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1 /*-------------------------------------------------------------------- |
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2 *© Portions copyright (c) 2006 Nokia Corporation. All rights reserved. |
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3 *-------------------------------------------------------------------- |
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4 */ |
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5 /* e_j0f.c -- float version of e_j0.c. |
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6 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com. |
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7 */ |
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8 |
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9 /* |
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10 * ==================================================== |
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11 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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12 * |
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13 * Developed at SunPro, a Sun Microsystems, Inc. business. |
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14 * Permission to use, copy, modify, and distribute this |
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15 * software is freely granted, provided that this notice |
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16 * is preserved. |
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17 * ==================================================== |
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18 */ |
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19 #ifndef __SYMBIAN32__ |
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20 #ifndef lint |
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21 static char rcsid[] = "$FreeBSD: src/lib/msun/src/e_j0f.c,v 1.7 2002/05/28 18:15:03 alfred Exp $"; |
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22 #endif |
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23 #endif //__SYMBIAN32__ |
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24 |
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25 #include <math.h> |
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26 #include "math_private.h" |
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27 |
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28 static float pzerof(float), qzerof(float); |
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29 |
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30 static const float |
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31 huge = 1e30, |
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32 one = 1.0, |
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33 invsqrtpi= 5.6418961287e-01, /* 0x3f106ebb */ |
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34 tpi = 6.3661974669e-01, /* 0x3f22f983 */ |
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35 /* R0/S0 on [0, 2.00] */ |
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36 R02 = 1.5625000000e-02, /* 0x3c800000 */ |
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37 R03 = -1.8997929874e-04, /* 0xb947352e */ |
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38 R04 = 1.8295404516e-06, /* 0x35f58e88 */ |
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39 R05 = -4.6183270541e-09, /* 0xb19eaf3c */ |
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40 S01 = 1.5619102865e-02, /* 0x3c7fe744 */ |
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41 S02 = 1.1692678527e-04, /* 0x38f53697 */ |
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42 S03 = 5.1354652442e-07, /* 0x3509daa6 */ |
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43 S04 = 1.1661400734e-09; /* 0x30a045e8 */ |
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44 |
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45 static const float zero = 0.0; |
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46 |
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47 EXPORT_C float |
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48 __ieee754_j0f(float x) |
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49 { |
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50 float z, s,c,ss,cc,r,u,v; |
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51 int32_t hx,ix; |
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52 |
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53 GET_FLOAT_WORD(hx,x); |
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54 ix = hx&0x7fffffff; |
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55 if(ix>=0x7f800000) return one/(x*x); |
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56 x = fabsf(x); |
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57 if(ix >= 0x40000000) { /* |x| >= 2.0 */ |
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58 s = sinf(x); |
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59 c = cosf(x); |
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60 ss = s-c; |
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61 cc = s+c; |
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62 if(ix<0x7f000000) { /* make sure x+x not overflow */ |
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63 z = -cosf(x+x); |
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64 if ((s*c)<zero) cc = z/ss; |
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65 else ss = z/cc; |
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66 } |
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67 /* |
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68 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) |
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69 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) |
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70 */ |
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71 if(ix>0x80000000) z = (invsqrtpi*cc)/sqrtf(x); |
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72 else { |
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73 u = pzerof(x); v = qzerof(x); |
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74 z = invsqrtpi*(u*cc-v*ss)/sqrtf(x); |
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75 } |
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76 return z; |
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77 } |
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78 if(ix<0x39000000) { /* |x| < 2**-13 */ |
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79 if(huge+x>one) { /* raise inexact if x != 0 */ |
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80 if(ix<0x32000000) return one; /* |x|<2**-27 */ |
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81 else return one - (float)0.25*x*x; |
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82 } |
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83 } |
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84 z = x*x; |
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85 r = z*(R02+z*(R03+z*(R04+z*R05))); |
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86 s = one+z*(S01+z*(S02+z*(S03+z*S04))); |
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87 if(ix < 0x3F800000) { /* |x| < 1.00 */ |
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88 return one + z*((float)-0.25+(r/s)); |
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89 } else { |
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90 u = (float)0.5*x; |
