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111
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1 -- CXG2016.A
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2 --
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3 -- Grant of Unlimited Rights
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4 --
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5 -- Under contracts F33600-87-D-0337, F33600-84-D-0280, MDA903-79-C-0687,
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6 -- F08630-91-C-0015, and DCA100-97-D-0025, the U.S. Government obtained
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7 -- unlimited rights in the software and documentation contained herein.
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8 -- Unlimited rights are defined in DFAR 252.227-7013(a)(19). By making
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9 -- this public release, the Government intends to confer upon all
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10 -- recipients unlimited rights equal to those held by the Government.
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11 -- These rights include rights to use, duplicate, release or disclose the
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12 -- released technical data and computer software in whole or in part, in
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13 -- any manner and for any purpose whatsoever, and to have or permit others
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14 -- to do so.
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15 --
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16 -- DISCLAIMER
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17 --
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18 -- ALL MATERIALS OR INFORMATION HEREIN RELEASED, MADE AVAILABLE OR
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19 -- DISCLOSED ARE AS IS. THE GOVERNMENT MAKES NO EXPRESS OR IMPLIED
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20 -- WARRANTY AS TO ANY MATTER WHATSOEVER, INCLUDING THE CONDITIONS OF THE
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21 -- SOFTWARE, DOCUMENTATION OR OTHER INFORMATION RELEASED, MADE AVAILABLE
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22 -- OR DISCLOSED, OR THE OWNERSHIP, MERCHANTABILITY, OR FITNESS FOR A
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23 -- PARTICULAR PURPOSE OF SAID MATERIAL.
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24 --*
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25 --
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26 -- OBJECTIVE:
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27 -- Check that the ARCTAN function returns a
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28 -- result that is within the error bound allowed.
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29 --
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30 -- TEST DESCRIPTION:
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31 -- This test consists of a generic package that is
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32 -- instantiated to check both Float and a long float type.
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33 -- The test for each floating point type is divided into
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34 -- several parts:
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35 -- Special value checks where the result is a known constant.
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36 -- Exception checks.
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37 --
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38 -- SPECIAL REQUIREMENTS
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39 -- The Strict Mode for the numerical accuracy must be
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40 -- selected. The method by which this mode is selected
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41 -- is implementation dependent.
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42 --
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43 -- APPLICABILITY CRITERIA:
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44 -- This test applies only to implementations supporting the
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45 -- Numerics Annex.
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46 -- This test only applies to the Strict Mode for numerical
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47 -- accuracy.
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48 --
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49 --
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50 -- CHANGE HISTORY:
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51 -- 19 Mar 96 SAIC Initial release for 2.1
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52 -- 30 APR 96 SAIC Fixed optimization issue
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53 -- 17 AUG 96 SAIC Incorporated Reviewer's suggestions.
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54 -- 12 OCT 96 SAIC Incorporated Reviewer's suggestions.
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55 -- 02 DEC 97 EDS Remove procedure Identity_1_Test and calls to
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56 -- procedure.
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57 -- 29 JUN 98 EDS Replace -0.0 with call to ImpDef.Annex_G.Negative_Zero
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58 -- 28 APR 99 RLB Replaced comma accidentally deleted in above change.
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59 -- 15 DEC 99 RLB Added model range checking to "exact" results,
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60 -- in order to avoid too strictly requiring a specific
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61 -- result.
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62 --!
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63
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64 --
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65 -- References:
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66 --
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67 -- Software Manual for the Elementary Functions
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68 -- William J. Cody, Jr. and William Waite
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69 -- Prentice-Hall, 1980
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70 --
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71 -- CRC Standard Mathematical Tables
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72 -- 23rd Edition
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73 --
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74 -- Implementation and Testing of Function Software
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75 -- W. J. Cody
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76 -- Problems and Methodologies in Mathematical Software Production
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77 -- editors P. C. Messina and A. Murli
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78 -- Lecture Notes in Computer Science Volume 142
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79 -- Springer Verlag, 1982
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80 --
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81
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82 with System;
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83 with Report;
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84 with Ada.Numerics.Generic_Elementary_Functions;
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85 with Impdef.Annex_G;
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86 procedure CXG2016 is
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87 Verbose : constant Boolean := False;
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88 Max_Samples : constant := 1000;
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89
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90 -- CRC Standard Mathematical Tables; 23rd Edition; pg 738
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91 Sqrt2 : constant :=
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92 1.41421_35623_73095_04880_16887_24209_69807_85696_71875_37695;
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93 Sqrt3 : constant :=
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94 1.73205_08075_68877_29352_74463_41505_87236_69428_05253_81039;
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95
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96 Pi : constant := Ada.Numerics.Pi;
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97
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98 generic
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99 type Real is digits <>;
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100 Half_PI_Low : in Real; -- The machine number closest to, but not greater
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101 -- than PI/2.0.
