e+03, /* 0x40AA684E, 0x448E7C9A */
|
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324 | 5.54772909720722782367e+03, /* 0x40B5ABBA, 0xA61D54A6 */
|
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325 | 1.90311919338810798763e+03, /* 0x409DBC7A, 0x0DD4DF4B */
|
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326 | -1.35201191444307340817e+02, /* 0xC060E670, 0x290A311F */
|
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327 | };
|
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328 |
|
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329 | static const double qr2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
|
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330 | -1.78381727510958865572e-07, /* 0xBE87F126, 0x44C626D2 */
|
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331 | -1.02517042607985553460e-01, /* 0xBFBA3E8E, 0x9148B010 */
|
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332 | -2.75220568278187460720e+00, /* 0xC0060484, 0x69BB4EDA */
|
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333 | -1.96636162643703720221e+01, /* 0xC033A9E2, 0xC168907F */
|
---|
334 | -4.23253133372830490089e+01, /* 0xC04529A3, 0xDE104AAA */
|
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335 | -2.13719211703704061733e+01, /* 0xC0355F36, 0x39CF6E52 */
|
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336 | };
|
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337 | static const double qs2[6] = {
|
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338 | 2.95333629060523854548e+01, /* 0x403D888A, 0x78AE64FF */
|
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339 | 2.52981549982190529136e+02, /* 0x406F9F68, 0xDB821CBA */
|
---|
340 | 7.57502834868645436472e+02, /* 0x4087AC05, 0xCE49A0F7 */
|
---|
341 | 7.39393205320467245656e+02, /* 0x40871B25, 0x48D4C029 */
|
---|
342 | 1.55949003336666123687e+02, /* 0x40637E5E, 0x3C3ED8D4 */
|
---|
343 | -4.95949898822628210127e+00, /* 0xC013D686, 0xE71BE86B */
|
---|
344 | };
|
---|
345 |
|
---|
346 | static double qone(double x)
|
---|
347 | {
|
---|
348 | const double *p,*q;
|
---|
349 | double_t s,r,z;
|
---|
350 | uint32_t ix;
|
---|
351 |
|
---|
352 | GET_HIGH_WORD(ix, x);
|
---|
353 | ix &= 0x7fffffff;
|
---|
354 | if (ix >= 0x40200000){p = qr8; q = qs8;}
|
---|
355 | else if (ix >= 0x40122E8B){p = qr5; q = qs5;}
|
---|
356 | else if (ix >= 0x4006DB6D){p = qr3; q = qs3;}
|
---|
357 | else /*ix >= 0x40000000*/ {p = qr2; q = qs2;}
|
---|
358 | z = 1.0/(x*x);
|
---|
359 | r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
|
---|
360 | s = 1.0+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
|
---|
361 | return (.375 + r/s)/x;
|
---|
362 | }
|
---|