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-- 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;