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------------------------------------------------------------------------------

--                                                                          --

--                         GNAT RUN-TIME COMPONENTS                         --

--                                                                          --

--                     A D A . N U M E R I C S . A U X                      --

--                                                                          --

--                                 B o d y                                  --

--                          (Apple OS X Version)                            --

--                                                                          --

--          Copyright (C) 1998-2009, Free Software Foundation, Inc.         --

--                                                                          --

-- GNAT is free software;  you can  redistribute it  and/or modify it under --

-- terms of the  GNU General Public License as published  by the Free Soft- --

-- ware  Foundation;  either version 3,  or (at your option) any later ver- --

-- sion.  GNAT is distributed in the hope that it will be useful, but WITH- --

-- OUT ANY WARRANTY;  without even the  implied warranty of MERCHANTABILITY --

-- or FITNESS FOR A PARTICULAR PURPOSE.                                     --

--                                                                          --

-- As a special exception under Section 7 of GPL version 3, you are granted --

-- additional permissions described in the GCC Runtime Library Exception,   --

-- version 3.1, as published by the Free Software Foundation.               --

--                                                                          --

-- You should have received a copy of the GNU General Public License and    --

-- a copy of the GCC Runtime Library Exception along with this program;     --

-- see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see    --

-- <http://www.gnu.org/licenses/>.                                          --

--                                                                          --

-- GNAT was originally developed  by the GNAT team at  New York University. --

-- Extensive contributions were provided by Ada Core Technologies Inc.      --

--                                                                          --

------------------------------------------------------------------------------

--  File a-numaux.adb <- a-numaux-darwin.adb

package body Ada.Numerics.Aux is

   -----------------------

   -- Local subprograms --

   -----------------------

   procedure Reduce (X : in out Double; Q : out Natural);

   --  Implements reduction of X by Pi/2. Q is the quadrant of the final

   --  result in the range 0 .. 3. The absolute value of X is at most Pi/4.

   --  The following three functions implement Chebishev approximations

   --  of the trigonometric functions in their reduced domain.

   --  These approximations have been computed using Maple.

   function Sine_Approx (X : Double) return Double;

   function Cosine_Approx (X : Double) return Double;

   pragma Inline (Reduce);

   pragma Inline (Sine_Approx);

   pragma Inline (Cosine_Approx);

   function Cosine_Approx (X : Double) return Double is

      XX : constant Double := X * X;

   begin

      return (((((16#8.DC57FBD05F640#E-08 * XX

              - 16#4.9F7D00BF25D80#E-06) * XX

              + 16#1.A019F7FDEFCC2#E-04) * XX

              - 16#5.B05B058F18B20#E-03) * XX

              + 16#A.AAAAAAAA73FA8#E-02) * XX

              - 16#7.FFFFFFFFFFDE4#E-01) * XX

              - 16#3.655E64869ECCE#E-14 + 1.0;

   end Cosine_Approx;

   function Sine_Approx (X : Double) return Double is

      XX : constant Double := X * X;

   begin

      return (((((16#A.EA2D4ABE41808#E-09 * XX

              - 16#6.B974C10F9D078#E-07) * XX

              + 16#2.E3BC673425B0E#E-05) * XX

              - 16#D.00D00CCA7AF00#E-04) * XX

              + 16#2.222222221B190#E-02) * XX

              - 16#2.AAAAAAAAAAA44#E-01) * (XX * X) + X;

   end Sine_Approx;

   ------------

   -- Reduce --

   ------------

   procedure Reduce (X : in out Double; Q : out Natural) is

      Half_Pi     : constant := Pi / 2.0;

      Two_Over_Pi : constant := 2.0 / Pi;

      HM : constant := Integer'Min (Double'Machine_Mantissa / 2, Natural'Size);

      M  : constant Double := 0.5 + 2.0**(1 - HM); -- Splitting constant

      P1 : constant Double := Double'Leading_Part (Half_Pi, HM);

      P2 : constant Double := Double'Leading_Part (Half_Pi - P1, HM);

      P3 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2, HM);

      P4 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2 - P3, HM);

      P5 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2 - P3

                                                                 - P4, HM);

      P6 : constant Double := Double'Model (Half_Pi - P1 - P2 - P3 - P4 - P5);

      K  : Double;

   begin

      --  For X < 2.0**HM, all products below are computed exactly.

      --  Due to cancellation effects all subtractions are exact as well.

      --  As no double extended floating-point number has more than 75

      --  zeros after the binary point, the result will be the correctly

      --  rounded result of X - K * (Pi / 2.0).

      K := X * Two_Over_Pi;

      while abs K >= 2.0 ** HM loop

         K := K * M - (K * M - K);

         X :=

           (((((X - K * P1) - K * P2) - K * P3) - K * P4) - K * P5) - K * P6;

         K := X * Two_Over_Pi;

      end loop;

      --  If K is not a number (because X was not finite) raise exception

      if K /= K then

         raise Constraint_Error;

      end if;

      K := Double'Rounding (K);

      Q := Integer (K) mod 4;

      X := (((((X - K * P1) - K * P2) - K * P3)

                  - K * P4) - K * P5) - K * P6;

   end Reduce;

   ---------

   -- Cos --

   ---------

   function Cos (X : Double) return Double is

      Reduced_X : Double := abs X;

      Quadrant  : Natural range 0 .. 3;

   begin

      if Reduced_X > Pi / 4.0 then

         Reduce (Reduced_X, Quadrant);

         case Quadrant is

            when 0 =>

               return Cosine_Approx (Reduced_X);

            when 1 =>

               return Sine_Approx (-Reduced_X);

            when 2 =>

               return -Cosine_Approx (Reduced_X);

            when 3 =>

               return Sine_Approx (Reduced_X);

         end case;

      end if;

      return Cosine_Approx (Reduced_X);

   end Cos;

   ---------

   -- Sin --

   ---------

   function Sin (X : Double) return Double is

      Reduced_X : Double := X;

      Quadrant  : Natural range 0 .. 3;

   begin

      if abs X > Pi / 4.0 then

         Reduce (Reduced_X, Quadrant);

         case Quadrant is

            when 0 =>

               return Sine_Approx (Reduced_X);

            when 1 =>

               return Cosine_Approx (Reduced_X);

            when 2 =>

               return Sine_Approx (-Reduced_X);

            when 3 =>

               return -Cosine_Approx (Reduced_X);

         end case;

      end if;

      return Sine_Approx (Reduced_X);

   end Sin;

end Ada.Numerics.Aux;