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2757
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1 /* @(#)k_rem_pio2.c 5.1 93/09/24 */
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2 /*
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3 * ====================================================
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4 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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5 *
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6 * Developed at SunPro, a Sun Microsystems, Inc. business.
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7 * Permission to use, copy, modify, and distribute this
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8 * software is freely granted, provided that this notice
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9 * is preserved.
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10 * ====================================================
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11 */
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12
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13 #if defined(LIBM_SCCS) && !defined(lint)
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3162
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14 static const char rcsid[] =
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2757
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15 "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $";
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16 #endif
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17
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18 /*
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19 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
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20 * double x[],y[]; int e0,nx,prec; int ipio2[];
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21 *
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22 * __kernel_rem_pio2 return the last three digits of N with
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23 * y = x - N*pi/2
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24 * so that |y| < pi/2.
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25 *
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26 * The method is to compute the integer (mod 8) and fraction parts of
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27 * (2/pi)*x without doing the full multiplication. In general we
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28 * skip the part of the product that are known to be a huge integer (
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29 * more accurately, = 0 mod 8 ). Thus the number of operations are
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30 * independent of the exponent of the input.
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31 *
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32 * (2/pi) is represented by an array of 24-bit integers in ipio2[].
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33 *
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34 * Input parameters:
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35 * x[] The input value (must be positive) is broken into nx
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36 * pieces of 24-bit integers in double precision format.
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37 * x[i] will be the i-th 24 bit of x. The scaled exponent
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38 * of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
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39 * match x's up to 24 bits.
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40 *
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41 * Example of breaking a double positive z into x[0]+x[1]+x[2]:
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42 * e0 = ilogb(z)-23
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43 * z = scalbn(z,-e0)
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44 * for i = 0,1,2
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45 * x[i] = floor(z)
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46 * z = (z-x[i])*2**24
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47 *
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48 *
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49 * y[] ouput result in an array of double precision numbers.
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50 * The dimension of y[] is:
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51 * 24-bit precision 1
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52 * 53-bit precision 2
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53 * 64-bit precision 2
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54 * 113-bit precision 3
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55 * The actual value is the sum of them. Thus for 113-bit
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56 * precison, one may have to do something like:
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57 *
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58 * long double t,w,r_head, r_tail;
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59 * t = (long double)y[2] + (long double)y[1];
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60 * w = (long double)y[0];
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61 * r_head = t+w;
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62 * r_tail = w - (r_head - t);
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63 *
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64 * e0 The exponent of x[0]
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65 *
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66 * nx dimension of x[]
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67 *
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68 * prec an integer indicating the precision:
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69 * 0 24 bits (single)
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70 * 1 53 bits (double)
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71 * 2 64 bits (extended)
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72 * 3 113 bits (quad)
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73 *
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74 * ipio2[]
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75 * integer array, contains the (24*i)-th to (24*i+23)-th
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76 * bit of 2/pi after binary point. The corresponding
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77 * floating value is
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78 *
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79 * ipio2[i] * 2^(-24(i+1)).
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80 *
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81 * External function:
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82 * double scalbn(), floor();
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83 *
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84 *
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85 * Here is the description of some local variables:
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86 *
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87 * jk jk+1 is the initial number of terms of ipio2[] needed
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88 * in the computation. The recommended value is 2,3,4,
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89 * 6 for single, double, extended,and quad.
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90 *
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91 * jz local integer variable indicating the number of
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92 * terms of ipio2[] used.
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93 *
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94 * jx nx - 1
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95 *
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96 * jv index for pointing to the suitable ipio2[] for the
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97 * computation. In general, we want
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98 * ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
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99 * is an integer. Thus
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100 * e0-3-24*jv >= 0 or (e0-3)/24 >= jv
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101 * Hence jv = max(0,(e0-3)/24).
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102 *
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103 * jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
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104 *
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105 * q[] double array with integral value, representing the
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106 * 24-bits chunk of the product of x and 2/pi.
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107 *
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108 * q0 the corresponding exponent of q[0]. Note that the
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109 * exponent for q[i] would be q0-24*i.
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110 *
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111 * PIo2[] double precision array, obtained by cutting pi/2
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112 * into 24 bits chunks.
