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1 /* |
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2 * Copyright 2001-2005 Sun Microsystems, Inc. All Rights Reserved. |
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3 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. |
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4 * |
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5 * This code is free software; you can redistribute it and/or modify it |
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6 * under the terms of the GNU General Public License version 2 only, as |
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7 * published by the Free Software Foundation. |
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8 * |
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9 * This code is distributed in the hope that it will be useful, but WITHOUT |
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10 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
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11 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
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12 * version 2 for more details (a copy is included in the LICENSE file that |
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13 * accompanied this code). |
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14 * |
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15 * You should have received a copy of the GNU General Public License version |
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16 * 2 along with this work; if not, write to the Free Software Foundation, |
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17 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. |
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18 * |
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19 * Please contact Sun Microsystems, Inc., 4150 Network Circle, Santa Clara, |
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20 * CA 95054 USA or visit www.sun.com if you need additional information or |
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21 * have any questions. |
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22 * |
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23 */ |
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24 |
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25 #include "incls/_precompiled.incl" |
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26 #include "incls/_sharedRuntimeTrig.cpp.incl" |
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27 |
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28 // This file contains copies of the fdlibm routines used by |
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29 // StrictMath. It turns out that it is almost always required to use |
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30 // these runtime routines; the Intel CPU doesn't meet the Java |
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31 // specification for sin/cos outside a certain limited argument range, |
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32 // and the SPARC CPU doesn't appear to have sin/cos instructions. It |
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33 // also turns out that avoiding the indirect call through function |
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34 // pointer out to libjava.so in SharedRuntime speeds these routines up |
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35 // by roughly 15% on both Win32/x86 and Solaris/SPARC. |
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36 |
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37 // Enabling optimizations in this file causes incorrect code to be |
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38 // generated; can not figure out how to turn down optimization for one |
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39 // file in the IDE on Windows |
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40 #ifdef WIN32 |
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41 # pragma optimize ( "", off ) |
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42 #endif |
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43 |
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44 #include <math.h> |
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45 |
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46 // VM_LITTLE_ENDIAN is #defined appropriately in the Makefiles |
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47 // [jk] this is not 100% correct because the float word order may different |
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48 // from the byte order (e.g. on ARM) |
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49 #ifdef VM_LITTLE_ENDIAN |
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50 # define __HI(x) *(1+(int*)&x) |
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51 # define __LO(x) *(int*)&x |
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52 #else |
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53 # define __HI(x) *(int*)&x |
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54 # define __LO(x) *(1+(int*)&x) |
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55 #endif |
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56 |
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57 static double copysignA(double x, double y) { |
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58 __HI(x) = (__HI(x)&0x7fffffff)|(__HI(y)&0x80000000); |
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59 return x; |
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60 } |
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61 |
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62 /* |
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63 * scalbn (double x, int n) |
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64 * scalbn(x,n) returns x* 2**n computed by exponent |
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65 * manipulation rather than by actually performing an |
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66 * exponentiation or a multiplication. |
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67 */ |
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68 |
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69 static const double |
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70 two54 = 1.80143985094819840000e+16, /* 0x43500000, 0x00000000 */ |
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71 twom54 = 5.55111512312578270212e-17, /* 0x3C900000, 0x00000000 */ |
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72 hugeX = 1.0e+300, |
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73 tiny = 1.0e-300; |
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74 |
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75 static double scalbnA (double x, int n) { |
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76 int k,hx,lx; |
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77 hx = __HI(x); |
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78 lx = __LO(x); |
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79 k = (hx&0x7ff00000)>>20; /* extract exponent */ |
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80 if (k==0) { /* 0 or subnormal x */ |
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81 if ((lx|(hx&0x7fffffff))==0) return x; /* +-0 */ |
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82 x *= two54; |
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83 hx = __HI(x); |
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84 k = ((hx&0x7ff00000)>>20) - 54; |
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85 if (n< -50000) return tiny*x; /*underflow*/ |
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86 } |
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87 if (k==0x7ff) return x+x; /* NaN or Inf */ |
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88 k = k+n; |
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89 if (k > 0x7fe) return hugeX*copysignA(hugeX,x); /* overflow */ |
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90 if (k > 0) /* normal result */ |
