| -- CXG2016.A |
| -- |
| -- Grant of Unlimited Rights |
| -- |
| -- Under contracts F33600-87-D-0337, F33600-84-D-0280, MDA903-79-C-0687, |
| -- F08630-91-C-0015, and DCA100-97-D-0025, the U.S. Government obtained |
| -- unlimited rights in the software and documentation contained herein. |
| -- Unlimited rights are defined in DFAR 252.227-7013(a)(19). By making |
| -- this public release, the Government intends to confer upon all |
| -- recipients unlimited rights equal to those held by the Government. |
| -- These rights include rights to use, duplicate, release or disclose the |
| -- released technical data and computer software in whole or in part, in |
| -- any manner and for any purpose whatsoever, and to have or permit others |
| -- to do so. |
| -- |
| -- DISCLAIMER |
| -- |
| -- ALL MATERIALS OR INFORMATION HEREIN RELEASED, MADE AVAILABLE OR |
| -- DISCLOSED ARE AS IS. THE GOVERNMENT MAKES NO EXPRESS OR IMPLIED |
| -- WARRANTY AS TO ANY MATTER WHATSOEVER, INCLUDING THE CONDITIONS OF THE |
| -- SOFTWARE, DOCUMENTATION OR OTHER INFORMATION RELEASED, MADE AVAILABLE |
| -- OR DISCLOSED, OR THE OWNERSHIP, MERCHANTABILITY, OR FITNESS FOR A |
| -- PARTICULAR PURPOSE OF SAID MATERIAL. |
| --* |
| -- |
| -- OBJECTIVE: |
| -- Check that the ARCTAN function returns a |
| -- result that is within the error bound allowed. |
| -- |
| -- TEST DESCRIPTION: |
| -- This test consists of a generic package that is |
| -- instantiated to check both Float and a long float type. |
| -- The test for each floating point type is divided into |
| -- several parts: |
| -- Special value checks where the result is a known constant. |
| -- Exception checks. |
| -- |
| -- SPECIAL REQUIREMENTS |
| -- The Strict Mode for the numerical accuracy must be |
| -- selected. The method by which this mode is selected |
| -- is implementation dependent. |
| -- |
| -- APPLICABILITY CRITERIA: |
| -- This test applies only to implementations supporting the |
| -- Numerics Annex. |
| -- This test only applies to the Strict Mode for numerical |
| -- accuracy. |
| -- |
| -- |
| -- CHANGE HISTORY: |
| -- 19 Mar 96 SAIC Initial release for 2.1 |
| -- 30 APR 96 SAIC Fixed optimization issue |
| -- 17 AUG 96 SAIC Incorporated Reviewer's suggestions. |
| -- 12 OCT 96 SAIC Incorporated Reviewer's suggestions. |
| -- 02 DEC 97 EDS Remove procedure Identity_1_Test and calls to |
| -- procedure. |
| -- 29 JUN 98 EDS Replace -0.0 with call to ImpDef.Annex_G.Negative_Zero |
| -- 28 APR 99 RLB Replaced comma accidentally deleted in above change. |
| -- 15 DEC 99 RLB Added model range checking to "exact" results, |
| -- in order to avoid too strictly requiring a specific |
| -- result. |
| --! |
| |
| -- |
| -- References: |
| -- |
| -- Software Manual for the Elementary Functions |
| -- William J. Cody, Jr. and William Waite |
| -- Prentice-Hall, 1980 |
| -- |
| -- CRC Standard Mathematical Tables |
| -- 23rd Edition |
| -- |
| -- Implementation and Testing of Function Software |
| -- W. J. Cody |
| -- Problems and Methodologies in Mathematical Software Production |
| -- editors P. C. Messina and A. Murli |
| -- Lecture Notes in Computer Science Volume 142 |
| -- Springer Verlag, 1982 |
| -- |
| |
| with System; |
| with Report; |
| with Ada.Numerics.Generic_Elementary_Functions; |
| with Impdef.Annex_G; |
| procedure CXG2016 is |
