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1 /* S_EXPM1.C |
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2 * |
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3 * Portions Copyright (c) 1993-2005 Nokia Corporation and/or its subsidiary(-ies). |
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4 * All rights reserved. |
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5 */ |
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6 |
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7 |
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8 /* @(#)s_expm1.c 5.1 93/09/24 */ |
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9 /* |
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10 * ==================================================== |
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11 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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12 * |
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13 * Developed at SunPro, a Sun Microsystems, Inc. business. |
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14 * Permission to use, copy, modify, and distribute this |
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15 * software is freely granted, provided that this notice |
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16 * is preserved. |
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17 * ==================================================== |
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18 */ |
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19 |
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20 /* |
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21 FUNCTION |
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22 <<expm1>>, <<expm1f>>---exponential minus 1 |
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23 INDEX |
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24 expm1 |
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25 INDEX |
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26 expm1f |
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27 |
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28 ANSI_SYNOPSIS |
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29 #include <math.h> |
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30 double expm1(double <[x]>); |
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31 float expm1f(float <[x]>); |
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32 |
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33 TRAD_SYNOPSIS |
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34 #include <math.h> |
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35 double expm1(<[x]>); |
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36 double <[x]>; |
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37 |
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38 float expm1f(<[x]>); |
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39 float <[x]>; |
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40 |
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41 DESCRIPTION |
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42 <<expm1>> and <<expm1f>> calculate the exponential of <[x]> |
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43 and subtract 1, that is, |
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44 @ifinfo |
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45 e raised to the power <[x]> minus 1 (where e |
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46 @end ifinfo |
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47 @tex |
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48 $e^x - 1$ (where $e$ |
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49 @end tex |
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50 is the base of the natural system of logarithms, approximately |
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51 2.71828). The result is accurate even for small values of |
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52 <[x]>, where using <<exp(<[x]>)-1>> would lose many |
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53 significant digits. |
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54 |
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55 RETURNS |
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56 e raised to the power <[x]>, minus 1. |
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57 |
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58 PORTABILITY |
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59 Neither <<expm1>> nor <<expm1f>> is required by ANSI C or by |
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60 the System V Interface Definition (Issue 2). |
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61 */ |
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62 |
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63 /* expm1(x) |
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64 * Returns exp(x)-1, the exponential of x minus 1. |
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65 * |
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66 * Method |
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67 * 1. Argument reduction: |
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68 * Given x, find r and integer k such that |
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69 * |
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70 * x = k*ln2 + r, |r| <= 0.5*ln2 ~ 0.34658 |
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71 * |
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72 * Here a correction term c will be computed to compensate |
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73 * the error in r when rounded to a floating-point number. |
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74 * |
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75 * 2. Approximating expm1(r) by a special rational function on |
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76 * the interval [0,0.34658]: |
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77 * Since |
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78 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 - r^4/360 + ... |
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79 * we define R1(r*r) by |
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80 * r*(exp(r)+1)/(exp(r)-1) = 2+ r^2/6 * R1(r*r) |
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81 * That is, |
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82 * R1(r**2) = 6/r *((exp(r)+1)/(exp(r)-1) - 2/r) |
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83 * = 6/r * ( 1 + 2.0*(1/(exp(r)-1) - 1/r)) |
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84 * = 1 - r^2/60 + r^4/2520 - r^6/100800 + ... |
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85 * We use a special Reme algorithm on [0,0.347] to generate |
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86 * a polynomial of degree 5 in r*r to approximate R1. The |
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87 * maximum error of this polynomial approximation is bounded |
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88 * by 2**-61. In other words, |
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89 * R1(z) ~ 1.0 + Q1*z + Q2*z**2 + Q3*z**3 + Q4*z**4 + Q5*z**5 |
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90 * where Q1 = -1.6666666666666567384E-2, |
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91 * Q2 = 3.9682539681370365873E-4, |
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92 * Q3 = -9.9206344733435987357E-6, |
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93 * Q4 = 2.5051361420808517002E-7, |
