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-- CXG2021.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 SIN and COS functions return

--      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:

--      27 Mar 96   SAIC    Initial release for 2.1

--      22 Aug 96   SAIC    No longer skips test for systems with

--                          more than 20 digits of precision.

--

--!

--

-- 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 CXG2021 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 Sin (X : Complex) return Complex renames CEF.Sin;

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

      -- 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;

      -- the E_Factor is an additional amount added to the Expected

      -- value prior to computing the maximum relative error.

      -- This is needed because the error analysis (Cody pg 17-20)

      -- requires this additional allowance.

      procedure Check (Actual, Expected : Real;

                       Test_Name : String;

                       MRE : Real;

                       E_Factor : Real := 0.0) 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 * Real'Model_Epsilon * (abs Expected + E_Factor);

         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) &

                           " efactor:" & Real'Image (E_Factor) );

         elsif Verbose then

            if Actual = Expected then

               Report.Comment (Test_Name & "  exact result");

            else

               Report.Comment (Test_Name & "  passed" &

                           " actual: " & Real'Image (Actual) &

                           " expected: " & Real'Image (Expected) &

                           " difference: " & Real'Image (Actual - Expected) &

                           " max err:" & Real'Image (Max_Error) &

                           " efactor:" & Real'Image (E_Factor) );

            end if;

         end if;

      end Check;

      procedure Check (Actual, Expected : Complex;

                       Test_Name : String;

                       MRE : Real;

                       R_Factor, I_Factor : Real := 0.0) is

      begin

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

                MRE, R_Factor);

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

                MRE, I_Factor);

      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 if the argument is exact.

         -- Since the argument involves Pi, we must allow for this

         -- inexact argument.

         Minimum_Error : constant := 11.0;

      begin

         Check (Sin (Pi/2.0 + 0.0*i),

                1.0 + 0.0*i,

                "sin(pi/2+0i)",

                Minimum_Error + 1.0);

         Check (Cos (Pi/2.0 + 0.0*i),

                0.0 + 0.0*i,

                "cos(pi/2+0i)",

                Minimum_Error + 1.0);

      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 (Sin(0.0 + 0.0*i),  0.0 + 0.0 * i, "sin(0+0i)", No_Error);

         Check (Cos(0.0 + 0.0*i),  1.0 + 0.0 * i, "cos(0+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 (RA, RB, IA, IB : Real) is

      -- Tests an identity over a range of values specified

      -- by the 4 parameters.  RA and RB denote the range for the

      -- real part while IA and IB denote the range for the

      -- imaginary part.

      --

      -- For this test we use the identity

      --    Sin(Z) = Sin(Z-W) * Cos(W) + Cos(Z-W) * Sin(W)

      -- and

      --    Cos(Z) = Cos(Z-W) * Cos(W) - Sin(Z-W) * Sin(W)

      --

         X, Y : Real;

         Z   : Complex;

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

         ZmW : Complex;  -- Z - W

         Sin_ZmW,

         Cos_ZmW : Complex;

         Actual1, Actual2 : Complex;

         R_Factor : Real;  -- additional real error factor

         I_Factor : Real;  -- additional imaginary error factor

         Sin_W : constant Complex := (6.2581348413276935585E-2,

                                      6.2418588008436587236E-2);

         -- numeric stability is enhanced by using Cos(W) - 1.0 instead of

         -- Cos(W) in the computation.

         Cos_W_m_1 : constant Complex := (-2.5431314180235545803E-6,

                                            -3.9062493377261771826E-3);

      begin

         if Real'Digits > 20 then

            -- constants used here accurate to 20 digits.  Allow 1

            -- additional digit of error for computation.

