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-- CXG2018.A

--

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--                                    DISCLAIMER

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--     ALL MATERIALS OR INFORMATION HEREIN RELEASED, MADE AVAILABLE OR

--     DISCLOSED ARE AS IS.  THE GOVERNMENT MAKES NO EXPRESS OR IMPLIED

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--     OR DISCLOSED, OR THE OWNERSHIP, MERCHANTABILITY, OR FITNESS FOR A

--     PARTICULAR PURPOSE OF SAID MATERIAL.

--*

--

-- OBJECTIVE:

--      Check that the complex EXP 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 complex numbers based upon

--      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.

--

-- 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:

--      21 Mar 96   SAIC    Initial release for 2.1

--      17 Aug 96   SAIC    Incorporated reviewer comments.

--      27 Aug 99   RLB     Repair on the error result of checks.

--      02 Apr 03   RLB     Added code to discard excess precision in the

--                          construction of the test value for the

--                          Identity_Test.

--

--!

--

-- References:

--

-- W. J. Cody

-- CELEFUNT: A Portable Test Package for Complex Elementary Functions

-- Algorithm 714, Collected Algorithms from ACM.

-- Published in Transactions On Mathematical Software,

-- Vol. 19, No. 1, March, 1993, pp. 1-21.

--

-- CRC Standard Mathematical Tables

-- 23rd Edition

--

with System;

with Report;

with Ada.Numerics.Generic_Complex_Types;

with Ada.Numerics.Generic_Complex_Elementary_Functions;

procedure CXG2018 is

   Verbose : constant Boolean := False;

   -- Note that Max_Samples is the number of samples taken in

   -- both the real and imaginary directions.  Thus, for Max_Samples

   -- of 100 the number of values checked is 10000.

   Max_Samples : constant := 100;

   E  : constant := Ada.Numerics.E;

   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 Complex_Type is new

           Ada.Numerics.Generic_Complex_Types (Real);

      use Complex_Type;

      package CEF is new

           Ada.Numerics.Generic_Complex_Elementary_Functions (Complex_Type);

      function Exp (X : Complex) return Complex renames CEF.Exp;

      function Exp (X : Imaginary) return Complex renames CEF.Exp;

      -- 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_Small instead

         -- of Model_Epsilon and Expected.

         Rel_Error := MRE * abs Expected * Real'Model_Epsilon;

         Abs_Error := MRE * Real'Model_Small;

         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 Check (Actual, Expected : Complex;

                       Test_Name : String;

                       MRE : Real) is

      begin

         Check (Actual.Re, Expected.Re, Test_Name & " real part", MRE);

         Check (Actual.Im, Expected.Im, Test_Name & " imaginary part", MRE);

      end Check;

      procedure Special_Value_Test is

         -- In the following tests the expected result is accurate

         -- to the machine precision so the minimum guaranteed error

         -- bound can be used.

         --

         -- The error bounds given assumed z is  exact.  When using

         -- pi there is an extra error of 1.0ME.

         -- The pi inside the exp call requires that the complex

         -- component have an extra error allowance of 1.0*angle*ME.

         -- Thus for pi/2,the Minimum_Error_I is

         -- (2.0 + 1.0(pi/2))ME <= 3.6ME.

         -- For pi, it is (2.0 + 1.0*pi)ME <= 5.2ME,

         -- and for 2pi, it is (2.0 + 1.0(2pi))ME <= 8.3ME.

         -- The addition of 1 or i to a result is so that neither of

         -- the components of an expected result is 0.  This is so

         -- that a reasonable relative error is allowed.