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94 * Q5 = -6.2843505682382617102E-9; |
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95 * (where z=r*r, and the values of Q1 to Q5 are listed below) |
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96 * with error bounded by |
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97 * | 5 | -61 |
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98 * | 1.0+Q1*z+...+Q5*z - R1(z) | <= 2 |
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99 * | | |
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100 * |
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101 * expm1(r) = exp(r)-1 is then computed by the following |
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102 * specific way which minimize the accumulation rounding error: |
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103 * 2 3 |
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104 * r r [ 3 - (R1 + R1*r/2) ] |
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105 * expm1(r) = r + --- + --- * [--------------------] |
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106 * 2 2 [ 6 - r*(3 - R1*r/2) ] |
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107 * |
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108 * To compensate the error in the argument reduction, we use |
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109 * expm1(r+c) = expm1(r) + c + expm1(r)*c |
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110 * ~ expm1(r) + c + r*c |
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111 * Thus c+r*c will be added in as the correction terms for |
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112 * expm1(r+c). Now rearrange the term to avoid optimization |
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113 * screw up: |
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114 * ( 2 2 ) |
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115 * ({ ( r [ R1 - (3 - R1*r/2) ] ) } r ) |
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116 * expm1(r+c)~r - ({r*(--- * [--------------------]-c)-c} - --- ) |
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117 * ({ ( 2 [ 6 - r*(3 - R1*r/2) ] ) } 2 ) |
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118 * ( ) |
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119 * |
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120 * = r - E |
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121 * 3. Scale back to obtain expm1(x): |
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122 * From step 1, we have |
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123 * expm1(x) = either 2^k*[expm1(r)+1] - 1 |
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124 * = or 2^k*[expm1(r) + (1-2^-k)] |
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125 * 4. Implementation notes: |
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126 * (A). To save one multiplication, we scale the coefficient Qi |
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127 * to Qi*2^i, and replace z by (x^2)/2. |
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128 * (B). To achieve maximum accuracy, we compute expm1(x) by |
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129 * (i) if x < -56*ln2, return -1.0, (raise inexact if x!=inf) |
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130 * (ii) if k=0, return r-E |
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131 * (iii) if k=-1, return 0.5*(r-E)-0.5 |
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132 * (iv) if k=1 if r < -0.25, return 2*((r+0.5)- E) |
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133 * else return 1.0+2.0*(r-E); |
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134 * (v) if (k<-2||k>56) return 2^k(1-(E-r)) - 1 (or exp(x)-1) |
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135 * (vi) if k <= 20, return 2^k((1-2^-k)-(E-r)), else |
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136 * (vii) return 2^k(1-((E+2^-k)-r)) |
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137 * |
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138 * Special cases: |
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139 * expm1(INF) is INF, expm1(NaN) is NaN; |
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140 * expm1(-INF) is -1, and |
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141 * for finite argument, only expm1(0)=0 is exact. |
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142 * |
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143 * Accuracy: |
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144 * according to an error analysis, the error is always less than |
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145 * 1 ulp (unit in the last place). |
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146 * |
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147 * Misc. info. |
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148 * For IEEE double |
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149 * if x > 7.09782712893383973096e+02 then expm1(x) overflow |
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150 * |
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151 * Constants: |
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152 * The hexadecimal values are the intended ones for the following |
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153 * constants. The decimal values may be used, provided that the |
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154 * compiler will convert from decimal to binary accurately enough |
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155 * to produce the hexadecimal values shown. |
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156 */ |
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157 |
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158 #include "FDLIBM.H" |
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159 |
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160 static const double |
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161 one = 1.0, |
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162 huge = 1.0e+300, |
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163 tiny = 1.0e-300, |
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164 o_threshold = 7.09782712893383973096e+02,/* 0x40862E42, 0xFEFA39EF */ |
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165 ln2_hi = 6.93147180369123816490e-01,/* 0x3fe62e42, 0xfee00000 */ |
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166 ln2_lo = 1.90821492927058770002e-10,/* 0x3dea39ef, 0x35793c76 */ |
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167 invln2 = 1.44269504088896338700e+00,/* 0x3ff71547, 0x652b82fe */ |
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168 /* scaled coefficients related to expm1 */ |
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169 Q1 = -3.33333333333331316428e-02, /* BFA11111 111110F4 */ |
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170 Q2 = 1.58730158725481460165e-03, /* 3F5A01A0 19FE5585 */ |
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171 Q3 = -7.93650757867487942473e-05, /* BF14CE19 9EAADBB7 */ |
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172 Q4 = 4.00821782732936239552e-06, /* 3ED0CFCA 86E65239 */ |
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173 Q5 = -2.01099218183624371326e-07; /* BE8AFDB7 6E09C32D */ |
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174 |
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175 /** |
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176 Calculate the exponential of x and subtract 1 |
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177 that is raised to the power x minus 1 |
