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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;
|
|
|
363 for (i=1;i<=jz;i++) fw += fq[i];
|
|
|
364 y[1] = (ih==0)? fw: -fw;
|
|
|
365 break;
|
|
|
366 case 3: /* painful */
|
|
|
367 for (i=jz;i>0;i--) {
|
|
|
368 fw = fq[i-1]+fq[i];
|
|
|
369 fq[i] += fq[i-1]-fw;
|
|
|
370 fq[i-1] = fw;
|
|
|
371 }
|
|
|
372 for (i=jz;i>1;i--) {
|
|
|
373 fw = fq[i-1]+fq[i];
|
|
|
374 fq[i] += fq[i-1]-fw;
|
|
|
375 fq[i-1] = fw;
|
|
|
376 }
|
|
|
377 for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
|
|
|
378 if(ih==0) {
|
|
|
379 y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
|
|
|
380 } else {
|
|
|
381 y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
|
|
|
382 }
|
|
|
383 }
|
|
|
384 return n&7;
|
|
|
385 }
|
|
|
386
|
|
|
387
|
|
|
388 /*
|
|
|
389 * ====================================================
|
|
|
390 * Copyright 13 Dec 1993 Sun Microsystems, Inc. All Rights Reserved.
|
|
|
391 *
|
|
|
392 * Developed at SunPro, a Sun Microsystems, Inc. business.
|
|
|
393 * Permission to use, copy, modify, and distribute this
|
|
|
394 * software is freely granted, provided that this notice
|
|
|
395 * is preserved.
|
|
|
396 * ====================================================
|
|
|
397 *
|
|
|
398 */
|
|
|
399
|
|
|
400 /* __ieee754_rem_pio2(x,y)
|
|
|
401 *
|
|
|
402 * return the remainder of x rem pi/2 in y[0]+y[1]
|
|
|
403 * use __kernel_rem_pio2()
|
|
|
404 */
|
|
|
405
|
|
|
406 /*
|
|
|
407 * Table of constants for 2/pi, 396 Hex digits (476 decimal) of 2/pi
|
|
|
408 */
|
|
|
409 static const int two_over_pi[] = {
|
|
|
410 0xA2F983, 0x6E4E44, 0x1529FC, 0x2757D1, 0xF534DD, 0xC0DB62,
|
|
|
411 0x95993C, 0x439041, 0xFE5163, 0xABDEBB, 0xC561B7, 0x246E3A,
|
|
|
412 0x424DD2, 0xE00649, 0x2EEA09, 0xD1921C, 0xFE1DEB, 0x1CB129,
|
|
|
413 0xA73EE8, 0x8235F5, 0x2EBB44, 0x84E99C, 0x7026B4, 0x5F7E41,
|
|
|
414 0x3991D6, 0x398353, 0x39F49C, 0x845F8B, 0xBDF928, 0x3B1FF8,
|
|
|
415 0x97FFDE, 0x05980F, 0xEF2F11, 0x8B5A0A, 0x6D1F6D, 0x367ECF,
|
|
|
416 0x27CB09, 0xB74F46, 0x3F669E, 0x5FEA2D, 0x7527BA, 0xC7EBE5,
|
|
|
417 0xF17B3D, 0x0739F7, 0x8A5292, 0xEA6BFB, 0x5FB11F, 0x8D5D08,
|
|
|
418 0x560330, 0x46FC7B, 0x6BABF0, 0xCFBC20, 0x9AF436, 0x1DA9E3,
|
|
|
419 0x91615E, 0xE61B08, 0x659985, 0x5F14A0, 0x68408D, 0xFFD880,
|
|
|
420 0x4D7327, 0x310606, 0x1556CA, 0x73A8C9, 0x60E27B, 0xC08C6B,
|
|
|
421 };
|
|
|
422
|
|
|
423 static const int npio2_hw[] = {
|
|
|
424 0x3FF921FB, 0x400921FB, 0x4012D97C, 0x401921FB, 0x401F6A7A, 0x4022D97C,
|
|
|
425 0x4025FDBB, 0x402921FB, 0x402C463A, 0x402F6A7A, 0x4031475C, 0x4032D97C,
|
|
|
426 0x40346B9C, 0x4035FDBB, 0x40378FDB, 0x403921FB, 0x403AB41B, 0x403C463A,
|
|
|
427 0x403DD85A, 0x403F6A7A, 0x40407E4C, 0x4041475C, 0x4042106C, 0x4042D97C,
|
|
|
428 0x4043A28C, 0x40446B9C, 0x404534AC, 0x4045FDBB, 0x4046C6CB, 0x40478FDB,
|
|
|
429 0x404858EB, 0x404921FB,
|
|
|
430 };
|
|
|
431
|
|
|
432 /*
|
|
|
433 * invpio2: 53 bits of 2/pi
|
|
|
434 * pio2_1: first 33 bit of pi/2
|
|
|
435 * pio2_1t: pi/2 - pio2_1
|
|
|
436 * pio2_2: second 33 bit of pi/2
|
|
|
437 * pio2_2t: pi/2 - (pio2_1+pio2_2)
|
|
|
438 * pio2_3: third 33 bit of pi/2
|
|
|
439 * pio2_3t: pi/2 - (pio2_1+pio2_2+pio2_3)
|
|
|
440 */
|
|
|
441
|
|
|
442 static const double
|
|
|
443 zeroA = 0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
|
|
|
444 half = 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */
|
|
|
445 two24A = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
|
|
|
446 invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
|
|
|
447 pio2_1 = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
|
|
|
448 pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
|
|
|