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102 Half_PI_High : in Real;-- The machine number closest to, but not less
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103 -- than PI/2.0.
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104 PI_Low : in Real; -- The machine number closest to, but not greater
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105 -- than PI.
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106 PI_High : in Real; -- The machine number closest to, but not less
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107 -- than PI.
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108 package Generic_Check is
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109 procedure Do_Test;
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110 end Generic_Check;
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111
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112 package body Generic_Check is
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113 package Elementary_Functions is new
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114 Ada.Numerics.Generic_Elementary_Functions (Real);
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115
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116 function Arctan (Y : Real;
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117 X : Real := 1.0) return Real renames
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118 Elementary_Functions.Arctan;
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119 function Arctan (Y : Real;
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120 X : Real := 1.0;
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121 Cycle : Real) return Real renames
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122 Elementary_Functions.Arctan;
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123
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124 -- flag used to terminate some tests early
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125 Accuracy_Error_Reported : Boolean := False;
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126
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127 -- The following value is a lower bound on the accuracy
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128 -- required. It is normally 0.0 so that the lower bound
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129 -- is computed from Model_Epsilon. However, for tests
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130 -- where the expected result is only known to a certain
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131 -- amount of precision this bound takes on a non-zero
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132 -- value to account for that level of precision.
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133 Error_Low_Bound : Real := 0.0;
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134
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135 procedure Check (Actual, Expected : Real;
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136 Test_Name : String;
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137 MRE : Real) is
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138 Max_Error : Real;
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139 Rel_Error : Real;
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140 Abs_Error : Real;
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141 begin
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142 -- In the case where the expected result is very small or 0
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143 -- we compute the maximum error as a multiple of Model_Epsilon
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144 -- instead of Model_Epsilon and Expected.
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145 Rel_Error := MRE * abs Expected * Real'Model_Epsilon;
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146 Abs_Error := MRE * Real'Model_Epsilon;
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147 if Rel_Error > Abs_Error then
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148 Max_Error := Rel_Error;
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149 else
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150 Max_Error := Abs_Error;
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151 end if;
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152
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153 -- take into account the low bound on the error
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154 if Max_Error < Error_Low_Bound then
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155 Max_Error := Error_Low_Bound;
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156 end if;
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157
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158 if abs (Actual - Expected) > Max_Error then
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159 Accuracy_Error_Reported := True;
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160 Report.Failed (Test_Name &
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161 " actual: " & Real'Image (Actual) &
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162 " expected: " & Real'Image (Expected) &
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163 " difference: " & Real'Image (Actual - Expected) &
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164 " max err:" & Real'Image (Max_Error) );
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165 elsif Verbose then
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166 if Actual = Expected then
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167 Report.Comment (Test_Name & " exact result");
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168 else
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169 Report.Comment (Test_Name & " passed");
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170 end if;
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171 end if;
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172 end Check;
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173
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174
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175 procedure Special_Value_Test is
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176 -- If eta is very small, arctan(x + eta) ~= arctan(x) + eta/(1+x*x).
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177 --
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178 -- For tests 4 and 5, there is an error of 4.0ME for arctan + an
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179 -- additional error of 1.0ME because pi is not exact for a total of 5.0ME.
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180 --
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181 -- In test 3 there is the error for pi plus an additional error
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182 -- of (1.0ME)/4 since sqrt3 is not exact, for a total of 5.25ME.
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183 --
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184 -- In test 2 there is the error for pi plus an additional error
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185 -- of (3/4)(1.0ME) since sqrt3 is not exact, for a total of 5.75ME.
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186
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187
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188 type Data_Point is
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189 record
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190 Degrees,
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191 Radians,
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192 Tangent,
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193 Allowed_Error : Real;
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194 end record;
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195
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196 type Test_Data_Type is array (Positive range <>) of Data_Point;
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197
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198 -- the values in the following table only involve static
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199 -- expressions so no additional loss of precision occurs.