| Verbose : constant Boolean := False; |
| Max_Samples : constant := 1000; |
| |
| -- CRC Standard Mathematical Tables; 23rd Edition; pg 738 |
| Sqrt2 : constant := |
| 1.41421_35623_73095_04880_16887_24209_69807_85696_71875_37695; |
| Sqrt3 : constant := |
| 1.73205_08075_68877_29352_74463_41505_87236_69428_05253_81039; |
| |
| Pi : constant := Ada.Numerics.Pi; |
| |
| generic |
| type Real is digits <>; |
| Half_PI_Low : in Real; -- The machine number closest to, but not greater |
| -- than PI/2.0. |
| Half_PI_High : in Real;-- The machine number closest to, but not less |
| -- than PI/2.0. |
| PI_Low : in Real; -- The machine number closest to, but not greater |
| -- than PI. |
| PI_High : in Real; -- The machine number closest to, but not less |
| -- than PI. |
| package Generic_Check is |
| procedure Do_Test; |
| end Generic_Check; |
| |
| package body Generic_Check is |
| package Elementary_Functions is new |
| Ada.Numerics.Generic_Elementary_Functions (Real); |
| |
| function Arctan (Y : Real; |
| X : Real := 1.0) return Real renames |
| Elementary_Functions.Arctan; |
| function Arctan (Y : Real; |
| X : Real := 1.0; |
| Cycle : Real) return Real renames |
| Elementary_Functions.Arctan; |
| |
| -- flag used to terminate some tests early |
| Accuracy_Error_Reported : Boolean := False; |
| |
| -- The following value is a lower bound on the accuracy |
| -- required. It is normally 0.0 so that the lower bound |
| -- is computed from Model_Epsilon. However, for tests |
| -- where the expected result is only known to a certain |
| -- amount of precision this bound takes on a non-zero |
| -- value to account for that level of precision. |
| Error_Low_Bound : Real := 0.0; |
| |
| procedure Check (Actual, Expected : Real; |
| Test_Name : String; |
| MRE : Real) is |
| Max_Error : Real; |
| Rel_Error : Real; |
| Abs_Error : Real; |
| begin |
| -- In the case where the expected result is very small or 0 |
| -- we compute the maximum error as a multiple of Model_Epsilon |
| -- instead of Model_Epsilon and Expected. |
| Rel_Error := MRE * abs Expected * Real'Model_Epsilon; |
| Abs_Error := MRE * Real'Model_Epsilon; |
| if Rel_Error > Abs_Error then |
| Max_Error := Rel_Error; |
| else |
| Max_Error := Abs_Error; |
| end if; |
| |
| -- take into account the low bound on the error |
| if Max_Error < Error_Low_Bound then |
| Max_Error := Error_Low_Bound; |
| end if; |
| |
| if abs (Actual - Expected) > Max_Error then |
| Accuracy_Error_Reported := True; |
| Report.Failed (Test_Name & |
| " actual: " & Real'Image (Actual) & |
| " expected: " & Real'Image (Expected) & |
| " difference: " & Real'Image (Actual - Expected) & |
| " max err:" & Real'Image (Max_Error) ); |
| elsif Verbose then |
| if Actual = Expected then |
| Report.Comment (Test_Name & " exact result"); |
| else |
| Report.Comment (Test_Name & " passed"); |
| end if; |
| end if; |
| end Check; |
| |
| |
| procedure Special_Value_Test is |
| -- If eta is very small, arctan(x + eta) ~= arctan(x) + eta/(1+x*x). |
| -- |
| -- For tests 4 and 5, there is an error of 4.0ME for arctan + an |
| -- additional error of 1.0ME because pi is not exact for a total of 5.0ME. |
| -- |
| -- In test 3 there is the error for pi plus an additional error |
| -- of (1.0ME)/4 since sqrt3 is not exact, for a total of 5.25ME. |
| -- |
| -- In test 2 there is the error for pi plus an additional error |