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91 {__HI(x) = (hx&0x800fffff)|(k<<20); return x;} |
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92 if (k <= -54) { |
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93 if (n > 50000) /* in case integer overflow in n+k */ |
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94 return hugeX*copysignA(hugeX,x); /*overflow*/ |
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95 else return tiny*copysignA(tiny,x); /*underflow*/ |
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96 } |
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97 k += 54; /* subnormal result */ |
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98 __HI(x) = (hx&0x800fffff)|(k<<20); |
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99 return x*twom54; |
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100 } |
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101 |
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102 /* |
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103 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2) |
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104 * double x[],y[]; int e0,nx,prec; int ipio2[]; |
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105 * |
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106 * __kernel_rem_pio2 return the last three digits of N with |
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107 * y = x - N*pi/2 |
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108 * so that |y| < pi/2. |
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109 * |
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110 * The method is to compute the integer (mod 8) and fraction parts of |
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111 * (2/pi)*x without doing the full multiplication. In general we |
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112 * skip the part of the product that are known to be a huge integer ( |
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113 * more accurately, = 0 mod 8 ). Thus the number of operations are |
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114 * independent of the exponent of the input. |
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115 * |
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116 * (2/pi) is represented by an array of 24-bit integers in ipio2[]. |
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117 * |
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118 * Input parameters: |
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119 * x[] The input value (must be positive) is broken into nx |
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120 * pieces of 24-bit integers in double precision format. |
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121 * x[i] will be the i-th 24 bit of x. The scaled exponent |
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122 * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0 |
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123 * match x's up to 24 bits. |
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124 * |
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125 * Example of breaking a double positive z into x[0]+x[1]+x[2]: |
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126 * e0 = ilogb(z)-23 |
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127 * z = scalbn(z,-e0) |
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128 * for i = 0,1,2 |
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129 * x[i] = floor(z) |
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130 * z = (z-x[i])*2**24 |
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131 * |
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132 * |
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133 * y[] ouput result in an array of double precision numbers. |
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134 * The dimension of y[] is: |
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135 * 24-bit precision 1 |
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136 * 53-bit precision 2 |
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137 * 64-bit precision 2 |
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138 * 113-bit precision 3 |
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139 * The actual value is the sum of them. Thus for 113-bit |
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140 * precsion, one may have to do something like: |
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141 * |
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142 * long double t,w,r_head, r_tail; |
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143 * t = (long double)y[2] + (long double)y[1]; |
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144 * w = (long double)y[0]; |
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145 * r_head = t+w; |
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146 * r_tail = w - (r_head - t); |
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147 * |
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148 * e0 The exponent of x[0] |
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149 * |
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150 * nx dimension of x[] |
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151 * |
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152 * prec an interger indicating the precision: |
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153 * 0 24 bits (single) |
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154 * 1 53 bits (double) |
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155 * 2 64 bits (extended) |
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156 * 3 113 bits (quad) |
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157 * |
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158 * ipio2[] |
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159 * integer array, contains the (24*i)-th to (24*i+23)-th |
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160 * bit of 2/pi after binary point. The corresponding |
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161 * floating value is |
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162 * |
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163 * ipio2[i] * 2^(-24(i+1)). |
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164 * |
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165 * External function: |
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166 * double scalbn(), floor(); |
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167 * |
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168 * |
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169 * Here is the description of some local variables: |
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170 * |
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171 * jk jk+1 is the initial number of terms of ipio2[] needed |
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172 * in the computation. The recommended value is 2,3,4, |
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173 * 6 for single, double, extended,and quad. |
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174 * |
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175 * jz local integer variable indicating the number of |
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176 * terms of ipio2[] used. |
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177 * |
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178 * jx nx - 1 |
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179 * |
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180 * jv index for pointing to the suitable ipio2[] for the |
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181 * computation. In general, we want |
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182 * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8 |
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183 * is an integer. Thus |
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184 * e0-3-24*jv >= 0 or (e0-3)/24 >= jv |
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185 * Hence jv = max(0,(e0-3)/24). |
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186 * |