         Minimum_Error_C : constant := 7.0;   -- for exp(Complex)

         Minimum_Error_I : constant := 2.0;   -- for exp(Imaginary)

      begin

         Check (Exp (1.0 + 0.0*i) + i,

                E + i,

                "exp(1+0i)",

                Minimum_Error_C);

         Check (Exp ((Pi / 2.0) * i) + 1.0,

                1.0 + 1.0*i,

                "exp(pi/2*i)",

                3.6);

         Check (Exp (Pi * i) + i,

                -1.0 + 1.0*i,

                "exp(pi*i)",

                5.2);

         Check (Exp (Pi * 2.0 * i) + i,

                1.0 + i,

                "exp(2pi*i)",

                8.3);

      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 Exact_Result_Test is

         No_Error : constant := 0.0;

      begin

         -- G.1.2(36);6.0

         Check (Exp(0.0 + 0.0*i),  1.0 + 0.0 * i, "exp(0+0i)", No_Error);

         Check (Exp(      0.0*i),  1.0 + 0.0 * i, "exp(0i)", 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 Identity_Test (A, B : Real) is

      -- For this test we use the identity

      --    Exp(Z) = Exp(Z-W) * Exp (W)

      -- where W = (1+i)/16

      --

      -- The second part of this test checks the identity

      --    Exp(Z) * Exp(-Z)  = 1

      --

         X, Y : Complex;

         Actual1, Actual2 : Complex;

         W : constant Complex := (0.0625, 0.0625);

         -- the following constant was taken from the CELEFUNC EXP test.

         -- This is the value EXP(W) - 1

         C : constant Complex := (6.2416044877018563681e-2,

                                  6.6487597751003112768e-2);

      begin

         if Real'Digits > 20 then

            -- constant ExpW is accurate to 20 digits.

            -- The low bound is 19 * 10**-20

            Error_Low_Bound := 0.00000_00000_00019;

            Report.Comment ("complex exp accuracy checked to 20 digits");

         end if;

         Accuracy_Error_Reported := False;  -- reset

         for II in 1..Max_Samples loop

            X.Re :=  Real'Machine ((B - A) * Real (II) / Real (Max_Samples)

                        + A);

            for J in 1..Max_Samples loop

               X.Im :=  Real'Machine ((B - A) * Real (J) / Real (Max_Samples)

                           + A);

               Actual1 := Exp(X);

               -- Exp(X) = Exp(X-W) * Exp (W)

               --        = Exp(X-W) * (1 - (1-Exp(W))

               --        = Exp(X-W) * (1 + (Exp(W) - 1))

               --        = Exp(X-W) * (1 + C)

               Y := X - W;

               Actual2 := Exp(Y);

               Actual2 := Actual2 + Actual2 * C;

               Check (Actual1, Actual2,

                      "Identity_1_Test " & Integer'Image (II) &

                         Integer'Image (J) & ": Exp((" &

                         Real'Image (X.Re) & ", " &

                         Real'Image (X.Im) & ")) ",

                      20.0);   -- 2 exp and 1 multiply and 1 add = 2*7+1*5+1

                    -- Note: The above is not strictly correct, as multiply

                    -- has a box error, rather than a relative error.

                    -- Supposedly, the interval is chosen to avoid the need

                    -- to worry about this.

               -- Exp(X) * Exp(-X) + i  = 1 + i

               -- The addition of i is to allow a reasonable relative

               -- error in the imaginary part

               Actual2 := (Actual1 * Exp(-X)) + i;

               Check (Actual2, (1.0, 1.0),

                      "Identity_2_Test " & Integer'Image (II) &

                         Integer'Image (J) & ": Exp((" &

                         Real'Image (X.Re) & ", " &

                         Real'Image (X.Im) & ")) ",

                      20.0);   -- 2 exp and 1 multiply and one add = 2*7+1*5+1

               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;

         end loop;

         Error_Low_Bound := 0.0;

      exception

         when Constraint_Error =>

            Report.Failed

               ("Constraint_Error raised in Identity_Test" &

                " for X=(" & Real'Image (X.Re) &

                ", " & Real'Image (X.Im) & ")");

         when others =>

            Report.Failed ("exception in Identity_Test" &

                " for X=(" & Real'Image (X.Re) &

                ", " & Real'Image (X.Im) & ")");

      end Identity_Test;

      procedure Do_Test is

      begin

         Special_Value_Test;

         Exact_Result_Test;

            -- test regions where we can avoid cancellation error problems

            -- See Cody page 10.

         Identity_Test (0.0625, 1.0);

         Identity_Test (15.0, 17.0);

         Identity_Test (1.625, 3.0);

      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 ("CXG2018",

                "Check the accuracy of the complex EXP 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 CXG2018;