| -- of (3/4)(1.0ME) since sqrt3 is not exact, for a total of 5.75ME. |
| |
| |
| type Data_Point is |
| record |
| Degrees, |
| Radians, |
| Tangent, |
| Allowed_Error : Real; |
| end record; |
| |
| type Test_Data_Type is array (Positive range <>) of Data_Point; |
| |
| -- the values in the following table only involve static |
| -- expressions so no additional loss of precision occurs. |
| Test_Data : constant Test_Data_Type := ( |
| -- degrees radians tangent error test # |
| ( 0.0, 0.0, 0.0, 4.0 ), -- 1 |
| ( 30.0, Pi/6.0, Sqrt3/3.0, 5.75), -- 2 |
| ( 60.0, Pi/3.0, Sqrt3, 5.25), -- 3 |
| ( 45.0, Pi/4.0, 1.0, 5.0 ), -- 4 |
| (-45.0, -Pi/4.0, -1.0, 5.0 ) ); -- 5 |
| |
| begin |
| for I in Test_Data'Range loop |
| Check (Arctan (Test_Data (I).Tangent), |
| Test_Data (I).Radians, |
| "special value test" & Integer'Image (I) & |
| " arctan(" & |
| Real'Image (Test_Data (I).Tangent) & |
| ")", |
| Test_Data (I).Allowed_Error); |
| Check (Arctan (Test_Data (I).Tangent, Cycle => 360.0), |
| Test_Data (I).Degrees, |
| "special value test" & Integer'Image (I) & |
| " arctan(" & |
| Real'Image (Test_Data (I).Tangent) & |
| ", cycle=>360)", |
| Test_Data (I).Allowed_Error); |
| end loop; |
| |
| exception |
| when Constraint_Error => |
| Report.Failed ("Constraint_Error raised in special value test"); |
| when others => |
| Report.Failed ("exception in special value test"); |
| end Special_Value_Test; |
| |
| |
| |
| procedure Check_Exact (Actual, Expected_Low, Expected_High : Real; |
| Test_Name : String) is |
| -- If the expected result is not a model number, then Expected_Low is |
| -- the first machine number less than the (exact) expected |
| -- result, and Expected_High is the first machine number greater than |
| -- the (exact) expected result. If the expected result is a model |
| -- number, Expected_Low = Expected_High = the result. |
| Model_Expected_Low : Real := Expected_Low; |
| Model_Expected_High : Real := Expected_High; |
| begin |
| -- Calculate the first model number nearest to, but below (or equal) |
| -- to the expected result: |
| while Real'Model (Model_Expected_Low) /= Model_Expected_Low loop |
| -- Try the next machine number lower: |
| Model_Expected_Low := Real'Adjacent(Model_Expected_Low, 0.0); |
| end loop; |
| -- Calculate the first model number nearest to, but above (or equal) |
| -- to the expected result: |
| while Real'Model (Model_Expected_High) /= Model_Expected_High loop |
| -- Try the next machine number higher: |
| Model_Expected_High := Real'Adjacent(Model_Expected_High, 100.0); |
| end loop; |
| |
| if Actual < Model_Expected_Low or Actual > Model_Expected_High then |
| Accuracy_Error_Reported := True; |
| if Actual < Model_Expected_Low then |
| Report.Failed (Test_Name & |
| " actual: " & Real'Image (Actual) & |
| " expected low: " & Real'Image (Model_Expected_Low) & |
| " expected high: " & Real'Image (Model_Expected_High) & |
| " difference: " & Real'Image (Actual - Expected_Low)); |
| else |
| Report.Failed (Test_Name & |
| " actual: " & Real'Image (Actual) & |
| " expected low: " & Real'Image (Model_Expected_Low) & |
| " expected high: " & Real'Image (Model_Expected_High) & |
| " difference: " & Real'Image (Expected_High - Actual)); |
| end if; |
| elsif Verbose then |
| Report.Comment (Test_Name & " passed"); |
| end if; |
| end Check_Exact; |