449 pio2_2 = 6.07710050630396597660e-11, /* 0x3DD0B461, 0x1A600000 */
|
|
|
450 pio2_2t = 2.02226624879595063154e-21, /* 0x3BA3198A, 0x2E037073 */
|
|
|
451 pio2_3 = 2.02226624871116645580e-21, /* 0x3BA3198A, 0x2E000000 */
|
|
|
452 pio2_3t = 8.47842766036889956997e-32; /* 0x397B839A, 0x252049C1 */
|
|
|
453
|
|
|
454 static int __ieee754_rem_pio2(double x, double *y) {
|
|
|
455 double z,w,t,r,fn;
|
|
|
456 double tx[3];
|
|
|
457 int e0,i,j,nx,n,ix,hx,i0;
|
|
|
458
|
|
|
459 i0 = ((*(int*)&two24A)>>30)^1; /* high word index */
|
|
|
460 hx = *(i0+(int*)&x); /* high word of x */
|
|
|
461 ix = hx&0x7fffffff;
|
|
|
462 if(ix<=0x3fe921fb) /* |x| ~<= pi/4 , no need for reduction */
|
|
|
463 {y[0] = x; y[1] = 0; return 0;}
|
|
|
464 if(ix<0x4002d97c) { /* |x| < 3pi/4, special case with n=+-1 */
|
|
|
465 if(hx>0) {
|
|
|
466 z = x - pio2_1;
|
|
|
467 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
|
|
|
468 y[0] = z - pio2_1t;
|
|
|
469 y[1] = (z-y[0])-pio2_1t;
|
|
|
470 } else { /* near pi/2, use 33+33+53 bit pi */
|
|
|
471 z -= pio2_2;
|
|
|
472 y[0] = z - pio2_2t;
|
|
|
473 y[1] = (z-y[0])-pio2_2t;
|
|
|
474 }
|
|
|
475 return 1;
|
|
|
476 } else { /* negative x */
|
|
|
477 z = x + pio2_1;
|
|
|
478 if(ix!=0x3ff921fb) { /* 33+53 bit pi is good enough */
|
|
|
479 y[0] = z + pio2_1t;
|
|
|
480 y[1] = (z-y[0])+pio2_1t;
|
|
|
481 } else { /* near pi/2, use 33+33+53 bit pi */
|
|
|
482 z += pio2_2;
|
|
|
483 y[0] = z + pio2_2t;
|
|
|
484 y[1] = (z-y[0])+pio2_2t;
|
|
|
485 }
|
|
|
486 return -1;
|
|
|
487 }
|
|
|
488 }
|
|
|
489 if(ix<=0x413921fb) { /* |x| ~<= 2^19*(pi/2), medium size */
|
|
|
490 t = fabsd(x);
|
|
|
491 n = (int) (t*invpio2+half);
|
|
|
492 fn = (double)n;
|
|
|
493 r = t-fn*pio2_1;
|
|
|
494 w = fn*pio2_1t; /* 1st round good to 85 bit */
|
|
|
495 if(n<32&&ix!=npio2_hw[n-1]) {
|
|
|
496 y[0] = r-w; /* quick check no cancellation */
|
|
|
497 } else {
|
|
|
498 j = ix>>20;
|
|
|
499 y[0] = r-w;
|
|
|
500 i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff);
|
|
|
501 if(i>16) { /* 2nd iteration needed, good to 118 */
|
|
|
502 t = r;
|
|
|
503 w = fn*pio2_2;
|
|
|
504 r = t-w;
|
|
|
505 w = fn*pio2_2t-((t-r)-w);
|
|
|
506 y[0] = r-w;
|
|
|
507 i = j-(((*(i0+(int*)&y[0]))>>20)&0x7ff);
|
|
|
508 if(i>49) { /* 3rd iteration need, 151 bits acc */
|
|
|
509 t = r; /* will cover all possible cases */
|
|
|
510 w = fn*pio2_3;
|
|
|
511 r = t-w;
|
|
|
512 w = fn*pio2_3t-((t-r)-w);
|
|
|
513 y[0] = r-w;
|
|
|
514 }
|
|
|
515 }
|
|
|
516 }
|
|
|
517 y[1] = (r-y[0])-w;
|
|
|
518 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
|
|
|
519 else return n;
|
|
|
520 }
|
|
|
521 /*
|
|
|
522 * all other (large) arguments
|
|
|
523 */
|
|
|
524 if(ix>=0x7ff00000) { /* x is inf or NaN */
|
|
|
525 y[0]=y[1]=x-x; return 0;
|
|
|
526 }
|
|
|
527 /* set z = scalbn(|x|,ilogb(x)-23) */
|
|
|
528 *(1-i0+(int*)&z) = *(1-i0+(int*)&x);
|
|
|
529 e0 = (ix>>20)-1046; /* e0 = ilogb(z)-23; */
|
|
|
530 *(i0+(int*)&z) = ix - (e0<<20);
|
|
|
531 for(i=0;i<2;i++) {
|
|
|
532 tx[i] = (double)((int)(z));
|
|
|
533 z = (z-tx[i])*two24A;
|
|
|
534 }
|
|
|
535 tx[2] = z;
|
|
|
536 nx = 3;
|
|
|
537 while(tx[nx-1]==zeroA) nx--; /* skip zero term */
|
|
|
538 n = __kernel_rem_pio2(tx,y,e0,nx,2,two_over_pi);
|
|
|
539 if(hx<0) {y[0] = -y[0]; y[1] = -y[1]; return -n;}
|
|
|
540 return n;
|
|
|
541 }
|
|
|
542
|
|
|
543
|
|
|
544 /* __kernel_sin( x, y, iy)
|
|
|
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.
|
|
|
547 * Input y is the tail of x.
|
|
|
548 * Input iy indicates whether y is 0. (if iy=0, y assume to be 0).
|
|
|
549 *
|
|
|
550 * Algorithm
|
|
|
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
|
|
|
554 * [0,pi/4]
|
|
|
555 * 3 13
|
|
|
556 * sin(x) ~ x + S1*x + ... + S6*x
