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