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200 Test_Data : constant Test_Data_Type := (
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201 -- degrees radians tangent error test #
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202 ( 0.0, 0.0, 0.0, 4.0 ), -- 1
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203 ( 30.0, Pi/6.0, Sqrt3/3.0, 5.75), -- 2
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204 ( 60.0, Pi/3.0, Sqrt3, 5.25), -- 3
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205 ( 45.0, Pi/4.0, 1.0, 5.0 ), -- 4
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206 (-45.0, -Pi/4.0, -1.0, 5.0 ) ); -- 5
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207
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208 begin
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209 for I in Test_Data'Range loop
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210 Check (Arctan (Test_Data (I).Tangent),
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211 Test_Data (I).Radians,
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212 "special value test" & Integer'Image (I) &
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213 " arctan(" &
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214 Real'Image (Test_Data (I).Tangent) &
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215 ")",
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216 Test_Data (I).Allowed_Error);
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217 Check (Arctan (Test_Data (I).Tangent, Cycle => 360.0),
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218 Test_Data (I).Degrees,
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219 "special value test" & Integer'Image (I) &
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220 " arctan(" &
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221 Real'Image (Test_Data (I).Tangent) &
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222 ", cycle=>360)",
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223 Test_Data (I).Allowed_Error);
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224 end loop;
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225
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226 exception
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227 when Constraint_Error =>
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228 Report.Failed ("Constraint_Error raised in special value test");
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229 when others =>
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230 Report.Failed ("exception in special value test");
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231 end Special_Value_Test;
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232
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233
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234
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235 procedure Check_Exact (Actual, Expected_Low, Expected_High : Real;
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236 Test_Name : String) is
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237 -- If the expected result is not a model number, then Expected_Low is
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238 -- the first machine number less than the (exact) expected
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239 -- result, and Expected_High is the first machine number greater than
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240 -- the (exact) expected result. If the expected result is a model
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241 -- number, Expected_Low = Expected_High = the result.
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242 Model_Expected_Low : Real := Expected_Low;
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243 Model_Expected_High : Real := Expected_High;
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244 begin
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245 -- Calculate the first model number nearest to, but below (or equal)
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246 -- to the expected result:
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247 while Real'Model (Model_Expected_Low) /= Model_Expected_Low loop
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248 -- Try the next machine number lower:
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249 Model_Expected_Low := Real'Adjacent(Model_Expected_Low, 0.0);
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250 end loop;
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251 -- Calculate the first model number nearest to, but above (or equal)
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252 -- to the expected result:
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253 while Real'Model (Model_Expected_High) /= Model_Expected_High loop
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254 -- Try the next machine number higher:
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255 Model_Expected_High := Real'Adjacent(Model_Expected_High, 100.0);
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256 end loop;
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257
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258 if Actual < Model_Expected_Low or Actual > Model_Expected_High then
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259 Accuracy_Error_Reported := True;
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260 if Actual < Model_Expected_Low then
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261 Report.Failed (Test_Name &
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262 " actual: " & Real'Image (Actual) &
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263 " expected low: " & Real'Image (Model_Expected_Low) &
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264 " expected high: " & Real'Image (Model_Expected_High) &
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265 " difference: " & Real'Image (Actual - Expected_Low));
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266 else
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267 Report.Failed (Test_Name &
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268 " actual: " & Real'Image (Actual) &
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269 " expected low: " & Real'Image (Model_Expected_Low) &
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270 " expected high: " & Real'Image (Model_Expected_High) &
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271 " difference: " & Real'Image (Expected_High - Actual));
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272 end if;
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273 elsif Verbose then
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274 Report.Comment (Test_Name & " passed");
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275 end if;
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276 end Check_Exact;
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277
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278
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279 procedure Exact_Result_Test is
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280 begin
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281 -- A.5.1(40);6.0
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282 Check_Exact (Arctan (0.0, 1.0), 0.0, 0.0, "arctan(0,1)");
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283 Check_Exact (Arctan (0.0, 1.0, 27.0), 0.0, 0.0, "arctan(0,1,27)");
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284
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285 -- G.2.4(11-13);6.0
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286
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287 Check_Exact (Arctan (1.0, 0.0), Half_PI_Low, Half_PI_High,
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288 "arctan(1,0)");
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289 Check_Exact (Arctan (1.0, 0.0, 360.0), 90.0, 90.0, "arctan(1,0,360)");
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290
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291 Check_Exact (Arctan (-1.0, 0.0), -Half_PI_High, -Half_PI_Low,