|
|
|
557 * where
|
|
|
558 *
|
|
|
559 * |sin(x) 2 4 6 8 10 12 | -58
|
|
|
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
|
|
|
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
|
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670 * 1. Since tan(-x) = -tan(x), we need only to consider positive x.
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671 * 2. if x < 2^-28 (hx<0x3e300000 0), return x with inexact if x!=0.
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|
672 * 3. tan(x) is approximated by a odd polynomial of degree 27 on
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|
673 * [0,0.67434]
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674 * 3 27
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675 * tan(x) ~ x + T1*x + ... + T13*x
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676 * where
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677 *
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678 * |tan(x) 2 4 26 | -59.2
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679 * |----- - (1+T1*x +T2*x +.... +T13*x )| <= 2
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680 * | x |
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681 *
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|
682 * Note: tan(x+y) = tan(x) + tan'(x)*y
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683 * ~ tan(x) + (1+x*x)*y
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|
684 * Therefore, for better accuracy in computing tan(x+y), let
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|
685 * 3 2 2 2 2
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686 * r = x *(T2+x *(T3+x *(...+x *(T12+x *T13))))
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|
687 * then
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688 * 3 2
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|
689 * tan(x+y) = x + (T1*x + (x *(r+y)+y))
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690 *
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691 * 4. For x in [0.67434,pi/4], let y = pi/4 - x, then
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|
692 * tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
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693 * = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
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694 */
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695
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696 static const double
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697 pio4 = 7.85398163397448278999e-01, /* 0x3FE921FB, 0x54442D18 */
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698 pio4lo= 3.06161699786838301793e-17, /* 0x3C81A626, 0x33145C07 */
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|
699 T[] = {
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|
700 3.33333333333334091986e-01, /* 0x3FD55555, 0x55555563 */
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|
|
701 1.33333333333201242699e-01, /* 0x3FC11111, 0x1110FE7A */
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|
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)
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|
716 {
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|
|
717 double z,r,v,w,s;
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|
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]
|
|
|
850 * __ieee754_rem_pio2 ... argument reduction routine
|
|
|
851 *
|
|
|
852 * Method.
|
|
|
853 * Let S,C and T denote the sin, cos and tan respectively on
|
|
|
854 * [-PI/4, +PI/4]. Reduce the argument x to y1+y2 = x-k*pi/2
|
|
|
855 * in [-pi/4 , +pi/4], and let n = k mod 4.
