-- CXG2018.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 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;