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91 return((one+u)*(one-u)+z*(r/s)); |
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92 } |
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93 } |
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94 |
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95 static const float |
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96 u00 = -7.3804296553e-02, /* 0xbd9726b5 */ |
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97 u01 = 1.7666645348e-01, /* 0x3e34e80d */ |
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98 u02 = -1.3818567619e-02, /* 0xbc626746 */ |
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99 u03 = 3.4745343146e-04, /* 0x39b62a69 */ |
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100 u04 = -3.8140706238e-06, /* 0xb67ff53c */ |
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101 u05 = 1.9559013964e-08, /* 0x32a802ba */ |
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102 u06 = -3.9820518410e-11, /* 0xae2f21eb */ |
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103 v01 = 1.2730483897e-02, /* 0x3c509385 */ |
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104 v02 = 7.6006865129e-05, /* 0x389f65e0 */ |
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105 v03 = 2.5915085189e-07, /* 0x348b216c */ |
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106 v04 = 4.4111031494e-10; /* 0x2ff280c2 */ |
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107 |
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108 EXPORT_C float __ieee754_y0f(float x) |
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109 { |
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110 float z, s,c,ss,cc,u,v; |
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111 int32_t hx,ix; |
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112 |
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113 GET_FLOAT_WORD(hx,x); |
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114 ix = 0x7fffffff&hx; |
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115 /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0 */ |
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116 if(ix>=0x7f800000) return one/(x+x*x); |
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117 if(ix==0) return -one/zero; |
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118 #ifdef __SYMBIAN32__ |
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119 if(hx<0) return -one/zero; |
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120 #else |
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121 if(hx<0) return zero/zero; |
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122 #endif //__SYMBIAN32__ |
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123 |
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124 if(ix >= 0x40000000) { /* |x| >= 2.0 */ |
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125 /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0)) |
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126 * where x0 = x-pi/4 |
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127 * Better formula: |
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128 * cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4) |
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129 * = 1/sqrt(2) * (sin(x) + cos(x)) |
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130 * sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4) |
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131 * = 1/sqrt(2) * (sin(x) - cos(x)) |
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132 * To avoid cancellation, use |
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133 * sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x)) |
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134 * to compute the worse one. |
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135 */ |
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136 s = sinf(x); |
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137 c = cosf(x); |
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138 ss = s-c; |
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139 cc = s+c; |
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140 /* |
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141 * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x) |
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142 * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x) |
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143 */ |
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144 if(ix<0x7f000000) { /* make sure x+x not overflow */ |
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145 z = -cosf(x+x); |
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146 if ((s*c)<zero) cc = z/ss; |
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147 else ss = z/cc; |
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148 } |
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149 if(ix>0x80000000) z = (invsqrtpi*ss)/sqrtf(x); |
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150 else { |
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151 u = pzerof(x); v = qzerof(x); |
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152 z = invsqrtpi*(u*ss+v*cc)/sqrtf(x); |
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153 } |
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154 return z; |
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155 } |
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156 if(ix<=0x32000000) { /* x < 2**-27 */ |
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157 return(u00 + tpi*__ieee754_logf(x)); |
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158 } |
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159 z = x*x; |
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160 u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06))))); |
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161 v = one+z*(v01+z*(v02+z*(v03+z*v04))); |
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162 return(u/v + tpi*(__ieee754_j0f(x)*__ieee754_logf(x))); |
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163 } |
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164 |
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165 /* The asymptotic expansions of pzero is |
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166 * 1 - 9/128 s^2 + 11025/98304 s^4 - ..., where s = 1/x. |
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167 * For x >= 2, We approximate pzero by |
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168 * pzero(x) = 1 + (R/S) |
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169 * where R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10 |
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170 * S = 1 + pS0*s^2 + ... + pS4*s^10 |
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171 * and |
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172 * | pzero(x)-1-R/S | <= 2 ** ( -60.26) |
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173 */ |
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174 static const float pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
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175 0.0000000000e+00, /* 0x00000000 */ |
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176 -7.0312500000e-02, /* 0xbd900000 */ |
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177 -8.0816707611e+00, /* 0xc1014e86 */ |