|
|
|
856 * We have
|
|
|
857 *
|
|
|
858 * n sin(x) cos(x) tan(x)
|
|
|
859 * ----------------------------------------------------------
|
|
|
860 * 0 S C T
|
|
|
861 * 1 C -S -1/T
|
|
|
862 * 2 -S -C T
|
|
|
863 * 3 -C S -1/T
|
|
|
864 * ----------------------------------------------------------
|
|
|
865 *
|
|
|
866 * Special cases:
|
|
|
867 * Let trig be any of sin, cos, or tan.
|
|
|
868 * trig(+-INF) is NaN, with signals;
|
|
|
869 * trig(NaN) is that NaN;
|
|
|
870 *
|
|
|
871 * Accuracy:
|
|
|
872 * TRIG(x) returns trig(x) nearly rounded
|
|
|
873 */
|
|
|
874
|
|
|
875 JRT_LEAF(jdouble, SharedRuntime::dcos(jdouble x))
|
|
|
876 double y[2],z=0.0;
|
|
|
877 int n, ix;
|
|
|
878
|
|
|
879 /* High word of x. */
|
|
|
880 ix = __HI(x);
|
|
|
881
|
|
|
882 /* |x| ~< pi/4 */
|
|
|
883 ix &= 0x7fffffff;
|
|
|
884 if(ix <= 0x3fe921fb) return __kernel_cos(x,z);
|
|
|
885
|
|
|
886 /* cos(Inf or NaN) is NaN */
|
|
|
887 else if (ix>=0x7ff00000) return x-x;
|
|
|
888
|
|
|
889 /* argument reduction needed */
|
|
|
890 else {
|
|
|
891 n = __ieee754_rem_pio2(x,y);
|
|
|
892 switch(n&3) {
|
|
|
893 case 0: return __kernel_cos(y[0],y[1]);
|
|
|
894 case 1: return -__kernel_sin(y[0],y[1],1);
|
|
|
895 case 2: return -__kernel_cos(y[0],y[1]);
|
|
|
896 default:
|
|
|
897 return __kernel_sin(y[0],y[1],1);
|
|
|
898 }
|
|
|
899 }
|
|
|
900 JRT_END
|
|
|
901
|
|
|
902 /* tan(x)
|
|
|
903 * Return tangent function of x.
|
|
|
904 *
|
|
|
905 * kernel function:
|
|
|
906 * __kernel_tan ... tangent function on [-pi/4,pi/4]
|
|
|
907 * __ieee754_rem_pio2 ... argument reduction routine
|
|
|
908 *
|
|
|
909 * Method.
|
|
|
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 *
|
|
|
915 * n sin(x) cos(x) tan(x)
|
|
|
916 * ----------------------------------------------------------
|
|
|
917 * 0 S C T
|
|
|
918 * 1 C -S -1/T
|
|
|
919 * 2 -S -C T
|
|
|
920 * 3 -C S -1/T
|
|
|
921 * ----------------------------------------------------------
|
|
|
922 *
|
|
|
923 * Special cases:
|
|
|
924 * Let trig be any of sin, cos, or tan.
|
|
|
925 * trig(+-INF) is NaN, with signals;
|
|
|
926 * trig(NaN) is that NaN;
|
|
|
927 *
|
|
|
928 * Accuracy:
|
|
|
929 * TRIG(x) returns trig(x) nearly rounded
|
|
|
930 */
|
|
|
931
|
|
|
932 JRT_LEAF(jdouble, SharedRuntime::dtan(jdouble x))
|
|
|
933 double y[2],z=0.0;
|
|
|
934 int n, ix;
|
|
|
935
|
|
|
936 /* High word of x. */
|
|
|
937 ix = __HI(x);
|
|
|
938
|
|
|
939 /* |x| ~< pi/4 */
|
|
|
940 ix &= 0x7fffffff;
|
|
|
941 if(ix <= 0x3fe921fb) return __kernel_tan(x,z,1);
|
|
|
942
|
|
|
943 /* tan(Inf or NaN) is NaN */
|
|
|
944 else if (ix>=0x7ff00000) return x-x; /* NaN */
|
|
|
945
|
|
|
946 /* argument reduction needed */
|
|
|
947 else {
|
|
|
948 n = __ieee754_rem_pio2(x,y);
|
|
|
949 return __kernel_tan(y[0],y[1],1-((n&1)<<1)); /* 1 -- n even
|
|
|
950 -1 -- n odd */
|
|
|
951 }
|
|
|
952 JRT_END
|
|
|
953
|
|
|
954
|
|
|
955 #ifdef WIN32
|
|
|
956 # pragma optimize ( "", on )
|
|
|
957 #endif
|