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113 *
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114 * f[] ipio2[] in floating point
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115 *
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116 * iq[] integer array by breaking up q[] in 24-bits chunk.
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117 *
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118 * fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
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119 *
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120 * ih integer. If >0 it indicates q[] is >= 0.5, hence
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121 * it also indicates the *sign* of the result.
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122 *
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123 */
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124
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125
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126 /*
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127 * Constants:
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128 * The hexadecimal values are the intended ones for the following
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129 * constants. The decimal values may be used, provided that the
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130 * compiler will convert from decimal to binary accurately enough
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131 * to produce the hexadecimal values shown.
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132 */
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133
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134 #include "math.h"
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135 #include "math_private.h"
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136
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137 libm_hidden_proto(scalbn)
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138 libm_hidden_proto(floor)
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139 #ifdef __STDC__
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140 static const int init_jk[] = { 2, 3, 4, 6 }; /* initial value for jk */
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141 #else
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142 static int init_jk[] = { 2, 3, 4, 6 };
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143 #endif
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144
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145 #ifdef __STDC__
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146 static const double PIo2[] = {
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147 #else
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148 static double PIo2[] = {
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149 #endif
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150 1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
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151 7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
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152 5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
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153 3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
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154 1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
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155 1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
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156 2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
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157 2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
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158 };
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159
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160 #ifdef __STDC__
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161 static const double
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162 #else
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163 static double
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164 #endif
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165 zero = 0.0, one = 1.0, two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
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166 twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
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167
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168 #ifdef __STDC__
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169 int attribute_hidden
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170 __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec,
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171 const int32_t * ipio2)
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172 #else
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173 int attribute_hidden
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174 __kernel_rem_pio2(x, y, e0, nx, prec, ipio2)
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175 double x[], y[];
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176 int e0, nx, prec;
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177 int32_t ipio2[];
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178 #endif
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179 {
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180 int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
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181 double z, fw, f[20], fq[20], q[20];
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182
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183 /* initialize jk */
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184 jk = init_jk[prec];
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185 jp = jk;
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186
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187 /* determine jx,jv,q0, note that 3>q0 */
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188 jx = nx - 1;
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189 jv = (e0 - 3) / 24;
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190 if (jv < 0)
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191 jv = 0;
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192 q0 = e0 - 24 * (jv + 1);
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193
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194 /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
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195 j = jv - jx;
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196 m = jx + jk;
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197 for (i = 0; i <= m; i++, j++)
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198 f[i] = (j < 0) ? zero : (double) ipio2[j];
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199
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200 /* compute q[0],q[1],...q[jk] */
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201 for (i = 0; i <= jk; i++) {
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202 for (j = 0, fw = 0.0; j <= jx; j++)
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203 fw += x[j] * f[jx + i - j];
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204 q[i] = fw;
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205 }
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206
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207 jz = jk;
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208 recompute:
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209 /* distill q[] into iq[] reversingly */
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210 for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) {
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211 fw = (double) ((int32_t) (twon24 * z));
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212 iq[i] = (int32_t) (z - two24 * fw);
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213 z = q[j - 1] + fw;
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214 }
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215
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216 /* compute n */
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217 z = scalbn(z, q0); /* actual value of z */
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218 z -= 8.0 * floor(z * 0.125); /* trim off integer >= 8 */
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219 n = (int32_t) z;
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220 z -= (double) n;
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221 ih = 0;
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222 if (q0 > 0) { /* need iq[jz-1] to determine n */
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223 i = (iq[jz - 1] >> (24 - q0));
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224 n += i;
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225 iq[jz - 1] -= i << (24 - q0);
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226 ih = iq[jz - 1] >> (23 - q0);
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227 } else if (q0 == 0)
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228 ih = iq[jz - 1] >> 23;
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229 else if (z >= 0.5)
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230 ih = 2;