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187 * jp jp+1 is the number of terms in PIo2[] needed, jp = jk. |
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188 * |
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189 * q[] double array with integral value, representing the |
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190 * 24-bits chunk of the product of x and 2/pi. |
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191 * |
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192 * q0 the corresponding exponent of q[0]. Note that the |
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193 * exponent for q[i] would be q0-24*i. |
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194 * |
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195 * PIo2[] double precision array, obtained by cutting pi/2 |
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196 * into 24 bits chunks. |
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197 * |
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198 * f[] ipio2[] in floating point |
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199 * |
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200 * iq[] integer array by breaking up q[] in 24-bits chunk. |
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201 * |
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202 * fq[] final product of x*(2/pi) in fq[0],..,fq[jk] |
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203 * |
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204 * ih integer. If >0 it indicats q[] is >= 0.5, hence |
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205 * it also indicates the *sign* of the result. |
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206 * |
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207 */ |
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208 |
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209 |
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210 /* |
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211 * Constants: |
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212 * The hexadecimal values are the intended ones for the following |
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213 * constants. The decimal values may be used, provided that the |
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214 * compiler will convert from decimal to binary accurately enough |
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215 * to produce the hexadecimal values shown. |
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216 */ |
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217 |
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218 |
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219 static const int init_jk[] = {2,3,4,6}; /* initial value for jk */ |
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220 |
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221 static const double PIo2[] = { |
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222 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */ |
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223 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */ |
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224 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */ |
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225 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */ |
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226 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */ |
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227 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */ |
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228 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */ |
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229 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */ |
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230 }; |
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231 |
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232 static const double |
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233 zeroB = 0.0, |
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234 one = 1.0, |
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235 two24B = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ |
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236 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */ |
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237 |
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238 static int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int *ipio2) { |
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239 int jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih; |
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240 double z,fw,f[20],fq[20],q[20]; |
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241 |
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242 /* initialize jk*/ |
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243 jk = init_jk[prec]; |
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244 jp = jk; |
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245 |
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246 /* determine jx,jv,q0, note that 3>q0 */ |
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247 jx = nx-1; |
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248 jv = (e0-3)/24; if(jv<0) jv=0; |
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249 q0 = e0-24*(jv+1); |
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250 |
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251 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */ |
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252 j = jv-jx; m = jx+jk; |
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253 for(i=0;i<=m;i++,j++) f[i] = (j<0)? zeroB : (double) ipio2[j]; |
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254 |
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255 /* compute q[0],q[1],...q[jk] */ |
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256 for (i=0;i<=jk;i++) { |
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257 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw; |
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258 } |
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259 |
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260 jz = jk; |
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261 recompute: |
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262 /* distill q[] into iq[] reversingly */ |
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263 for(i=0,j=jz,z=q[jz];j>0;i++,j--) { |
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264 fw = (double)((int)(twon24* z)); |
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265 iq[i] = (int)(z-two24B*fw); |
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266 z = q[j-1]+fw; |
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267 } |
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268 |
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269 /* compute n */ |
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270 z = scalbnA(z,q0); /* actual value of z */ |
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271 z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */ |
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272 n = (int) z; |
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273 z -= (double)n; |
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274 ih = 0; |
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275 if(q0>0) { /* need iq[jz-1] to determine n */ |
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276 i = (iq[jz-1]>>(24-q0)); n += i; |
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277 iq[jz-1] -= i<<(24-q0); |
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278 ih = iq[jz-1]>>(23-q0); |
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279 } |
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280 else if(q0==0) ih = iq[jz-1]>>23; |
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281 else if(z>=0.5) ih=2; |
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282 |