| |
| |
| procedure Exact_Result_Test is |
| begin |
| -- A.5.1(40);6.0 |
| Check_Exact (Arctan (0.0, 1.0), 0.0, 0.0, "arctan(0,1)"); |
| Check_Exact (Arctan (0.0, 1.0, 27.0), 0.0, 0.0, "arctan(0,1,27)"); |
| |
| -- G.2.4(11-13);6.0 |
| |
| Check_Exact (Arctan (1.0, 0.0), Half_PI_Low, Half_PI_High, |
| "arctan(1,0)"); |
| Check_Exact (Arctan (1.0, 0.0, 360.0), 90.0, 90.0, "arctan(1,0,360)"); |
| |
| Check_Exact (Arctan (-1.0, 0.0), -Half_PI_High, -Half_PI_Low, |
| "arctan(-1,0)"); |
| Check_Exact (Arctan (-1.0, 0.0, 360.0), -90.0, -90.0, |
| "arctan(-1,0,360)"); |
| |
| if Real'Signed_Zeros then |
| Check_Exact (Arctan (0.0, -1.0), PI_Low, PI_High, "arctan(+0,-1)"); |
| Check_Exact (Arctan (0.0, -1.0, 360.0), 180.0, 180.0, |
| "arctan(+0,-1,360)"); |
| Check_Exact (Arctan ( Real ( ImpDef.Annex_G.Negative_Zero ), -1.0), |
| -PI_High, -PI_Low, "arctan(-0,-1)"); |
| Check_Exact (Arctan ( Real ( ImpDef.Annex_G.Negative_Zero ), -1.0, |
| 360.0), -180.0, -180.0, "arctan(-0,-1,360)"); |
| else |
| Check_Exact (Arctan (0.0, -1.0), PI_Low, PI_High, "arctan(0,-1)"); |
| Check_Exact (Arctan (0.0, -1.0, 360.0), 180.0, 180.0, |
| "arctan(0,-1,360)"); |
| end if; |
| exception |
| when Constraint_Error => |
| Report.Failed ("Constraint_Error raised in Exact_Result Test"); |
| when others => |
| Report.Failed ("Exception in Exact_Result Test"); |
| end Exact_Result_Test; |
| |
| |
| procedure Taylor_Series_Test is |
| -- This test checks the Arctan by using a taylor series expansion that |
| -- will produce a result accurate to 19 decimal digits for |
| -- the range under test. |
| -- |
| -- The maximum relative error bound for this test is |
| -- 4 for the arctan operation and 2 for the Taylor series |
| -- for a total of 6 * Model_Epsilon |
| |
| A : constant := -1.0/16.0; |
| B : constant := 1.0/16.0; |
| X : Real; |
| Actual, Expected : Real; |
| Sum, Em, X_Squared : Real; |
| begin |
| if Real'Digits > 19 then |
| -- Taylor series calculation produces result accurate to 19 |
| -- digits. If type being tested has more digits then set |
| -- the error low bound to account for this. |
| -- The error low bound is conservatively set to 6*10**-19 |
| Error_Low_Bound := 0.00000_00000_00000_0006; |
| Report.Comment ("arctan accuracy checked to 19 digits"); |
| end if; |
| |
| Accuracy_Error_Reported := False; -- reset |
| for I in 0..Max_Samples loop |
| X := (B - A) * Real (I) / Real (Max_Samples) + A; |
| X_Squared := X * X; |
| Em := 17.0; |
| Sum := X_Squared / Em; |
| |
| for II in 1 .. 7 loop |
| Em := Em - 2.0; |
| Sum := (1.0 / Em - Sum) * X_Squared; |
| end loop; |
| Sum := -X * Sum; |
| Expected := X + Sum; |
| Sum := (X - Expected) + Sum; |
| if not Real'Machine_Rounds then |
| Expected := Expected + (Sum + Sum); |
| end if; |
| |
| Actual := Arctan (X); |
| |
| Check (Actual, Expected, |
| "Taylor_Series_Test " & Integer'Image (I) & ": arctan(" & |
| Real'Image (X) & ") ", |
| 6.0); |
| |
| if Accuracy_Error_Reported then |
| -- only report the first error in this test in order to keep |
| -- lots of failures from producing a huge error log |
| return; |
| end if; |
| |
| end loop; |
| Error_Low_Bound := 0.0; -- reset |
| exception |
| when Constraint_Error => |
| Report.Failed |
| ("Constraint_Error raised in Taylor_Series_Test"); |
| when others => |
| Report.Failed ("exception in Taylor_Series_Test"); |