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178 @return e raised to the power x, minus 1. |
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179 @param x e's power. |
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180 */ |
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181 EXPORT_C double expm1(double x) __SOFTFP |
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182 { |
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183 double y,hi,lo,c = 0.0,t,e,hxs,hfx,r1; |
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184 __int32_t k,xsb; |
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185 __uint32_t hx; |
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186 |
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187 GET_HIGH_WORD(hx,x); |
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188 xsb = hx&0x80000000; /* sign bit of x */ |
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189 if(xsb==0) y=x; else y= -x; /* y = |x| */ |
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190 hx &= 0x7fffffff; /* high word of |x| */ |
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191 |
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192 /* filter out huge and non-finite argument */ |
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193 if(hx >= 0x4043687A) { /* if |x|>=56*ln2 */ |
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194 if(hx >= 0x40862E42) { /* if |x|>=709.78... */ |
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195 if(hx>=0x7ff00000) { |
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196 __uint32_t low; |
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197 GET_LOW_WORD(low,x); |
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198 if(((hx&0xfffff)|low)!=0) |
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199 return x+x; /* NaN */ |
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200 else return (xsb==0)? x:-1.0;/* exp(+-inf)={inf,-1} */ |
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201 } |
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202 if(x > o_threshold) return huge*huge; /* overflow */ |
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203 } |
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204 if(xsb!=0) { /* x < -56*ln2, return -1.0 with inexact */ |
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205 if(x+tiny<0.0) /* raise inexact */ |
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206 return tiny-one; /* return -1 */ |
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207 } |
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208 } |
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209 |
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210 /* argument reduction */ |
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211 if(hx > 0x3fd62e42) { /* if |x| > 0.5 ln2 */ |
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212 if(hx < 0x3FF0A2B2) { /* and |x| < 1.5 ln2 */ |
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213 if(xsb==0) |
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214 {hi = x - ln2_hi; lo = ln2_lo; k = 1;} |
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215 else |
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216 {hi = x + ln2_hi; lo = -ln2_lo; k = -1;} |
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217 } else { |
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218 k = invln2*x+((xsb==0)?0.5:-0.5); |
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219 t = k; |
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220 hi = x - t*ln2_hi; /* t*ln2_hi is exact here */ |
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221 lo = t*ln2_lo; |
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222 } |
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223 x = hi - lo; |
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224 c = (hi-x)-lo; |
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225 } |
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226 else if(hx < 0x3c900000) { /* when |x|<2**-54, return x */ |
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227 t = huge+x; /* return x with inexact flags when x!=0 */ |
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228 return x - (t-(huge+x)); |
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229 } |
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230 else k = 0; |
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231 |
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232 /* x is now in primary range */ |
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233 hfx = 0.5*x; |
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234 hxs = x*hfx; |
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235 r1 = one+hxs*(Q1+hxs*(Q2+hxs*(Q3+hxs*(Q4+hxs*Q5)))); |
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236 t = 3.0-r1*hfx; |
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237 e = hxs*((r1-t)/(6.0 - x*t)); |
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238 if(k==0) return x - (x*e-hxs); /* c is 0 */ |
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239 else { |
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240 e = (x*(e-c)-c); |
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241 e -= hxs; |
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242 if(k== -1) return 0.5*(x-e)-0.5; |
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243 if(k==1) { |
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244 if(x < -0.25) return -2.0*(e-(x+0.5)); |
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245 else return one+2.0*(x-e); |
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246 } |
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247 if (k <= -2 || k>56) { /* suffice to return exp(x)-1 */ |
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248 __uint32_t high; |
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249 y = one-(e-x); |
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250 GET_HIGH_WORD(high,y); |
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251 SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ |
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252 return y-one; |
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253 } |
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254 t = one; |
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255 if(k<20) { |
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256 __uint32_t high; |
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257 SET_HIGH_WORD(t,0x3ff00000 - (0x200000>>k)); /* t=1-2^-k */ |
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258 y = t-(e-x); |
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259 GET_HIGH_WORD(high,y); |
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260 SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ |
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261 } else { |
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262 __uint32_t high; |
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263 SET_HIGH_WORD(t,((0x3ff-k)<<20)); /* 2^-k */ |
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264 y = x-(e+t); |
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265 y += one; |
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266 GET_HIGH_WORD(high,y); |
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267 SET_HIGH_WORD(y,high+(k<<20)); /* add k to y's exponent */ |
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268 } |
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269 } |
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270 return y; |
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271 } |