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231
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232 if (ih > 0) { /* q > 0.5 */
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233 n += 1;
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234 carry = 0;
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235 for (i = 0; i < jz; i++) { /* compute 1-q */
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236 j = iq[i];
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237 if (carry == 0) {
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238 if (j != 0) {
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239 carry = 1;
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240 iq[i] = 0x1000000 - j;
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241 }
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242 } else
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243 iq[i] = 0xffffff - j;
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244 }
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245 if (q0 > 0) { /* rare case: chance is 1 in 12 */
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246 switch (q0) {
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247 case 1:
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248 iq[jz - 1] &= 0x7fffff;
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249 break;
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250 case 2:
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251 iq[jz - 1] &= 0x3fffff;
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252 break;
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253 }
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254 }
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255 if (ih == 2) {
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256 z = one - z;
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257 if (carry != 0)
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258 z -= scalbn(one, q0);
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259 }
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260 }
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261
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262 /* check if recomputation is needed */
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263 if (z == zero) {
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264 j = 0;
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265 for (i = jz - 1; i >= jk; i--)
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266 j |= iq[i];
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267 if (j == 0) { /* need recomputation */
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268 for (k = 1; iq[jk - k] == 0; k++); /* k = no. of terms needed */
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269
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270 for (i = jz + 1; i <= jz + k; i++) { /* add q[jz+1] to q[jz+k] */
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271 f[jx + i] = (double) ipio2[jv + i];
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272 for (j = 0, fw = 0.0; j <= jx; j++)
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273 fw += x[j] * f[jx + i - j];
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274 q[i] = fw;
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275 }
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276 jz += k;
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277 goto recompute;
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278 }
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279 }
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280
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281 /* chop off zero terms */
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282 if (z == 0.0) {
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283 jz -= 1;
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284 q0 -= 24;
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285 while (iq[jz] == 0) {
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286 jz--;
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287 q0 -= 24;
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288 }
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289 } else { /* break z into 24-bit if necessary */
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290 z = scalbn(z, -q0);
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291 if (z >= two24) {
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292 fw = (double) ((int32_t) (twon24 * z));
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293 iq[jz] = (int32_t) (z - two24 * fw);
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294 jz += 1;
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295 q0 += 24;
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296 iq[jz] = (int32_t) fw;
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297 } else
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298 iq[jz] = (int32_t) z;
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299 }
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300
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301 /* convert integer "bit" chunk to floating-point value */
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302 fw = scalbn(one, q0);
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303 for (i = jz; i >= 0; i--) {
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304 q[i] = fw * (double) iq[i];
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305 fw *= twon24;
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306 }
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307
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308 /* compute PIo2[0,...,jp]*q[jz,...,0] */
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309 for (i = jz; i >= 0; i--) {
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310 for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++)
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311 fw += PIo2[k] * q[i + k];
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312 fq[jz - i] = fw;
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313 }
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314
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315 /* compress fq[] into y[] */
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316 switch (prec) {
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317 case 0:
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318 fw = 0.0;
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319 for (i = jz; i >= 0; i--)
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320 fw += fq[i];
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321 y[0] = (ih == 0) ? fw : -fw;
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322 break;
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323 case 1:
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324 case 2:
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325 fw = 0.0;
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326 for (i = jz; i >= 0; i--)
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327 fw += fq[i];
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328 y[0] = (ih == 0) ? fw : -fw;
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329 fw = fq[0] - fw;
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330 for (i = 1; i <= jz; i++)
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331 fw += fq[i];
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332 y[1] = (ih == 0) ? fw : -fw;
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333 break;
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334 case 3: /* painful */
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335 for (i = jz; i > 0; i--) {
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336 fw = fq[i - 1] + fq[i];
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337 fq[i] += fq[i - 1] - fw;
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338 fq[i - 1] = fw;
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339 }
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340 for (i = jz; i > 1; i--) {
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341 fw = fq[i - 1] + fq[i];
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342 fq[i] += fq[i - 1] - fw;
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343 fq[i - 1] = fw;
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344 }
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345 for (fw = 0.0, i = jz; i >= 2; i--)
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346 fw += fq[i];
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347 if (ih == 0) {
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348 y[0] = fq[0];
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349 y[1] = fq[1];
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350 y[2] = fw;
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351 } else {
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352 y[0] = -fq[0];
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353 y[1] = -fq[1];
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354 y[2] = -fw;
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355 }
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356 }
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357 return n & 7;
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358 }
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