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292 "arctan(-1,0)");
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293 Check_Exact (Arctan (-1.0, 0.0, 360.0), -90.0, -90.0,
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294 "arctan(-1,0,360)");
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295
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296 if Real'Signed_Zeros then
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297 Check_Exact (Arctan (0.0, -1.0), PI_Low, PI_High, "arctan(+0,-1)");
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298 Check_Exact (Arctan (0.0, -1.0, 360.0), 180.0, 180.0,
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299 "arctan(+0,-1,360)");
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300 Check_Exact (Arctan ( Real ( ImpDef.Annex_G.Negative_Zero ), -1.0),
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301 -PI_High, -PI_Low, "arctan(-0,-1)");
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302 Check_Exact (Arctan ( Real ( ImpDef.Annex_G.Negative_Zero ), -1.0,
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303 360.0), -180.0, -180.0, "arctan(-0,-1,360)");
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304 else
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305 Check_Exact (Arctan (0.0, -1.0), PI_Low, PI_High, "arctan(0,-1)");
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306 Check_Exact (Arctan (0.0, -1.0, 360.0), 180.0, 180.0,
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307 "arctan(0,-1,360)");
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308 end if;
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309 exception
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310 when Constraint_Error =>
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311 Report.Failed ("Constraint_Error raised in Exact_Result Test");
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312 when others =>
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313 Report.Failed ("Exception in Exact_Result Test");
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314 end Exact_Result_Test;
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315
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316
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317 procedure Taylor_Series_Test is
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318 -- This test checks the Arctan by using a taylor series expansion that
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319 -- will produce a result accurate to 19 decimal digits for
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320 -- the range under test.
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321 --
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322 -- The maximum relative error bound for this test is
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323 -- 4 for the arctan operation and 2 for the Taylor series
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324 -- for a total of 6 * Model_Epsilon
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325
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326 A : constant := -1.0/16.0;
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327 B : constant := 1.0/16.0;
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328 X : Real;
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329 Actual, Expected : Real;
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330 Sum, Em, X_Squared : Real;
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331 begin
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332 if Real'Digits > 19 then
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333 -- Taylor series calculation produces result accurate to 19
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334 -- digits. If type being tested has more digits then set
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335 -- the error low bound to account for this.
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336 -- The error low bound is conservatively set to 6*10**-19
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337 Error_Low_Bound := 0.00000_00000_00000_0006;
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338 Report.Comment ("arctan accuracy checked to 19 digits");
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339 end if;
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340
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341 Accuracy_Error_Reported := False; -- reset
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342 for I in 0..Max_Samples loop
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343 X := (B - A) * Real (I) / Real (Max_Samples) + A;
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344 X_Squared := X * X;
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345 Em := 17.0;
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346 Sum := X_Squared / Em;
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347
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348 for II in 1 .. 7 loop
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349 Em := Em - 2.0;
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350 Sum := (1.0 / Em - Sum) * X_Squared;
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351 end loop;
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352 Sum := -X * Sum;
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353 Expected := X + Sum;
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354 Sum := (X - Expected) + Sum;
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355 if not Real'Machine_Rounds then
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356 Expected := Expected + (Sum + Sum);
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357 end if;
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358
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359 Actual := Arctan (X);
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360
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361 Check (Actual, Expected,
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362 "Taylor_Series_Test " & Integer'Image (I) & ": arctan(" &
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363 Real'Image (X) & ") ",
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364 6.0);
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365
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366 if Accuracy_Error_Reported then
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367 -- only report the first error in this test in order to keep
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368 -- lots of failures from producing a huge error log
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369 return;
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370 end if;
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371
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372 end loop;
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373 Error_Low_Bound := 0.0; -- reset
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374 exception
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375 when Constraint_Error =>
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376 Report.Failed
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377 ("Constraint_Error raised in Taylor_Series_Test");
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378 when others =>
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379 Report.Failed ("exception in Taylor_Series_Test");
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380 end Taylor_Series_Test;
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381
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382
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383 procedure Exception_Test is
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384 X1, X2, X3 : Real := 0.0;
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385 begin
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386
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387 begin -- A.5.1(20);6.0
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388 X1 := Arctan(0.0, Cycle => 0.0);
