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| author | kono |
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| date | Fri, 27 Oct 2017 22:46:09 +0900 |
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-- CXG2015.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 ARCSIN and ARCCOS 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 in a specific range where a Taylor series can be -- used to compute an accurate result for comparison. -- Exception checks. -- The Taylor series tests are a direct translation of the -- FORTRAN code found in the reference. -- -- 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: -- 18 Mar 96 SAIC Initial release for 2.1 -- 24 Apr 96 SAIC Fixed error bounds. -- 17 Aug 96 SAIC Added reference information and improved -- checking for machines with more than 23 -- digits of precision. -- 03 Feb 97 PWB.CTA Removed checks with explicit Cycle => 2.0*Pi -- 22 Dec 99 RLB Added model range checking to "exact" results, -- in order to avoid too strictly requiring a specific -- result, and too weakly checking results. -- -- 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 -- -- CELEFUNT: A Portable Test Package for Complex Elementary Functions -- ACM Collected Algorithms number 714 with System; with Report; with Ada.Numerics.Generic_Elementary_Functions; procedure CXG2015 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; -- relative error bound from G.2.4(7);6.0 Minimum_Error : constant := 4.0; 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 Arcsin (X : Real) return Real renames Elementary_Functions.Arcsin; function Arcsin (X, Cycle : Real) return Real renames Elementary_Functions.Arcsin; function Arccos (X : Real) return Real renames Elementary_Functions.ArcCos; function Arccos (X, Cycle : Real) return Real renames Elementary_Functions.ArcCos; -- needed for support function Log (X, Base : Real) return Real renames Elementary_Functions.Log; -- 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 -- In the following tests the expected result is accurate -- to the machine precision so the minimum guaranteed error -- bound can be used. type Data_Point is record Degrees, Radians, Argument, Error_Bound : Real; end record; type Test_Data_Type is array (Positive range <>) of Data_Point; -- the values in the following tables only involve static -- expressions so no loss of precision occurs. However, -- rounding can be an issue with expressions involving Pi -- and square roots. The error bound specified in the -- table takes the sqrt error into account but not the -- error due to Pi. The Pi error is added in in the -- radians test below. Arcsin_Test_Data : constant Test_Data_Type := ( -- degrees radians sine error_bound test # --( 0.0, 0.0, 0.0, 0.0 ), -- 1 - In Exact_Result_Test. ( 30.0, Pi/6.0, 0.5, 4.0 ), -- 2 ( 60.0, Pi/3.0, Sqrt3/2.0, 5.0 ), -- 3 --( 90.0, Pi/2.0, 1.0, 4.0 ), -- 4 - In Exact_Result_Test. --(-90.0, -Pi/2.0, -1.0, 4.0 ), -- 5 - In Exact_Result_Test. (-60.0, -Pi/3.0, -Sqrt3/2.0, 5.0 ), -- 6 (-30.0, -Pi/6.0, -0.5, 4.0 ), -- 7 ( 45.0, Pi/4.0, Sqrt2/2.0, 5.0 ), -- 8 (-45.0, -Pi/4.0, -Sqrt2/2.0, 5.0 ) ); -- 9 Arccos_Test_Data : constant Test_Data_Type := ( -- degrees radians cosine error_bound test # --( 0.0, 0.0, 1.0, 0.0 ), -- 1 - In Exact_Result_Test. ( 30.0, Pi/6.0, Sqrt3/2.0, 5.0 ), -- 2 ( 60.0, Pi/3.0, 0.5, 4.0 ), -- 3 --( 90.0, Pi/2.0, 0.0, 4.0 ), -- 4 - In Exact_Result_Test. (120.0, 2.0*Pi/3.0, -0.5, 4.0 ), -- 5 (150.0, 5.0*Pi/6.0, -Sqrt3/2.0, 5.0 ), -- 6 --(180.0, Pi, -1.0, 4.0 ), -- 7 - In Exact_Result_Test. ( 45.0, Pi/4.0, Sqrt2/2.0, 5.0 ), -- 8 (135.0, 3.0*Pi/4.0, -Sqrt2/2.0, 5.0 ) ); -- 9 Cycle_Error, Radian_Error : Real; begin for I in Arcsin_Test_Data'Range loop -- note exact result requirements A.5.1(38);6.0 and -- G.2.4(12);6.0 if Arcsin_Test_Data (I).Error_Bound = 0.0 then Cycle_Error := 0.0; Radian_Error := 0.0; else Cycle_Error := Arcsin_Test_Data (I).Error_Bound; -- allow for rounding error in the specification of Pi Radian_Error := Cycle_Error + 1.0; end