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283 if(ih>0) { /* q > 0.5 */ |
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284 n += 1; carry = 0; |
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285 for(i=0;i<jz ;i++) { /* compute 1-q */ |
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286 j = iq[i]; |
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287 if(carry==0) { |
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288 if(j!=0) { |
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289 carry = 1; iq[i] = 0x1000000- j; |
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290 } |
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291 } else iq[i] = 0xffffff - j; |
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292 } |
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293 if(q0>0) { /* rare case: chance is 1 in 12 */ |
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294 switch(q0) { |
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295 case 1: |
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296 iq[jz-1] &= 0x7fffff; break; |
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297 case 2: |
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298 iq[jz-1] &= 0x3fffff; break; |
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299 } |
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300 } |
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301 if(ih==2) { |
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302 z = one - z; |
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303 if(carry!=0) z -= scalbnA(one,q0); |
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304 } |
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305 } |
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306 |
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307 /* check if recomputation is needed */ |
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308 if(z==zeroB) { |
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309 j = 0; |
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310 for (i=jz-1;i>=jk;i--) j |= iq[i]; |
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311 if(j==0) { /* need recomputation */ |
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312 for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */ |
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313 |
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314 for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */ |
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315 f[jx+i] = (double) ipio2[jv+i]; |
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316 for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; |
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317 q[i] = fw; |
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318 } |
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319 jz += k; |
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320 goto recompute; |
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321 } |
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322 } |
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323 |
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324 /* chop off zero terms */ |
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325 if(z==0.0) { |
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326 jz -= 1; q0 -= 24; |
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327 while(iq[jz]==0) { jz--; q0-=24;} |
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328 } else { /* break z into 24-bit if neccessary */ |
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329 z = scalbnA(z,-q0); |
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330 if(z>=two24B) { |
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331 fw = (double)((int)(twon24*z)); |
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332 iq[jz] = (int)(z-two24B*fw); |
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333 jz += 1; q0 += 24; |
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334 iq[jz] = (int) fw; |
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335 } else iq[jz] = (int) z ; |
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336 } |
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337 |
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338 /* convert integer "bit" chunk to floating-point value */ |
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339 fw = scalbnA(one,q0); |
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340 for(i=jz;i>=0;i--) { |
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341 q[i] = fw*(double)iq[i]; fw*=twon24; |
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342 } |
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343 |
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344 /* compute PIo2[0,...,jp]*q[jz,...,0] */ |
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345 for(i=jz;i>=0;i--) { |
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346 for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k]; |
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347 fq[jz-i] = fw; |
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348 } |
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349 |
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350 /* compress fq[] into y[] */ |
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351 switch(prec) { |
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352 case 0: |
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353 fw = 0.0; |
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354 for (i=jz;i>=0;i--) fw += fq[i]; |
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355 y[0] = (ih==0)? fw: -fw; |
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356 break; |
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357 case 1: |
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358 case 2: |
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359 fw = 0.0; |
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360 for (i=jz;i>=0;i--) fw += fq[i]; |
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361 y[0] = (ih==0)? fw: -fw; |
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362 fw = fq[0]-fw; |
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363 for (i=1;i<=jz;i++) fw += fq[i]; |
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364 y[1] = (ih==0)? fw: -fw; |
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365 break; |
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366 case 3: /* painful */ |
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367 for (i=jz;i>0;i--) { |
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368 fw = fq[i-1]+fq[i]; |
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369 fq[i] += fq[i-1]-fw; |
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370 fq[i-1] = fw; |
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371 } |
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372 for (i=jz;i>1;i--) { |
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373 fw = fq[i-1]+fq[i]; |
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374 fq[i] += fq[i-1]-fw; |
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375 fq[i-1] = fw; |
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376 } |
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377 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i]; |
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378 if(ih==0) { |
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379 y[0] = fq[0]; y[1] = fq[1]; y[2] = fw; |
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380 } else { |
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381 y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw; |
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382 } |
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383 } |
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384 return n&7; |
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385 } |
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386 |
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387 |
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388 /* |
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389 * ==================================================== |