| end Taylor_Series_Test; |
| |
| |
| procedure Exception_Test is |
| X1, X2, X3 : Real := 0.0; |
| begin |
| |
| begin -- A.5.1(20);6.0 |
| X1 := Arctan(0.0, Cycle => 0.0); |
| Report.Failed ("no exception for cycle = 0.0"); |
| exception |
| when Ada.Numerics.Argument_Error => null; |
| when others => |
| Report.Failed ("wrong exception for cycle = 0.0"); |
| end; |
| |
| begin -- A.5.1(20);6.0 |
| X2 := Arctan (0.0, Cycle => -1.0); |
| Report.Failed ("no exception for cycle < 0.0"); |
| exception |
| when Ada.Numerics.Argument_Error => null; |
| when others => |
| Report.Failed ("wrong exception for cycle < 0.0"); |
| end; |
| |
| begin -- A.5.1(25);6.0 |
| X3 := Arctan (0.0, 0.0); |
| Report.Failed ("no exception for arctan(0,0)"); |
| exception |
| when Ada.Numerics.Argument_Error => null; |
| when others => |
| Report.Failed ("wrong exception for arctan(0,0)"); |
| end; |
| |
| -- optimizer thwarting |
| if Report.Ident_Bool (False) then |
| Report.Comment (Real'Image (X1 + X2 + X3)); |
| end if; |
| end Exception_Test; |
| |
| |
| procedure Do_Test is |
| begin |
| Special_Value_Test; |
| Exact_Result_Test; |
| Taylor_Series_Test; |
| Exception_Test; |
| end Do_Test; |
| end Generic_Check; |
| |
| ----------------------------------------------------------------------- |
| ----------------------------------------------------------------------- |
| -- These expressions must be truly static, which is why we have to do them |
| -- outside of the generic, and we use the named numbers. Note that we know |
| -- that PI is not a machine number (it is irrational), and it should be |
| -- represented to more digits than supported by the target machine. |
| Float_Half_PI_Low : constant := Float'Adjacent(PI/2.0, 0.0); |
| Float_Half_PI_High : constant := Float'Adjacent(PI/2.0, 10.0); |
| Float_PI_Low : constant := Float'Adjacent(PI, 0.0); |
| Float_PI_High : constant := Float'Adjacent(PI, 10.0); |
| package Float_Check is new Generic_Check (Float, |
| Half_PI_Low => Float_Half_PI_Low, |
| Half_PI_High => Float_Half_PI_High, |
| PI_Low => Float_PI_Low, |
| PI_High => Float_PI_High); |
| |
| -- check the Floating point type with the most digits |
| type A_Long_Float is digits System.Max_Digits; |
| A_Long_Float_Half_PI_Low : constant := A_Long_Float'Adjacent(PI/2.0, 0.0); |
| A_Long_Float_Half_PI_High : constant := A_Long_Float'Adjacent(PI/2.0, 10.0); |
| A_Long_Float_PI_Low : constant := A_Long_Float'Adjacent(PI, 0.0); |
| A_Long_Float_PI_High : constant := A_Long_Float'Adjacent(PI, 10.0); |
| package A_Long_Float_Check is new Generic_Check (A_Long_Float, |
| Half_PI_Low => A_Long_Float_Half_PI_Low, |
| Half_PI_High => A_Long_Float_Half_PI_High, |
| PI_Low => A_Long_Float_PI_Low, |
| PI_High => A_Long_Float_PI_High); |
| |
| ----------------------------------------------------------------------- |
| ----------------------------------------------------------------------- |
| |
| |
| begin |
| Report.Test ("CXG2016", |
| "Check the accuracy of the ARCTAN function"); |
| |
| if Verbose then |
| Report.Comment ("checking Standard.Float"); |
| end if; |
| |
| Float_Check.Do_Test; |
| |
| if Verbose then |
| Report.Comment ("checking a digits" & |
| Integer'Image (System.Max_Digits) & |
| " floating point type"); |
| end if; |
| |
| A_Long_Float_Check.Do_Test; |
| |
| |
| Report.Result; |
| end CXG2016; |