if; Check (Arcsin (Arcsin_Test_Data (I).Argument), Arcsin_Test_Data (I).Radians, "test" & Integer'Image (I) & " arcsin(" & Real'Image (Arcsin_Test_Data (I).Argument) & ")", Radian_Error); --pwb-math Check (Arcsin (Arcsin_Test_Data (I).Argument, 2.0 * Pi), --pwb-math Arcsin_Test_Data (I).Radians, --pwb-math "test" & Integer'Image (I) & --pwb-math " arcsin(" & --pwb-math Real'Image (Arcsin_Test_Data (I).Argument) & --pwb-math ", 2pi)", --pwb-math Cycle_Error); Check (Arcsin (Arcsin_Test_Data (I).Argument, 360.0), Arcsin_Test_Data (I).Degrees, "test" & Integer'Image (I) & " arcsin(" & Real'Image (Arcsin_Test_Data (I).Argument) & ", 360)", Cycle_Error); end loop; for I in Arccos_Test_Data'Range loop -- note exact result requirements A.5.1(39);6.0 and -- G.2.4(12);6.0 if Arccos_Test_Data (I).Error_Bound = 0.0 then Cycle_Error := 0.0; Radian_Error := 0.0; else Cycle_Error := Arccos_Test_Data (I).Error_Bound; -- allow for rounding error in the specification of Pi Radian_Error := Cycle_Error + 1.0; end if; Check (Arccos (Arccos_Test_Data (I).Argument), Arccos_Test_Data (I).Radians, "test" & Integer'Image (I) & " arccos(" & Real'Image (Arccos_Test_Data (I).Argument) & ")", Radian_Error); --pwb-math Check (Arccos (Arccos_Test_Data (I).Argument, 2.0 * Pi), --pwb-math Arccos_Test_Data (I).Radians, --pwb-math "test" & Integer'Image (I) & --pwb-math " arccos(" & --pwb-math Real'Image (Arccos_Test_Data (I).Argument) & --pwb-math ", 2pi)", --pwb-math Cycle_Error); Check (Arccos (Arccos_Test_Data (I).Argument, 360.0), Arccos_Test_Data (I).Degrees, "test" & Integer'Image (I) & " arccos(" & Real'Image (Arccos_Test_Data (I).Argument) & ", 360)", Cycle_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(38) Check_Exact (Arcsin (0.0), 0.0, 0.0, "arcsin(0)"); Check_Exact (Arcsin (0.0, 45.0), 0.0, 0.0, "arcsin(0,45)"); -- A.5.1(39) Check_Exact (Arccos (1.0), 0.0, 0.0, "arccos(1)"); Check_Exact (Arccos (1.0, 75.0), 0.0, 0.0, "arccos(1,75)"); -- G.2.4(11-13) Check_Exact (Arcsin (1.0), Half_PI_Low, Half_PI_High, "arcsin(1)"); Check_Exact (Arcsin (1.0, 360.0), 90.0, 90.0, "arcsin(1,360)"); Check_Exact (Arcsin (-1.0), -Half_PI_High, -Half_PI_Low, "arcsin(-1)"); Check_Exact (Arcsin (-1.0, 360.0), -90.0, -90.0, "arcsin(-1,360)"); Check_Exact (Arccos (0.0), Half_PI_Low, Half_PI_High, "arccos(0)"); Check_Exact (Arccos (0.0, 360.0), 90.0, 90.0, "arccos(0,360)"); Check_Exact (Arccos (-1.0), PI_Low, PI_High, "arccos(-1)"); Check_Exact (Arccos (-1.0, 360.0), 180.0, 180.0, "arccos(-1,360)"); 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 Arcsin_Taylor_Series_Test is -- the following range is chosen so that the Taylor series -- used will produce a result accurate to machine precision. -- -- The following formula is used for the Taylor series: -- TS(x) = x { 1 + (xsq/2) [ (1/3) + (3/4)xsq { (1/5) + -- (5/6)xsq [ (1/7) + (7/8)xsq/9 ] } ] } -- where xsq = x * x -- A : constant := -0.125; B : constant := 0.125; X : Real; Y, Y_Sq : Real; Actual, Sum, Xm : Real; -- terms in Taylor series K : constant Integer := Integer ( Log ( Real (Real'Machine_Radix) ** Real'Machine_Mantissa, 10.0)) + 1; begin Accuracy_Error_Reported := False; -- reset for I in 1..Max_Samples loop -- make sure there is no error in x-1, x, and x+1 X := (B - A) * Real (I) / Real (Max_Samples) + A; Y := X; Y_Sq := Y * Y; Sum := 0.0; Xm := Real (K + K + 1); for M in 1 .. K loop Sum := Y_Sq * (Sum + 1.0/Xm); Xm := Xm - 2.0; Sum := Sum * (Xm /(Xm + 1.0)); end loop; Sum := Sum * Y; Actual := Y + Sum; Sum := (Y - Actual) + Sum; if not Real'Machine_Rounds then Actual := Actual + (Sum + Sum); end if; Check (Actual, Arcsin (X), "Taylor Series test" & Integer'Image (I) & ": arcsin(" & Real'Image (X) & ") ", Minimum_Error); 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 Arcsin_Taylor_Series_Test" & " for X=" & Real'Image (X)); when others => Report.Failed ("exception in Arcsin_Taylor_Series_Test" & " for X=" & Real'Image (X)); end Arcsin_Taylor_Series_Test; procedure Arccos_Taylor_Series_Test is -- the