-- CXG2013.A -- -- Grant of Unlimited Rights -- -- Under contracts F33600-87-D-0337, F33600-84-D-0280, MDA903-79-C-0687 and -- F08630-91-C-0015, 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 TAN and COT functions return -- results that are 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. -- Checks that use an identity for determining the result. -- 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: -- 11 Mar 96 SAIC Initial release for 2.1 -- 17 Aug 96 SAIC Commentary fixes. -- 03 Feb 97 PWB.CTA Removed checks with explicit Cycle => 2.0*Pi -- -- CHANGE NOTE: -- According to Ken Dritz, author of the Numerics Annex of the RM, -- one should never specify the cycle 2.0*Pi for the trigonometric -- functions. In particular, if the machine number for the first -- argument is not an exact multiple of the machine number for the -- explicit cycle, then the specified exact results cannot be -- reasonably expected. The affected checks in this test have been -- marked as comments, with the additional notation "pwb-math". -- Phil Brashear --! -- -- 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; procedure CXG2013 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 <>; 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 Sqrt (X : Real) return Real renames Elementary_Functions.Sqrt; function Tan (X : Real) return Real renames Elementary_Functions.Tan; function Cot (X : Real) return Real renames Elementary_Functions.Cot; function Tan (X, Cycle : Real) return Real renames Elementary_Functions.Tan; function Cot (X, Cycle : Real) return Real renames Elementary_Functions.Cot; -- flag used to terminate some tests early Accuracy_Error_Reported : Boolean := False; -- factor to be applied in computing MRE Maximum_Relative_Error : constant Real := 4.0; 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; 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_Angle_Test is Undef : constant := 1234.5; -- special flag type Data_Point is record Degrees, Radians, Tangent, Cotangent : 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 loss of precision occurs. -- However, since Pi isn't exact, this error of less than -- 1 digit in the argument affects the expected result as -- follows: -- Let rad be the value of the angle that we have. -- Let theta be the true value of the angle. -- Let eta = rad - theta. -- Then tan(rad) = tan (theta + eta) -- ~= tan(theta) + eta[ 1 - (tan(theta))**2] -- and |eta| <= 1.0Model_Epsilon|rad|. -- -- Hence, the maximum relative error is less than -- 4.0ME + 1.0ME|rad|*|1-(tan(theta))**2| / tan(rad) -- -- One could replace rad by theta or vice versa and still be OK. -- For cot, just replace tan by cot in the above. -- -- We simplify the above error bound to 6.0ME for all tests where -- the argument involves Pi by choosing our range so that the -- function is defined (avoiding poles). Test_Data : constant Test_Data_Type := ( -- degrees radians tangent cotangent test # ( 0.0, 0.0, 0.0, Undef ), -- 1 ( 30.0, Pi/6.0, Sqrt3/3.0, Sqrt3 ), -- 2 ( 60.0, Pi/3.0, Sqrt3, Sqrt3/3.0 ), -- 3 ( 90.0, Pi/2.0, Undef, 0.0 ), -- 4 (120.0, 2.0*Pi/3.0, -Sqrt3, -Sqrt3/3.0 ), -- 5 (150.0, 5.0*Pi/6.0, -Sqrt3/3.0, -Sqrt3 ), -- 6 (180.0, Pi, 0.0, Undef ), -- 7 (360.0, 2.0*Pi, 0.0, Undef ), -- 8 ( 45.0, Pi/4.0, 1.0, 1.0 ), -- 9 (135.0, 3.0*Pi/4.0, -1.0, -1.0 ), -- 10 (225.0, 5.0*Pi/4.0, 1.0, 1.0 ), -- 11 (315.0, 7.0*Pi/4.0, -1.0, -1.0 ), -- 12 (405.0, 9.0*Pi/4.0, 1.0, 1.0 ) ); -- 13 Y : Real; Allowed_Error : Real; begin for I in Test_Data'Range loop if I = Test_Data'First then Allowed_Error := 4.0; else Allowed_Error := 6.0; end if; if Test_Data (I).Tangent /= Undef then Y := Tan (Test_Data (I).Radians); Check (Y, Test_Data (I).Tangent, "test" & Integer'Image (I) & " tan(r)", Allowed_Error); Y := Tan (Test_Data (I).Degrees, 360.0); Check (Y, Test_Data (I).Tangent, "test" & Integer'Image (I) & " tan(d,360)", Allowed_Error); --pwb-math Y := Tan (Test_Data (I).Radians, 2.0*Pi); --pwb-math Check (Y, Test_Data (I).Tangent, --pwb-math "test" & Integer'Image (I) & " tan(r,2pi)", --pwb-math Allowed_Error); end if; if Test_Data (I).Cotangent /= Undef then Y := Cot (Test_Data (I).Radians); Check (Y, Test_Data (I).Cotangent, "test" & Integer'Image (I) & " cot(r)", Allowed_Error); Y := Cot (Test_Data (I).Degrees, 360.0); Check (Y, Test_Data (I).Cotangent, "test" & Integer'Image (I) & " cot(d,360)", Allowed_Error); --pwb-math Y := Cot (Test_Data (I).Radians, 2.0*Pi); --pwb-math Check (Y, Test_Data (I).Cotangent, --pwb-math "test" & Integer'Image (I) & " cot(r,2pi)", --pwb-math Allowed_Error); end if; end loop; exception when Constraint_Error => Report.Failed ("Constraint_Error raised in special angle test"); when others => Report.Failed ("exception in special angle test"); end Special_Angle_Test; procedure Exact_Result_Test is No_Error : constant := 0.0; begin -- A.5.1(38);6.0 Check (Tan (0.0), 0.0, "tan(0)", No_Error); -- A.5.1(41);6.0 Check (Tan (180.0, 360.0), 0.0, "tan(180,360)", No_Error); Check (Tan (360.0, 360.0), 0.0, "tan(360,360)", No_Error); Check (Tan (720.0, 360.0), 0.0, "tan(720,360)", No_Error); -- A.5.1(41);6.0 Check (Cot ( 90.0, 360.0), 0.0, "cot( 90,360)", No_Error); Check (Cot (270.0, 360.0), 0.0, "cot(270,360)", No_Error); Check (Cot (810.0, 360.0), 0.0, "cot(810,360)", No_Error); 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 Tan_Test (A, B : Real) is -- Use identity Tan(X) = [2*Tan(x/2)]/[1-Tan(x/2) ** 2] -- checks over the range -pi/4 .. pi/4 require no argument reduction -- checks over the range 7pi/8 .. 9pi/8 require argument reduction X, Y : Real; Actual1, Actual2 : Real; begin Accuracy_Error_Reported := False; -- reset for I in 1..Max_Samples loop X := (B - A) * Real (I) / Real (Max_Samples) + A; -- argument purification to insure x and x/2 are exact -- See Cody page 170. Y := Real'Machine (X*0.5); X := Real'Machine (Y + Y); Actual1 := Tan(X); Actual2 := (2.0 * Tan (Y)) / (1.0 - Tan (Y) ** 2); if abs (X - Pi) > ( (B-A)/Real(2*Max_Samples) ) then Check (Actual1, Actual2, "Tan_Test " & Integer'Image (I) & ": tan(" & Real'Image (X) & ") ", (1.0 + Sqrt2) * Maximum_Relative_Error); -- see Cody pg 165 for error bound info end if; 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; exception when Constraint_Error => Report.Failed ("Constraint_Error raised in Tan_Test"); when others => Report.Failed ("exception in Tan_Test"); end Tan_Test; procedure Cot_Test is -- Use identity Cot(X) = [Cot(X/2)**2 - 1]/[2*Cot(X/2)] A : constant := 6.0 * Pi; B : constant := 25.0 / 4.0 * Pi; X, Y : Real; Actual1, Actual2 : Real; begin Accuracy_Error_Reported := False; -- reset for I in 1..Max_Samples loop X := (B - A) * Real (I) / Real (Max_Samples) + A; -- argument purification to insure x and x/2 are exact. -- See Cody page 170. Y := Real'Machine (X*0.5); X := Real'Machine (Y + Y); Actual1 := Cot(X); Actual2 := (Cot (Y) ** 2 - 1.0) / (2.0 * Cot (Y)); Check (Actual1, Actual2, "Cot_Test " & Integer'Image (I) & ": cot(" & Real'Image (X) & ") ", (1.0 + Sqrt2) * Maximum_Relative_Error); -- see Cody pg 165 for error bound info 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; exception when Constraint_Error => Report.Failed ("Constraint_Error raised in Cot_Test"); when others => Report.Failed ("exception in Cot_Test"); end Cot_Test; procedure Exception_Test is X1, X2, X3, X4, X5 : Real := 0.0; begin begin -- A.5.1(20);6.0 X1 := Tan (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 := Cot (1.0, Cycle => -3.0); 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; -- the remaining tests only apply to machines that overflow if Real'Machine_Overflows then -- A.5.1(28);6.0 begin -- A.5.1(29);6.0 X3 := Cot (0.0); Report.Failed ("exception not raised for cot(0)"); exception when Constraint_Error => null; -- ok when others => Report.Failed ("wrong exception raised for cot(0)"); end; begin -- A.5.1(31);6.0 X4 := Tan (90.0, 360.0); Report.Failed ("exception not raised for tan(90,360)"); exception when Constraint_Error => null; -- ok when others => Report.Failed ("wrong exception raised for tan(90,360)"); end; begin -- A.5.1(32);6.0 X5 := Cot (180.0, 360.0); Report.Failed ("exception not raised for cot(180,360)"); exception when Constraint_Error => null; -- ok when others => Report.Failed ("wrong exception raised for cot(180,360)"); end; end if; -- optimizer thwarting if Report.Ident_Bool (False) then Report.Comment (Real'Image (X1+X2+X3+X4+X5)); end if; end Exception_Test; procedure Do_Test is begin Special_Angle_Test; Exact_Result_Test; Tan_Test (-Pi/4.0, Pi/4.0); Tan_Test (7.0*Pi/8.0, 9.0*Pi/8.0); Cot_Test; Exception_Test; end Do_Test; end Generic_Check; ----------------------------------------------------------------------- ----------------------------------------------------------------------- package Float_Check is new Generic_Check (Float); -- check the floating point type with the most digits type A_Long_Float is digits System.Max_Digits; package A_Long_Float_Check is new Generic_Check (A_Long_Float); ----------------------------------------------------------------------- ----------------------------------------------------------------------- begin Report.Test ("CXG2013", "Check the accuracy of the TAN and COT functions"); 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 CXG2013;