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389 Report.Failed ("no exception for cycle = 0.0");
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390 exception
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391 when Ada.Numerics.Argument_Error => null;
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392 when others =>
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393 Report.Failed ("wrong exception for cycle = 0.0");
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394 end;
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395
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396 begin -- A.5.1(20);6.0
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397 X2 := Arctan (0.0, Cycle => -1.0);
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398 Report.Failed ("no exception for cycle < 0.0");
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399 exception
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400 when Ada.Numerics.Argument_Error => null;
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401 when others =>
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402 Report.Failed ("wrong exception for cycle < 0.0");
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403 end;
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404
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405 begin -- A.5.1(25);6.0
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406 X3 := Arctan (0.0, 0.0);
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407 Report.Failed ("no exception for arctan(0,0)");
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408 exception
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409 when Ada.Numerics.Argument_Error => null;
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410 when others =>
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411 Report.Failed ("wrong exception for arctan(0,0)");
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412 end;
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413
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414 -- optimizer thwarting
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415 if Report.Ident_Bool (False) then
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416 Report.Comment (Real'Image (X1 + X2 + X3));
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417 end if;
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418 end Exception_Test;
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419
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420
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421 procedure Do_Test is
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422 begin
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423 Special_Value_Test;
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424 Exact_Result_Test;
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425 Taylor_Series_Test;
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426 Exception_Test;
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427 end Do_Test;
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428 end Generic_Check;
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429
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430 -----------------------------------------------------------------------
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431 -----------------------------------------------------------------------
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432 -- These expressions must be truly static, which is why we have to do them
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433 -- outside of the generic, and we use the named numbers. Note that we know
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434 -- that PI is not a machine number (it is irrational), and it should be
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435 -- represented to more digits than supported by the target machine.
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436 Float_Half_PI_Low : constant := Float'Adjacent(PI/2.0, 0.0);
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437 Float_Half_PI_High : constant := Float'Adjacent(PI/2.0, 10.0);
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438 Float_PI_Low : constant := Float'Adjacent(PI, 0.0);
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439 Float_PI_High : constant := Float'Adjacent(PI, 10.0);
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440 package Float_Check is new Generic_Check (Float,
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441 Half_PI_Low => Float_Half_PI_Low,
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442 Half_PI_High => Float_Half_PI_High,
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443 PI_Low => Float_PI_Low,
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444 PI_High => Float_PI_High);
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445
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446 -- check the Floating point type with the most digits
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447 type A_Long_Float is digits System.Max_Digits;
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448 A_Long_Float_Half_PI_Low : constant := A_Long_Float'Adjacent(PI/2.0, 0.0);
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449 A_Long_Float_Half_PI_High : constant := A_Long_Float'Adjacent(PI/2.0, 10.0);
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450 A_Long_Float_PI_Low : constant := A_Long_Float'Adjacent(PI, 0.0);
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451 A_Long_Float_PI_High : constant := A_Long_Float'Adjacent(PI, 10.0);
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452 package A_Long_Float_Check is new Generic_Check (A_Long_Float,
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453 Half_PI_Low => A_Long_Float_Half_PI_Low,
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454 Half_PI_High => A_Long_Float_Half_PI_High,
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455 PI_Low => A_Long_Float_PI_Low,
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456 PI_High => A_Long_Float_PI_High);
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457
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458 -----------------------------------------------------------------------
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459 -----------------------------------------------------------------------
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460
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461
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462 begin
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463 Report.Test ("CXG2016",
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464 "Check the accuracy of the ARCTAN function");
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465
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466 if Verbose then
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467 Report.Comment ("checking Standard.Float");
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468 end if;
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469
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470 Float_Check.Do_Test;
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471
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472 if Verbose then
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473 Report.Comment ("checking a digits" &
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474 Integer'Image (System.Max_Digits) &
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475 " floating point type");
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476 end if;
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477
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478 A_Long_Float_Check.Do_Test;
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479
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480
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481 Report.Result;
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482 end CXG2016;
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