following range is chosen so that the Taylor series -- used will produce a result accurate to machine precision. -- -- The following formula is used for the Taylor series: -- TS(x) = x { 1 + (xsq/2) [ (1/3) + (3/4)xsq { (1/5) + -- (5/6)xsq [ (1/7) + (7/8)xsq/9 ] } ] } -- arccos(x) = pi/2 - TS(x) A : constant := -0.125; B : constant := 0.125; C1, C2 : Real; X : Real; Y, Y_Sq : Real; Actual, Sum, Xm, S : Real; -- terms in Taylor series K : constant Integer := Integer ( Log ( Real (Real'Machine_Radix) ** Real'Machine_Mantissa, 10.0)) + 1; begin if Real'Digits > 23 then -- constants in this section only accurate to 23 digits Error_Low_Bound := 0.00000_00000_00000_00000_001; Report.Comment ("arctan accuracy checked to 23 digits"); end if; -- C1 + C2 equals Pi/2 accurate to 23 digits if Real'Machine_Radix = 10 then C1 := 1.57; C2 := 7.9632679489661923132E-4; else C1 := 201.0 / 128.0; C2 := 4.8382679489661923132E-4; end if; Accuracy_Error_Reported := False; -- reset for I in 1..Max_Samples loop -- make sure there is no error in x-1, x, and x+1 X := (B - A) * Real (I) / Real (Max_Samples) + A; Y := X; Y_Sq := Y * Y; Sum := 0.0; Xm := Real (K + K + 1); for M in 1 .. K loop Sum := Y_Sq * (Sum + 1.0/Xm); Xm := Xm - 2.0; Sum := Sum * (Xm /(Xm + 1.0)); end loop; Sum := Sum * Y; -- at this point we have arcsin(x). -- We compute arccos(x) = pi/2 - arcsin(x). -- The following code segment is translated directly from -- the CELEFUNT FORTRAN implementation S := C1 + C2; Sum := ((C1 - S) + C2) - Sum; Actual := S + Sum; Sum := ((S - Actual) + Sum) - Y; S := Actual; Actual := S + Sum; Sum := (S - Actual) + Sum; if not Real'Machine_Rounds then Actual := Actual + (Sum + Sum); end if; Check (Actual, Arccos (X), "Taylor Series test" & Integer'Image (I) & ": arccos(" & Real'Image (X) & ") ", Minimum_Error); -- only report the first error in this test in order to keep -- lots of failures from producing a huge error log exit when Accuracy_Error_Reported; end loop; Error_Low_Bound := 0.0; -- reset exception when Constraint_Error => Report.Failed ("Constraint_Error raised in Arccos_Taylor_Series_Test" & " for X=" & Real'Image (X)); when others => Report.Failed ("exception in Arccos_Taylor_Series_Test" & " for X=" & Real'Image (X)); end Arccos_Taylor_Series_Test; procedure Identity_Test is -- test the identity arcsin(-x) = -arcsin(x) -- range chosen to be most of the valid range of the argument. A : constant := -0.999; B : constant := 0.999; X : Real; begin Accuracy_Error_Reported := False; -- reset for I in 1..Max_Samples loop -- make sure there is no error in x-1, x, and x+1 X := (B - A) * Real (I) / Real (Max_Samples) + A; Check (Arcsin(-X), -Arcsin (X), "Identity test" & Integer'Image (I) & ": arcsin(" & Real'Image (X) & ") ", 8.0); -- 2 arcsin evaluations => twice the error bound 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 Identity_Test; procedure Exception_Test is X1, X2 : Real := 0.0; begin begin X1 := Arcsin (1.1); Report.Failed ("no exception for Arcsin (1.1)"); exception when Constraint_Error => Report.Failed ("Constraint_Error instead of " & "Argument_Error for Arcsin (1.1)"); when Ada.Numerics.Argument_Error => null; -- expected result when others => Report.Failed ("wrong exception for Arcsin(1.1)"); end; begin X2 := Arccos (-1.1); Report.Failed ("no exception for Arccos (-1.1)"); exception when Constraint_Error => Report.Failed ("Constraint_Error instead of " & "Argument_Error for Arccos (-1.1)"); when Ada.Numerics.Argument_Error => null; -- expected result when others => Report.Failed ("wrong exception for Arccos(-1.1)"); end; -- optimizer thwarting if Report.Ident_Bool (False) then Report.Comment (Real'Image (X1 + X2)); end if; end Exception_Test; procedure Do_Test is begin Special_Value_Test; Exact_Result_Test; Arcsin_Taylor_Series_Test; Arccos_Taylor_Series_Test; Identity_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 ("CXG2015", "Check the accuracy of the ARCSIN and ARCCOS 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 CXG2015;