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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;
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