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-- CXG2004.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 sin and cos 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
-- the following parts:
-- Special value checks where the result is a known constant.
-- Checks using an identity relationship.
--
-- 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:
-- 13 FEB 96 SAIC Initial release for 2.1
-- 22 APR 96 SAIC Changed to generic implementation.
-- 18 AUG 96 SAIC Improvements to commentary.
-- 23 OCT 96 SAIC Exact results are not required unless the
-- cycle is specified.
-- 28 FEB 97 PWB.CTA Removed checks where cycle 2.0*Pi is specified
-- 02 JUN 98 EDS Revised calculations to ensure that X is exactly
-- three times Y per advice of numerics experts.
--
-- 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
--
-- The sin and cos checks are translated directly from
-- the netlib FORTRAN code that was written by W. Cody.
--
with System;
with Report;
with Ada.Numerics.Generic_Elementary_Functions;
with Ada.Numerics.Elementary_Functions;
procedure CXG2004 is
Verbose : constant Boolean := False;
Number_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 <>;
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 Sin (X : Real) return Real renames
Elementary_Functions.Sin;
function Cos (X : Real) return Real renames
Elementary_Functions.Cos;
function Sin (X, Cycle : Real) return Real renames
Elementary_Functions.Sin;
function Cos (X, Cycle : Real) return Real renames
Elementary_Functions.Cos;
Accuracy_Error_Reported : Boolean := False;
procedure Check (Actual, Expected : Real;
Test_Name : String;
MRE : Real) is
Rel_Error,
Abs_Error,
Max_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;
-- in addition to the relative error checks we apply the
-- criteria of G.2.4(16)
if abs (Actual) > 1.0 then
Accuracy_Error_Reported := True;
Report.Failed (Test_Name & " result > 1.0");
elsif 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) &
" mre:" &
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 Sin_Check (A, B : Real;
Arg_Range : String) is
-- test a selection of
-- arguments selected from the range A to B.
--
-- This test uses the identity
-- sin(x) = sin(x/3)*(3 - 4 * sin(x/3)**2)
--
-- Note that in this test we must take into account the
-- error in the calculation of the expected result so
-- the maximum relative error is larger than the
-- accuracy required by the ARM.
X, Y, ZZ : Real;
Actual, Expected : Real;
MRE : Real;
Ran : Real;
begin
Accuracy_Error_Reported := False; -- reset
for I in 1 .. Number_Samples loop
-- Evenly distributed selection of arguments
Ran := Real (I) / Real (Number_Samples);
-- make sure x and x/3 are both exactly representable
-- on the machine. See "Implementation and Testing of
-- Function Software" page 44.
X := (B - A) * Ran + A;
Y := Real'Leading_Part
( X/3.0,
Real'Machine_Mantissa - Real'Exponent (3.0) );
X := Y * 3.0;
Actual := Sin (X);
ZZ := Sin(Y);
Expected := ZZ * (3.0 - 4.0 * ZZ * ZZ);
-- note that since the expected value is computed, we
-- must take the error in that computation into account.
-- See Cody pp 139-141.
MRE := 4.0;
Check (Actual, Expected,
"sin test of range" & Arg_Range &
Integer'Image (I),
MRE);
exit when Accuracy_Error_Reported;
end loop;
exception
when Constraint_Error =>
Report.Failed
("Constraint_Error raised in sin check");
when others =>
Report.Failed ("exception in sin check");
end Sin_Check;
procedure Cos_Check (A, B : Real;
Arg_Range : String) is
-- test a selection of
-- arguments selected from the range A to B.
--
-- This test uses the identity
-- cos(x) = cos(x/3)*(4 * cos(x/3)**2 - 3)
--
-- Note that in this test we must take into account the
-- error in the calculation of the expected result so
-- the maximum relative error is larger than the
-- accuracy required by the ARM.
X, Y, ZZ : Real;
Actual, Expected : Real;
MRE : Real;
Ran : Real;
begin
Accuracy_Error_Reported := False; -- reset
for I in 1 .. Number_Samples loop
-- Evenly distributed selection of arguments
Ran := Real (I) / Real (Number_Samples);
-- make sure x and x/3 are both exactly representable
-- on the machine. See "Implementation and Testing of
-- Function Software" page 44.
