-- 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;