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390 * Copyright 13 Dec 1993 Sun Microsystems, Inc. All Rights Reserved. |
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391 * |
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392 * Developed at SunPro, a Sun Microsystems, Inc. business. |
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393 * Permission to use, copy, modify, and distribute this |
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394 * software is freely granted, provided that this notice |
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395 * is preserved. |
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396 * ==================================================== |
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397 * |
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398 */ |
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399 |
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400 /* __ieee754_rem_pio2(x,y) |
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401 * |
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402 * return the remainder of x rem pi/2 in y[0]+y[1] |
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403 * use __kernel_rem_pio2() |
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404 */ |
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405 |
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406 /* |
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407 * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi |
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408 */ |
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409 static const int two_over_pi[] = { |
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410 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62, |
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411 0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A, |
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412 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129, |
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413 0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41, |
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414 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8, |
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415 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF, |
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416 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5, |
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417 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08, |
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418 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3, |
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419 0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880, |
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420 0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B, |
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421 }; |
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422 |
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423 static const int npio2_hw[] = { |
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424 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C, |
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425 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C, |
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426 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A, |
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427 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C, |
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428 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB, |
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429 0x404858EB, 0x404921FB, |
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430 }; |
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431 |
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432 /* |
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433 * invpio2: 53 bits of 2/pi |
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434 * pio2_1: first 33 bit of pi/2 |
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435 * pio2_1t: pi/2 - pio2_1 |
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436 * pio2_2: second 33 bit of pi/2 |
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437 * pio2_2t: pi/2 - (pio2_1+pio2_2) |
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438 * pio2_3: third 33 bit of pi/2 |
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439 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3) |
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440 */ |
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441 |
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442 static const double |
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443 zeroA = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */ |
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444 half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ |
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445 two24A = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */ |
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446 invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */ |
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447 pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */ |
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448 pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */ |
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449 pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */ |
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450 pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */ |
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451 pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */ |
|
452 pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */ |
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453 |
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454 static int __ieee754_rem_pio2(double x, double *y) { |
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455 double z,w,t,r,fn; |
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456 double tx[3]; |
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457 int e0,i,j,nx,n,ix,hx,i0; |
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458 |
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459 i0 = ((*(int*)&two24A)>>30)^1; /* high word index */ |
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460 hx = *(i0+(int*)&x); /* high word of x */ |
|
461 ix = hx&0x7fffffff; |
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462 if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */ |
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463 {y[0] = x; y[1] = 0; return 0;} |
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464 if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */ |
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465 if(hx>0) { |
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466 z = x - pio2_1; |
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467 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ |
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468 y[0] = z - pio2_1t; |
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469 y[1] = (z-y[0])-pio2_1t; |
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470 } else { /* near pi/2, use 33+33+53 bit pi */ |
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471 z -= pio2_2; |
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472 y[0] = z - pio2_2t; |
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473 y[1] = (z-y[0])-pio2_2t; |
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474 } |
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475 return 1; |
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476 } else { /* negative x */ |
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477 z = x + pio2_1; |
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478 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */ |
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479 y[0] = z + pio2_1t; |