X := (B - A) * Ran + A;
Y := Real'Leading_Part
( X/3.0,
Real'Machine_Mantissa - Real'Exponent (3.0) );
X := Y * 3.0;
Actual := Cos (X);
ZZ := Cos(Y);
Expected := ZZ * (4.0 * ZZ * ZZ - 3.0);
-- note that since the expected value is computed, we
-- must take the error in that computation into account.
-- See Cody pp 141-143.
MRE := 6.0;
Check (Actual, Expected,
"cos test of range" & Arg_Range &
Integer'Image (I),
MRE);
exit when Accuracy_Error_Reported;
end loop;
exception
when Constraint_Error =>
Report.Failed
("Constraint_Error raised in cos check");
when others =>
Report.Failed ("exception in cos check");
end Cos_Check;
procedure Special_Angle_Checks is
type Data_Point is
record
Degrees,
Radians,
Sine,
Cosine : Real;
Sin_Result_Error,
Cos_Result_Error : Boolean;
end record;
type Test_Data_Type is array (Positive range <>) of Data_Point;
-- the values in the following table only involve static
-- expressions to minimize any loss of precision. However,
-- there are two sources of error that must be accounted for
-- in the following tests.
-- First, when a cycle is not specified there can be a roundoff
-- error in the value of Pi used. This error does not apply
-- when a cycle of 2.0 * Pi is explicitly provided.
-- Second, the expected results that involve sqrt values also
-- have a potential roundoff error.
-- The amount of error due to error in the argument is computed
-- as follows:
-- sin(x+err) = sin(x)*cos(err) + cos(x)*sin(err)
-- ~= sin(x) + err * cos(x)
-- similarly for cos the error due to error in the argument is
-- computed as follows:
-- cos(x+err) = cos(x)*cos(err) - sin(x)*sin(err)
-- ~= cos(x) - err * sin(x)
-- In both cases the term "err" is bounded by 0.5 * argument.
Test_Data : constant Test_Data_Type := (
-- degrees radians sine cosine sin_er cos_er test #
( 0.0, 0.0, 0.0, 1.0, False, False ), -- 1
( 30.0, Pi/6.0, 0.5, Sqrt3/2.0, False, True ), -- 2
( 60.0, Pi/3.0, Sqrt3/2.0, 0.5, True, False ), -- 3
( 90.0, Pi/2.0, 1.0, 0.0, False, False ), -- 4
(120.0, 2.0*Pi/3.0, Sqrt3/2.0, -0.5, True, False ), -- 5
(150.0, 5.0*Pi/6.0, 0.5, -Sqrt3/2.0, False, True ), -- 6
(180.0, Pi, 0.0, -1.0, False, False ), -- 7
(210.0, 7.0*Pi/6.0, -0.5, -Sqrt3/2.0, False, True ), -- 8
(240.0, 8.0*Pi/6.0, -Sqrt3/2.0, -0.5, True, False ), -- 9
(270.0, 9.0*Pi/6.0, -1.0, 0.0, False, False ), -- 10
(300.0, 10.0*Pi/6.0, -Sqrt3/2.0, 0.5, True, False ), -- 11
(330.0, 11.0*Pi/6.0, -0.5, Sqrt3/2.0, False, True ), -- 12
(360.0, 2.0*Pi, 0.0, 1.0, False, False ), -- 13
( 45.0, Pi/4.0, Sqrt2/2.0, Sqrt2/2.0, True, True ), -- 14
(135.0, 3.0*Pi/4.0, Sqrt2/2.0, -Sqrt2/2.0, True, True ), -- 15
(225.0, 5.0*Pi/4.0, -Sqrt2/2.0, -Sqrt2/2.0, True, True ), -- 16
(315.0, 7.0*Pi/4.0, -Sqrt2/2.0, Sqrt2/2.0, True, True ), -- 17
(405.0, 9.0*Pi/4.0, Sqrt2/2.0, Sqrt2/2.0, True, True ) ); -- 18
Y : Real;
Sin_Arg_Err,
Cos_Arg_Err,
Sin_Result_Err,
Cos_Result_Err : Real;
begin
for I in Test_Data'Range loop
-- compute error components
Sin_Arg_Err := abs Test_Data (I).Cosine *
abs Test_Data (I).Radians / 2.0;
Cos_Arg_Err := abs Test_Data (I).Sine *
abs Test_Data (I).Radians / 2.0;