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178 -2.5706311035e+02, /* 0xc3808814 */ |
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179 -2.4852163086e+03, /* 0xc51b5376 */ |
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180 -5.2530439453e+03, /* 0xc5a4285a */ |
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181 }; |
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182 static const float pS8[5] = { |
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183 1.1653436279e+02, /* 0x42e91198 */ |
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184 3.8337448730e+03, /* 0x456f9beb */ |
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185 4.0597855469e+04, /* 0x471e95db */ |
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186 1.1675296875e+05, /* 0x47e4087c */ |
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187 4.7627726562e+04, /* 0x473a0bba */ |
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188 }; |
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189 static const float pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
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190 -1.1412546255e-11, /* 0xad48c58a */ |
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191 -7.0312492549e-02, /* 0xbd8fffff */ |
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192 -4.1596107483e+00, /* 0xc0851b88 */ |
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193 -6.7674766541e+01, /* 0xc287597b */ |
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194 -3.3123129272e+02, /* 0xc3a59d9b */ |
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195 -3.4643338013e+02, /* 0xc3ad3779 */ |
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196 }; |
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197 static const float pS5[5] = { |
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198 6.0753936768e+01, /* 0x42730408 */ |
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199 1.0512523193e+03, /* 0x44836813 */ |
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200 5.9789707031e+03, /* 0x45bad7c4 */ |
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201 9.6254453125e+03, /* 0x461665c8 */ |
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202 2.4060581055e+03, /* 0x451660ee */ |
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203 }; |
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204 |
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205 static const float pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ |
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206 -2.5470459075e-09, /* 0xb12f081b */ |
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207 -7.0311963558e-02, /* 0xbd8fffb8 */ |
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208 -2.4090321064e+00, /* 0xc01a2d95 */ |
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209 -2.1965976715e+01, /* 0xc1afba52 */ |
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210 -5.8079170227e+01, /* 0xc2685112 */ |
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211 -3.1447946548e+01, /* 0xc1fb9565 */ |
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212 }; |
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213 static const float pS3[5] = { |
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214 3.5856033325e+01, /* 0x420f6c94 */ |
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215 3.6151397705e+02, /* 0x43b4c1ca */ |
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216 1.1936077881e+03, /* 0x44953373 */ |
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217 1.1279968262e+03, /* 0x448cffe6 */ |
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218 1.7358093262e+02, /* 0x432d94b8 */ |
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219 }; |
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220 |
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221 static const float pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
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222 -8.8753431271e-08, /* 0xb3be98b7 */ |
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223 -7.0303097367e-02, /* 0xbd8ffb12 */ |
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224 -1.4507384300e+00, /* 0xbfb9b1cc */ |
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225 -7.6356959343e+00, /* 0xc0f4579f */ |
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226 -1.1193166733e+01, /* 0xc1331736 */ |
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227 -3.2336456776e+00, /* 0xc04ef40d */ |
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228 }; |
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229 static const float pS2[5] = { |
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230 2.2220300674e+01, /* 0x41b1c32d */ |
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231 1.3620678711e+02, /* 0x430834f0 */ |
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232 2.7047027588e+02, /* 0x43873c32 */ |
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233 1.5387539673e+02, /* 0x4319e01a */ |
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234 1.4657617569e+01, /* 0x416a859a */ |
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235 }; |
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236 |
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237 static float pzerof(float x) |
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238 { |
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239 const float *p,*q; |
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240 float z,r,s; |
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241 int32_t ix; |
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242 GET_FLOAT_WORD(ix,x); |
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243 ix &= 0x7fffffff; |
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244 if(ix>=0x41000000) {p = pR8; q= pS8;} |
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245 else if(ix>=0x40f71c58){p = pR5; q= pS5;} |
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246 else if(ix>=0x4036db68){p = pR3; q= pS3;} |
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247 else if(ix>=0x40000000){p = pR2; q= pS2;} |
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248 z = one/(x*x); |
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249 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); |
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250 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4])))); |
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251 return one+ r/s; |
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252 } |
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253 |
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254 |
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255 /* For x >= 8, the asymptotic expansions of qzero is |
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256 * -1/8 s + 75/1024 s^3 - ..., where s = 1/x. |
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257 * We approximate pzero by |
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258 * qzero(x) = s*(-1.25 + (R/S)) |
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259 * where R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10 |
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260 * S = 1 + qS0*s^2 + ... + qS5*s^12 |
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261 * and |
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262 * | qzero(x)/s +1.25-R/S | <= 2 ** ( -61.22) |