            Error_Low_Bound := 0.00000_00000_00000_0001;

            Report.Comment ("accuracy checked to 19 digits");

         end if;

         Accuracy_Error_Reported := False;  -- reset

         for II in 0..Max_Samples loop

            X :=  (RB - RA) * Real (II) / Real (Max_Samples) + RA;

            for J in 0..Max_Samples loop

               Y :=  (IB - IA) * Real (J) / Real (Max_Samples) + IA;

               Z := Compose_From_Cartesian(X,Y);

               ZmW := Z - W;

               Sin_ZmW := Sin (ZmW);

               Cos_ZmW := Cos (ZmW);

               -- now for the first identity

               --    Sin(Z) = Sin(Z-W) * Cos(W) + Cos(Z-W) * Sin(W)

               --           = Sin(Z-W) * (1+(Cos(W)-1)) + Cos(Z-W) * Sin(W)

               --           = Sin(Z-W) + Sin(Z-W)*(Cos(W)-1) + Cos(Z-W)*Sin(W)

               Actual1 := Sin (Z);

               Actual2 := Sin_ZmW + (Sin_ZmW * Cos_W_m_1 + Cos_ZmW * Sin_W);

               -- The computation of the additional error factors are taken

               -- from Cody pages 17-20.

               R_Factor := abs (Re (Sin_ZmW) * Re (1.0 - Cos_W_m_1)) +

                           abs (Im (Sin_ZmW) * Im (1.0 - Cos_W_m_1)) +

                           abs (Re (Cos_ZmW) * Re (Sin_W)) +

                           abs (Re (Cos_ZmW) * Re (1.0 - Cos_W_m_1));

               I_Factor := abs (Re (Sin_ZmW) * Im (1.0 - Cos_W_m_1)) +

                           abs (Im (Sin_ZmW) * Re (1.0 - Cos_W_m_1)) +

                           abs (Re (Cos_ZmW) * Im (Sin_W)) +

                           abs (Im (Cos_ZmW) * Re (1.0 - Cos_W_m_1));

               Check (Actual1, Actual2,

                      "Identity_1_Test " & Integer'Image (II) &

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

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

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

                      11.0, R_Factor, I_Factor);

               -- now for the second identity

               --    Cos(Z) = Cos(Z-W) * Cos(W) - Sin(Z-W) * Sin(W)

               --           = Cos(Z-W) * (1+(Cos(W)-1) - Sin(Z-W) * Sin(W)

               Actual1 := Cos (Z);

               Actual2 := Cos_ZmW + (Cos_ZmW * Cos_W_m_1 - Sin_ZmW * Sin_W);

               -- The computation of the additional error factors are taken

               -- from Cody pages 17-20.

               R_Factor := abs (Re (Sin_ZmW) * Re (Sin_W)) +

                           abs (Im (Sin_ZmW) * Im (Sin_W)) +

                           abs (Re (Cos_ZmW) * Re (1.0 - Cos_W_m_1)) +

                           abs (Im (Cos_ZmW) * Im (1.0 - Cos_W_m_1));

               I_Factor := abs (Re (Sin_ZmW) * Im (Sin_W)) +

                           abs (Im (Sin_ZmW) * Re (Sin_W)) +

                           abs (Re (Cos_ZmW) * Im (1.0 - Cos_W_m_1)) +

                           abs (Im (Cos_ZmW) * Re (1.0 - Cos_W_m_1));

               Check (Actual1, Actual2,

                      "Identity_2_Test " & Integer'Image (II) &

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

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

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

                      11.0, R_Factor, I_Factor);

               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

                 Error_Low_Bound := 0.0;  -- reset

                 return;

               end if;

            end loop;

         end loop;

         Error_Low_Bound := 0.0;  -- reset

      exception

         when Constraint_Error =>

            Report.Failed

               ("Constraint_Error raised in Identity_Test" &

                " for Z=(" & Real'Image (X) &

                ", " & Real'Image (Y) & ")");

         when others =>

            Report.Failed ("exception in Identity_Test" &

                " for Z=(" & Real'Image (X) &

                ", " & Real'Image (Y) & ")");

      end Identity_Test;

      procedure Do_Test is

      begin

         Special_Value_Test;

         Exact_Result_Test;

            -- test regions where sin and cos have the same sign and

            -- about the same magnitude.  This will minimize subtraction

            -- errors in the identities.

            -- See Cody page 17.

         Identity_Test (0.0625,   10.0,    0.0625,    10.0);

         Identity_Test (  16.0,   17.0,      16.0,    17.0);

      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 ("CXG2021",

                "Check the accuracy of the complex SIN and COS 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 CXG2021;