if Test_Data (I).Sin_Result_Error then
Sin_Result_Err := 0.5;
else
Sin_Result_Err := 0.0;
end if;
if Test_Data (I).Cos_Result_Error then
Cos_Result_Err := 1.0;
else
Cos_Result_Err := 0.0;
end if;
Y := Sin (Test_Data (I).Radians);
Check (Y, Test_Data (I).Sine,
"test" & Integer'Image (I) & " sin(r)",
2.0 + Sin_Arg_Err + Sin_Result_Err);
Y := Cos (Test_Data (I).Radians);
Check (Y, Test_Data (I).Cosine,
"test" & Integer'Image (I) & " cos(r)",
2.0 + Cos_Arg_Err + Cos_Result_Err);
Y := Sin (Test_Data (I).Degrees, 360.0);
Check (Y, Test_Data (I).Sine,
"test" & Integer'Image (I) & " sin(d,360)",
2.0 + Sin_Result_Err);
Y := Cos (Test_Data (I).Degrees, 360.0);
Check (Y, Test_Data (I).Cosine,
"test" & Integer'Image (I) & " cos(d,360)",
2.0 + Cos_Result_Err);
--pwb-math Y := Sin (Test_Data (I).Radians, 2.0*Pi);
--pwb-math Check (Y, Test_Data (I).Sine,
--pwb-math "test" & Integer'Image (I) & " sin(r,2pi)",
--pwb-math 2.0 + Sin_Result_Err);
--pwb-math Y := Cos (Test_Data (I).Radians, 2.0*Pi);
--pwb-math Check (Y, Test_Data (I).Cosine,
--pwb-math "test" & Integer'Image (I) & " cos(r,2pi)",
--pwb-math 2.0 + Cos_Result_Err);
end loop;
exception
when Constraint_Error =>
Report.Failed ("Constraint_Error raised in special angle test");
when others =>
Report.Failed ("exception in special angle test");
end Special_Angle_Checks;
-- check the rule of A.5.1(41);6.0 which requires that the
-- result be exact if the mathematical result is 0.0, 1.0,
-- or -1.0
procedure Exact_Result_Checks is
type Data_Point is
record
Degrees,
Sine,
Cosine : Real;
end record;
type Test_Data_Type is array (Positive range <>) of Data_Point;
Test_Data : constant Test_Data_Type := (
-- degrees sine cosine test #
( 0.0, 0.0, 1.0 ), -- 1
( 90.0, 1.0, 0.0 ), -- 2
(180.0, 0.0, -1.0 ), -- 3
(270.0, -1.0, 0.0 ), -- 4
(360.0, 0.0, 1.0 ), -- 5
( 90.0 + 360.0, 1.0, 0.0 ), -- 6
(180.0 + 360.0, 0.0, -1.0 ), -- 7
(270.0 + 360.0,-1.0, 0.0 ), -- 8
(360.0 + 360.0, 0.0, 1.0 ) ); -- 9
Y : Real;
begin
for I in Test_Data'Range loop
Y := Sin (Test_Data(I).Degrees, 360.0);
if Y /= Test_Data(I).Sine then
Report.Failed ("exact result for sin(" &
Real'Image (Test_Data(I).Degrees) &
", 360.0) is not" &
Real'Image (Test_Data(I).Sine) &
" Difference is " &
Real'Image (Y - Test_Data(I).Sine) );
end if;
Y := Cos (Test_Data(I).Degrees, 360.0);
if Y /= Test_Data(I).Cosine then
Report.Failed ("exact result for cos(" &
Real'Image (Test_Data(I).Degrees) &
", 360.0) is not" &
Real'Image (Test_Data(I).Cosine) &
" Difference is " &
Real'Image (Y - Test_Data(I).Cosine) );
end if;
end loop;
exception
when Constraint_Error =>
Report.Failed ("Constraint_Error raised in exact result check");
when others =>
Report.Failed ("exception in exact result check");
end Exact_Result_Checks;
procedure Do_Test is
begin
Special_Angle_Checks;
Sin_Check (0.0, Pi/2.0, "0..pi/2");
Sin_Check (6.0*Pi, 6.5*Pi, "6pi..6.5pi");
Cos_Check (7.0*Pi, 7.5*Pi, "7pi..7.5pi");
Exact_Result_Checks;
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 ("CXG2004",
"Check the accuracy of the 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 CXG2004;