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480 y[1] = (z-y[0])+pio2_1t; |
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481 } else { /* near pi/2, use 33+33+53 bit pi */ |
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482 z += pio2_2; |
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483 y[0] = z + pio2_2t; |
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484 y[1] = (z-y[0])+pio2_2t; |
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485 } |
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486 return -1; |
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487 } |
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488 } |
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489 if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */ |
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490 t = fabsd(x); |
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491 n = (int) (t*invpio2+half); |
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492 fn = (double)n; |
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493 r = t-fn*pio2_1; |
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494 w = fn*pio2_1t; /* 1st round good to 85 bit */ |
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495 if(n<32&&ix!=npio2_hw[n-1]) { |
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496 y[0] = r-w; /* quick check no cancellation */ |
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497 } else { |
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498 j = ix>>20; |
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499 y[0] = r-w; |
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500 i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff); |
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501 if(i>16) { /* 2nd iteration needed, good to 118 */ |
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502 t = r; |
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503 w = fn*pio2_2; |
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504 r = t-w; |
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505 w = fn*pio2_2t-((t-r)-w); |
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506 y[0] = r-w; |
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507 i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff); |
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508 if(i>49) { /* 3rd iteration need, 151 bits acc */ |
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509 t = r; /* will cover all possible cases */ |
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510 w = fn*pio2_3; |
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511 r = t-w; |
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512 w = fn*pio2_3t-((t-r)-w); |
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513 y[0] = r-w; |
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514 } |
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515 } |
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516 } |
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517 y[1] = (r-y[0])-w; |
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518 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} |
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519 else return n; |
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520 } |
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521 /* |
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522 * all other (large) arguments |
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523 */ |
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524 if(ix>=0x7ff00000) { /* x is inf or NaN */ |
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525 y[0]=y[1]=x-x; return 0; |
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526 } |
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527 /* set z = scalbn(|x|,ilogb(x)-23) */ |
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528 *(1-i0+(int*)&z) = *(1-i0+(int*)&x); |
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529 e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */ |
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530 *(i0+(int*)&z) = ix - (e0<<20); |
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531 for(i=0;i<2;i++) { |
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532 tx[i] = (double)((int)(z)); |
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533 z = (z-tx[i])*two24A; |
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534 } |
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535 tx[2] = z; |
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536 nx = 3; |
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537 while(tx[nx-1]==zeroA) nx--; /* skip zero term */ |
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538 n = __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi); |
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539 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;} |
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540 return n; |
|
541 } |
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542 |
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543 |
|
544 /* __kernel_sin( x, y, iy) |
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545 * kernel sin function on [-pi/4, pi/4], pi/4 ~ 0.7854 |
|
546 * Input x is assumed to be bounded by ~pi/4 in magnitude. |
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547 * Input y is the tail of x. |
|
548 * Input iy indicates whether y is 0. (if iy=0, y assume to be 0). |
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549 * |
|
550 * Algorithm |
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551 * 1. Since sin(-x) = -sin(x), we need only to consider positive x. |
|
552 * 2. if x < 2^-27 (hx<0x3e400000 0), return x with inexact if x!=0. |
|
553 * 3. sin(x) is approximated by a polynomial of degree 13 on |
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554 * [0,pi/4] |
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555 * 3 13 |
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556 * sin(x) ~ x + S1*x + ... + S6*x |
|
557 * where |
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558 * |
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559 * |sin(x) 2 4 6 8 10 12 | -58 |
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560 * |----- - (1+S1*x +S2*x +S3*x +S4*x +S5*x +S6*x )| <= 2 |
|
561 * | x | |
|
562 * |
|
563 * 4. sin(x+y) = sin(x) + sin'(x')*y |
|
564 * ~ sin(x) + (1-x*x/2)*y |
|
565 * For better accuracy, let |
|
566 * 3 2 2 2 2 |
|
567 * r = x *(S2+x *(S3+x *(S4+x *(S5+x *S6)))) |
|
568 * then 3 2 |
|
569 * sin(x) = x + (S1*x + (x *(r-y/2)+y)) |
|
570 */ |
|
571 |
|
572 static const double |
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573 S1 = -1.66666666666666324348e-01, /* 0xBFC55555, 0x55555549 */ |
|
574 S2 = 8.33333333332248946124e-03, /* 0x3F811111, 0x1110F8A6 */ |
|
575 S3 = -1.98412698298579493134e-04, /* 0xBF2A01A0, 0x19C161D5 */ |
|
576 S4 = 2.75573137070700676789e-06, /* 0x3EC71DE3, 0x57B1FE7D */ |
|
577 S5 = -2.50507602534068634195e-08, /* 0xBE5AE5E6, 0x8A2B9CEB */ |
|
578 S6 = 1.58969099521155010221e-10; /* 0x3DE5D93A, 0x5ACFD57C */ |
|
579 |
|
580 static double __kernel_sin(double x, double y, int iy) |
|
581 { |
|
582 double z,r,v; |
|
583 int ix; |
|
584 ix = __HI(x)&0x7fffffff; /* high word of x */ |
|
585 if(ix<0x3e400000) /* |x| < 2**-27 */ |
|
586 {if((int)x==0) return x;} /* generate inexact */ |
|
587 z = x*x; |
|
588 v = z*x; |
|
589 r = S2+z*(S3+z*(S4+z*(S5+z*S6))); |
|
590 if(iy==0) return x+v*(S1+z*r); |
|
591 else return x-((z*(half*y-v*r)-y)-v*S1); |
|
592 } |
|
593 |
|
594 /* |
|
595 * __kernel_cos( x, y ) |
|
596 * kernel cos function on [-pi/4, pi/4], pi/4 ~ 0.785398164 |
|
597 * Input x is assumed to be bounded by ~pi/4 in magnitude. |
|
598 * Input y is the tail of x. |
|
599 * |
|
600 * Algorithm |
|
601 * 1. Since cos(-x) = cos(x), we need only to consider positive x. |
|
602 * 2. if x < 2^-27 (hx<0x3e400000 0), return 1 with inexact if x!=0. |
|
603 * 3. cos(x) is approximated by a polynomial of degree 14 on |
|
604 * [0,pi/4] |
|
605 * 4 14 |
|
606 * cos(x) ~ 1 - x*x/2 + C1*x + ... + C6*x |
|
607 * where the remez error is |
|
608 * |
|
609 * | 2 4 6 8 10 12 14 | -58 |
|