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263 */ |
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264 static const float qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */ |
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265 0.0000000000e+00, /* 0x00000000 */ |
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266 7.3242187500e-02, /* 0x3d960000 */ |
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267 1.1768206596e+01, /* 0x413c4a93 */ |
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268 5.5767340088e+02, /* 0x440b6b19 */ |
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269 8.8591972656e+03, /* 0x460a6cca */ |
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270 3.7014625000e+04, /* 0x471096a0 */ |
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271 }; |
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272 static const float qS8[6] = { |
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273 1.6377603149e+02, /* 0x4323c6aa */ |
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274 8.0983447266e+03, /* 0x45fd12c2 */ |
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275 1.4253829688e+05, /* 0x480b3293 */ |
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276 8.0330925000e+05, /* 0x49441ed4 */ |
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277 8.4050156250e+05, /* 0x494d3359 */ |
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278 -3.4389928125e+05, /* 0xc8a7eb69 */ |
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279 }; |
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280 |
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281 static const float qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */ |
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282 1.8408595828e-11, /* 0x2da1ec79 */ |
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283 7.3242180049e-02, /* 0x3d95ffff */ |
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284 5.8356351852e+00, /* 0x40babd86 */ |
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285 1.3511157227e+02, /* 0x43071c90 */ |
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286 1.0272437744e+03, /* 0x448067cd */ |
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287 1.9899779053e+03, /* 0x44f8bf4b */ |
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288 }; |
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289 static const float qS5[6] = { |
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290 8.2776611328e+01, /* 0x42a58da0 */ |
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291 2.0778142090e+03, /* 0x4501dd07 */ |
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292 1.8847289062e+04, /* 0x46933e94 */ |
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293 5.6751113281e+04, /* 0x475daf1d */ |
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294 3.5976753906e+04, /* 0x470c88c1 */ |
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295 -5.3543427734e+03, /* 0xc5a752be */ |
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296 }; |
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297 |
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298 static const float qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */ |
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299 4.3774099900e-09, /* 0x3196681b */ |
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300 7.3241114616e-02, /* 0x3d95ff70 */ |
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301 3.3442313671e+00, /* 0x405607e3 */ |
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302 4.2621845245e+01, /* 0x422a7cc5 */ |
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303 1.7080809021e+02, /* 0x432acedf */ |
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304 1.6673394775e+02, /* 0x4326bbe4 */ |
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305 }; |
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306 static const float qS3[6] = { |
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307 4.8758872986e+01, /* 0x42430916 */ |
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308 7.0968920898e+02, /* 0x44316c1c */ |
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309 3.7041481934e+03, /* 0x4567825f */ |
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310 6.4604252930e+03, /* 0x45c9e367 */ |
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311 2.5163337402e+03, /* 0x451d4557 */ |
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312 -1.4924745178e+02, /* 0xc3153f59 */ |
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313 }; |
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314 |
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315 static const float qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */ |
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316 1.5044444979e-07, /* 0x342189db */ |
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317 7.3223426938e-02, /* 0x3d95f62a */ |
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318 1.9981917143e+00, /* 0x3fffc4bf */ |
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319 1.4495602608e+01, /* 0x4167edfd */ |
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320 3.1666231155e+01, /* 0x41fd5471 */ |
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321 1.6252708435e+01, /* 0x4182058c */ |
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322 }; |
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323 static const float qS2[6] = { |
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324 3.0365585327e+01, /* 0x41f2ecb8 */ |
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325 2.6934811401e+02, /* 0x4386ac8f */ |
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326 8.4478375244e+02, /* 0x44533229 */ |
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327 8.8293585205e+02, /* 0x445cbbe5 */ |
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328 2.1266638184e+02, /* 0x4354aa98 */ |
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329 -5.3109550476e+00, /* 0xc0a9f358 */ |
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330 }; |
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331 |
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332 static float qzerof(float x) |
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333 { |
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334 const float *p,*q; |
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335 float s,r,z; |
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336 int32_t ix; |
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337 GET_FLOAT_WORD(ix,x); |
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338 ix &= 0x7fffffff; |
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339 if(ix>=0x41000000) {p = qR8; q= qS8;} |
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340 else if(ix>=0x40f71c58){p = qR5; q= qS5;} |
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341 else if(ix>=0x4036db68){p = qR3; q= qS3;} |
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342 else if(ix>=0x40000000){p = qR2; q= qS2;} |
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343 z = one/(x*x); |
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344 r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5])))); |
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345 s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5]))))); |
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346 return (-(float).125 + r/s)/x; |
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347 } |