610 * |cos(x)-(1-.5*x +C1*x +C2*x +C3*x +C4*x +C5*x +C6*x )| <= 2 |
|
611 * | | |
|
612 * |
|
613 * 4 6 8 10 12 14 |
|
614 * 4. let r = C1*x +C2*x +C3*x +C4*x +C5*x +C6*x , then |
|
615 * cos(x) = 1 - x*x/2 + r |
|
616 * since cos(x+y) ~ cos(x) - sin(x)*y |
|
617 * ~ cos(x) - x*y, |
|
618 * a correction term is necessary in cos(x) and hence |
|
619 * cos(x+y) = 1 - (x*x/2 - (r - x*y)) |
|
620 * For better accuracy when x > 0.3, let qx = |x|/4 with |
|
621 * the last 32 bits mask off, and if x > 0.78125, let qx = 0.28125. |
|
622 * Then |
|
623 * cos(x+y) = (1-qx) - ((x*x/2-qx) - (r-x*y)). |
|
624 * Note that 1-qx and (x*x/2-qx) is EXACT here, and the |
|
625 * magnitude of the latter is at least a quarter of x*x/2, |
|
626 * thus, reducing the rounding error in the subtraction. |
|
627 */ |
|
628 |
|
629 static const double |
|
630 C1 = 4.16666666666666019037e-02, /* 0x3FA55555, 0x5555554C */ |
|
631 C2 = -1.38888888888741095749e-03, /* 0xBF56C16C, 0x16C15177 */ |
|
632 C3 = 2.48015872894767294178e-05, /* 0x3EFA01A0, 0x19CB1590 */ |
|
633 C4 = -2.75573143513906633035e-07, /* 0xBE927E4F, 0x809C52AD */ |
|
634 C5 = 2.08757232129817482790e-09, /* 0x3E21EE9E, 0xBDB4B1C4 */ |
|
635 C6 = -1.13596475577881948265e-11; /* 0xBDA8FAE9, 0xBE8838D4 */ |
|
636 |
|
637 static double __kernel_cos(double x, double y) |
|
638 { |
|
639 double a,hz,z,r,qx; |
|
640 int ix; |
|
641 ix = __HI(x)&0x7fffffff; /* ix = |x|'s high word*/ |
|
642 if(ix<0x3e400000) { /* if x < 2**27 */ |
|
643 if(((int)x)==0) return one; /* generate inexact */ |
|
644 } |
|
645 z = x*x; |
|
646 r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*C6))))); |
|
647 if(ix < 0x3FD33333) /* if |x| < 0.3 */ |
|
648 return one - (0.5*z - (z*r - x*y)); |
|
649 else { |
|
650 if(ix > 0x3fe90000) { /* x > 0.78125 */ |
|
651 qx = 0.28125; |
|
652 } else { |
|
653 __HI(qx) = ix-0x00200000; /* x/4 */ |
|
654 __LO(qx) = 0; |
|
655 } |
|
656 hz = 0.5*z-qx; |
|
657 a = one-qx; |
|
658 return a - (hz - (z*r-x*y)); |
|
659 } |
|
660 } |
|
661 |
|
662 /* __kernel_tan( x, y, k ) |
|
663 * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854 |
|
664 * Input x is assumed to be bounded by ~pi/4 in magnitude. |
|
665 * Input y is the tail of x. |
|
666 * Input k indicates whether tan (if k=1) or |
|
667 * -1/tan (if k= -1) is returned. |
|
668 * |
|
669 * Algorithm |
|
670 * 1. Since tan(-x) = -tan(x), we need only to consider positive x. |
|
671 * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0. |
|
672 * 3. tan(x) is approximated by a odd polynomial of degree 27 on |
|
673 * [0,0.67434] |
|
674 * 3 27 |
|
675 * tan(x) ~ x + T1*x + ... + T13*x |
|
676 * where |
|
677 * |
|
678 * |tan(x) 2 4 26 | -59.2 |
|
679 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2 |
|
680 * | x | |
|
681 * |
|
682 * Note: tan(x+y) = tan(x) + tan'(x)*y |
|
683 * ~ tan(x) + (1+x*x)*y |
|
684 * Therefore, for better accuracy in computing tan(x+y), let |
|
685 * 3 2 2 2 2 |
|
686 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13)))) |
|
687 * then |
|
688 * 3 2 |
|
689 * tan(x+y) = x + (T1*x + (x *(r+y)+y)) |
|
690 * |
|
691 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then |
|
692 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y)) |
|
693 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y))) |
|
694 */ |
|
695 |
|
696 static const double |
|
697 pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */ |
|
698 pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */ |
|
699 T[] = { |
|
700 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */ |
|
701 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */ |
|
702 5.39682539762260521377e-02, /* 0x3FABA1BA, 0x1BB341FE */ |
|
703 2.18694882948595424599e-02, /* 0x3F9664F4, 0x8406D637 */ |
|
704 8.86323982359930005737e-03, /* 0x3F8226E3, 0xE96E8493 */ |
|
705 3.59207910759131235356e-03, /* 0x3F6D6D22, 0xC9560328 */ |
|
706 1.45620945432529025516e-03, /* 0x3F57DBC8, 0xFEE08315 */ |
|
707 5.88041240820264096874e-04, /* 0x3F4344D8, 0xF2F26501 */ |
|
708 2.46463134818469906812e-04, /* 0x3F3026F7, 0x1A8D1068 */ |
|
709 7.81794442939557092300e-05, /* 0x3F147E88, 0xA03792A6 */ |
|
710 7.14072491382608190305e-05, /* 0x3F12B80F, 0x32F0A7E9 */ |
|
711 -1.85586374855275456654e-05, /* 0xBEF375CB, 0xDB605373 */ |
|
712 2.59073051863633712884e-05, /* 0x3EFB2A70, 0x74BF7AD4 */ |
|
713 }; |
|
714 |
|
715 static double __kernel_tan(double x, double y, int iy) |
|
716 { |
|
717 double z,r,v,w,s; |
|
718 int ix,hx; |
|
719 hx = __HI(x); /* high word of x */ |
|
720 ix = hx&0x7fffffff; /* high word of |x| */ |
|
721 if(ix<0x3e300000) { /* x < 2**-28 */ |
|
722 if((int)x==0) { /* generate inexact */ |
|
723 if (((ix | __LO(x)) | (iy + 1)) == 0) |
|
724 return one / fabsd(x); |
|
725 else { |
|
726 if (iy == 1) |
|
727 return x; |
|
728 else { /* compute -1 / (x+y) carefully */ |
|
729 double a, t; |
|
730 |
|
731 z = w = x + y; |
|
732 __LO(z) = 0; |
|
733 v = y - (z - x); |
|
734 t = a = -one / w; |
|
735 __LO(t) = 0; |
|
736 s = one + t * z; |
|
737 return t + a * (s + t * v); |
|
738 } |
|
739 } |
|
740 } |
|
741 } |
|
742 if(ix>=0x3FE59428) { /* |x|>=0.6744 */ |
|
743 if(hx<0) {x = -x; y = -y;} |
|
744 z = pio4-x; |
|
745 w = pio4lo-y; |
|
746 x = z+w; y = 0.0; |
|
747 } |
|
748 z = x*x; |
|
749 w = z*z; |
|
750 /* Break x^5*(T[1]+x^2*T[2]+...) into |
|
751 * x^5(T[1]+x^4*T[3]+...+x^20*T[11]) + |
|
752 * x^5(x^2*(T[2]+x^4*T[4]+...+x^22*[T12])) |
|
753 */ |
|
754 r = T[1]+w*(T[3]+w*(T[5]+w*(T[7]+w*(T[9]+w*T[11])))); |
|
755 v = z*(T[2]+w*(T[4]+w*(T[6]+w*(T[8]+w*(T[10]+w*T[12]))))); |
|
756 s = z*x; |
|
757 r = y + z*(s*(r+v)+y); |
|
758 r += T[0]*s; |
|
759 w = x+r; |
|
760 if(ix>=0x3FE59428) { |
|
761 v = (double)iy; |
|
762 return (double)(1-((hx>>30)&2))*(v-2.0*(x-(w*w/(w+v)-r))); |
|
763 } |
|
764 if(iy==1) return w; |
|
765 else { /* if allow error up to 2 ulp, |
|
766 simply return -1.0/(x+r) here */ |
|
767 /* compute -1.0/(x+r) accurately */ |
|
768 double a,t; |
|
769 z = w; |
|
770 __LO(z) = 0; |
|
771 v = r-(z - x); /* z+v = r+x */ |
|
772 t = a = -1.0/w; /* a = -1.0/w */ |
|
773 __LO(t) = 0; |
|
774 s = 1.0+t*z; |
|
775 return t+a*(s+t*v); |
|
776 } |
|
777 } |
|
778 |
|
779 |
|
780 //---------------------------------------------------------------------- |
|
781 // |
|
782 // Routines for new sin/cos implementation |
|
783 // |
|
784 //---------------------------------------------------------------------- |
|
785 |
|
786 /* sin(x) |
|
787 * Return sine function of x. |
|
788 * |
|
789 * kernel function: |
|
790 * __kernel_sin ... sine function on [-pi/4,pi/4] |
|
791 * __kernel_cos ... cose function on [-pi/4,pi/4] |
|
792 * __ieee754_rem_pio2 ... argument reduction routine |
|
793 * |
|
794 * Method. |
|
795 * Let S,C and T denote the sin, cos and tan respectively on |
|
796 * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 |
|
797 * in [-pi/4 , +pi/4], and let n = k mod 4. |
|
798 * We have |
|
799 * |
|
800 * n sin(x) cos(x) tan(x) |
|
801 * ---------------------------------------------------------- |
|
802 * 0 S C T |
|
803 * 1 C -S -1/T |
|
804 * 2 -S -C T |
|
805 * 3 -C S -1/T |
|
806 * ---------------------------------------------------------- |
|
807 * |
|
808 * Special cases: |
|
809 * Let trig be any of sin, cos, or tan. |
|
810 * trig(+-INF) is NaN, with signals; |
|
811 * trig(NaN) is that NaN; |
|
812 * |
|
813 * Accuracy: |
|
814 * TRIG(x) returns trig(x) nearly rounded |
|
815 */ |
|
816 |
|
817 JRT_LEAF(jdouble, SharedRuntime::dsin(jdouble x)) |
|
818 double y[2],z=0.0; |
|
819 int n, ix; |
|
820 |
|
821 /* High word of x. */ |
|
822 ix = __HI(x); |
|
823 |
|
824 /* |x| ~< pi/4 */ |
|
825 ix &= 0x7fffffff; |
|
826 if(ix <= 0x3fe921fb) return __kernel_sin(x,z,0); |
|
827 |
|
828 /* sin(Inf or NaN) is NaN */ |
|
829 else if (ix>=0x7ff00000) return x-x; |
|
830 |
|
831 /* argument reduction needed */ |
|
832 else { |
|
833 n = __ieee754_rem_pio2(x,y); |
|
834 switch(n&3) { |
|
835 case 0: return __kernel_sin(y[0],y[1],1); |
|
836 case 1: return __kernel_cos(y[0],y[1]); |
|
837 case 2: return -__kernel_sin(y[0],y[1],1); |
|
838 default: |
|
839 return -__kernel_cos(y[0],y[1]); |
|
840 } |
|
841 } |
|
842 JRT_END |
|
843 |
|
844 /* cos(x) |
|
845 * Return cosine function of x. |
|
846 * |
|
847 * kernel function: |
|
848 * __kernel_sin ... sine function on [-pi/4,pi/4] |
|
849 * __kernel_cos ... cosine function on [-pi/4,pi/4] |
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850 * __ieee754_rem_pio2 ... argument reduction routine |
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851 * |
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852 * Method. |
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853 * Let S,C and T denote the sin, cos and tan respectively on |
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854 * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 |
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855 * in [-pi/4 , +pi/4], and let n = k mod 4. |
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856 * We have |
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857 * |
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858 * n sin(x) cos(x) tan(x) |
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859 * ---------------------------------------------------------- |
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860 * 0 S C T |
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861 * 1 C -S -1/T |
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862 * 2 -S -C T |
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863 * 3 -C S -1/T |
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864 * ---------------------------------------------------------- |
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865 * |
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866 * Special cases: |
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867 * Let trig be any of sin, cos, or tan. |
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868 * trig(+-INF) is NaN, with signals; |
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869 * trig(NaN) is that NaN; |
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870 * |
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871 * Accuracy: |
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872 * TRIG(x) returns trig(x) nearly rounded |
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873 */ |
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874 |
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875 JRT_LEAF(jdouble, SharedRuntime::dcos(jdouble x)) |
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876 double y[2],z=0.0; |
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877 int n, ix; |
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878 |
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879 /* High word of x. */ |
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880 ix = __HI(x); |
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881 |
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882 /* |x| ~< pi/4 */ |
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883 ix &= 0x7fffffff; |
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884 if(ix <= 0x3fe921fb) return __kernel_cos(x,z); |
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885 |
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886 /* cos(Inf or NaN) is NaN */ |
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887 else if (ix>=0x7ff00000) return x-x; |
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888 |
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889 /* argument reduction needed */ |
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890 else { |
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891 n = __ieee754_rem_pio2(x,y); |
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892 switch(n&3) { |
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893 case 0: return __kernel_cos(y[0],y[1]); |
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894 case 1: return -__kernel_sin(y[0],y[1],1); |
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895 case 2: return -__kernel_cos(y[0],y[1]); |
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896 default: |
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897 return __kernel_sin(y[0],y[1],1); |
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898 } |
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899 } |
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900 JRT_END |
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901 |
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902 /* tan(x) |
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903 * Return tangent function of x. |
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904 * |
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905 * kernel function: |
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906 * __kernel_tan ... tangent function on [-pi/4,pi/4] |
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907 * __ieee754_rem_pio2 ... argument reduction routine |
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908 * |
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909 * Method. |
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910 * Let S,C and T denote the sin, cos and tan respectively on |
|
911 * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2 |
|
912 * in [-pi/4 , +pi/4], and let n = k mod 4. |
|
913 * We have |
|
914 * |
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915 * n sin(x) cos(x) tan(x) |
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916 * ---------------------------------------------------------- |
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917 * 0 S C T |
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918 * 1 C -S -1/T |
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919 * 2 -S -C T |
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920 * 3 -C S -1/T |
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921 * ---------------------------------------------------------- |
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922 * |
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923 * Special cases: |
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924 * Let trig be any of sin, cos, or tan. |
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925 * trig(+-INF) is NaN, with signals; |
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926 * trig(NaN) is that NaN; |
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927 * |
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928 * Accuracy: |
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929 * TRIG(x) returns trig(x) nearly rounded |
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930 */ |
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931 |
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932 JRT_LEAF(jdouble, SharedRuntime::dtan(jdouble x)) |
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933 double y[2],z=0.0; |
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934 int n, ix; |
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935 |
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936 /* High word of x. */ |
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937 ix = __HI(x); |
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938 |
|
939 /* |x| ~< pi/4 */ |
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940 ix &= 0x7fffffff; |
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941 if(ix <= 0x3fe921fb) return __kernel_tan(x,z,1); |
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942 |
|
943 /* tan(Inf or NaN) is NaN */ |
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944 else if (ix>=0x7ff00000) return x-x; /* NaN */ |
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945 |
|
946 /* argument reduction needed */ |
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947 else { |
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948 n = __ieee754_rem_pio2(x,y); |
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949 return __kernel_tan(y[0],y[1],1-((n&1)<<1)); /* 1 -- n even |
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950 -1 -- n odd */ |
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951 } |
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952 JRT_END |
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953 |
|
954 |
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955 #ifdef WIN32 |
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956 # pragma optimize ( "", on ) |
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957 #endif |