gcc/ada/a-numaux-darwin.adb

   1 ------------------------------------------------------------------------------
   2 --                                                                          --
   3 --                         GNAT RUN-TIME COMPONENTS                         --
   4 --                                                                          --
   5 --                     A D A . N U M E R I C S . A U X                      --
   6 --                                                                          --
   7 --                                 B o d y                                  --
   8 --                          (Apple OS X Version)                            --
   9 --                                                                          --
  10 --          Copyright (C) 1998-2009, Free Software Foundation, Inc.         --
  11 --                                                                          --
  12 -- GNAT is free software;  you can  redistribute it  and/or modify it under --
  13 -- terms of the  GNU General Public License as published  by the Free Soft- --
  14 -- ware  Foundation;  either version 3,  or (at your option) any later ver- --
  15 -- sion.  GNAT is distributed in the hope that it will be useful, but WITH- --
  16 -- OUT ANY WARRANTY;  without even the  implied warranty of MERCHANTABILITY --
  17 -- or FITNESS FOR A PARTICULAR PURPOSE.                                     --
  18 --                                                                          --
  19 -- As a special exception under Section 7 of GPL version 3, you are granted --
  20 -- additional permissions described in the GCC Runtime Library Exception,   --
  21 -- version 3.1, as published by the Free Software Foundation.               --
  22 --                                                                          --
  23 -- You should have received a copy of the GNU General Public License and    --
  24 -- a copy of the GCC Runtime Library Exception along with this program;     --
  25 -- see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see    --
  26 -- <http://www.gnu.org/licenses/>.                                          --
  27 --                                                                          --
  28 -- GNAT was originally developed  by the GNAT team at  New York University. --
  29 -- Extensive contributions were provided by Ada Core Technologies Inc.      --
  30 --                                                                          --
  31 ------------------------------------------------------------------------------
  32
  33 --  File a-numaux.adb <- a-numaux-darwin.adb
  34
  35 package body Ada.Numerics.Aux is
  36
  37    -----------------------
  38    -- Local subprograms --
  39    -----------------------
  40
  41    procedure Reduce (X : in out Double; Q : out Natural);
  42    --  Implements reduction of X by Pi/2. Q is the quadrant of the final
  43    --  result in the range 0 .. 3. The absolute value of X is at most Pi/4.
  44
  45    --  The following three functions implement Chebishev approximations
  46    --  of the trigonometric functions in their reduced domain.
  47    --  These approximations have been computed using Maple.
  48
  49    function Sine_Approx (X : Double) return Double;
  50    function Cosine_Approx (X : Double) return Double;
  51
  52    pragma Inline (Reduce);
  53    pragma Inline (Sine_Approx);
  54    pragma Inline (Cosine_Approx);
  55
  56    function Cosine_Approx (X : Double) return Double is
  57       XX : constant Double := X * X;
  58    begin
  59       return (((((16#8.DC57FBD05F640#E-08 * XX
  60               - 16#4.9F7D00BF25D80#E-06) * XX
  61               + 16#1.A019F7FDEFCC2#E-04) * XX
  62               - 16#5.B05B058F18B20#E-03) * XX
  63               + 16#A.AAAAAAAA73FA8#E-02) * XX
  64               - 16#7.FFFFFFFFFFDE4#E-01) * XX
  65               - 16#3.655E64869ECCE#E-14 + 1.0;
  66    end Cosine_Approx;
  67
  68    function Sine_Approx (X : Double) return Double is
  69       XX : constant Double := X * X;
  70    begin
  71       return (((((16#A.EA2D4ABE41808#E-09 * XX
  72               - 16#6.B974C10F9D078#E-07) * XX
  73               + 16#2.E3BC673425B0E#E-05) * XX
  74               - 16#D.00D00CCA7AF00#E-04) * XX
  75               + 16#2.222222221B190#E-02) * XX
  76               - 16#2.AAAAAAAAAAA44#E-01) * (XX * X) + X;
  77    end Sine_Approx;
  78
  79    ------------
  80    -- Reduce --
  81    ------------
  82
  83    procedure Reduce (X : in out Double; Q : out Natural) is
  84       Half_Pi     : constant := Pi / 2.0;
  85       Two_Over_Pi : constant := 2.0 / Pi;
  86
  87       HM : constant := Integer'Min (Double'Machine_Mantissa / 2, Natural'Size);
  88       M  : constant Double := 0.5 + 2.0**(1 - HM); -- Splitting constant
  89       P1 : constant Double := Double'Leading_Part (Half_Pi, HM);
  90       P2 : constant Double := Double'Leading_Part (Half_Pi - P1, HM);
  91       P3 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2, HM);
  92       P4 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2 - P3, HM);
  93       P5 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2 - P3
  94                                                                  - P4, HM);
  95       P6 : constant Double := Double'Model (Half_Pi - P1 - P2 - P3 - P4 - P5);
  96       K  : Double;
  97
  98    begin
  99       --  For X < 2.0**HM, all products below are computed exactly.
 100       --  Due to cancellation effects all subtractions are exact as well.
 101       --  As no double extended floating-point number has more than 75
 102       --  zeros after the binary point, the result will be the correctly
 103       --  rounded result of X - K * (Pi / 2.0).
 104
 105       K := X * Two_Over_Pi;
 106       while abs K >= 2.0 ** HM loop
 107          K := K * M - (K * M - K);
 108          X :=
 109            (((((X - K * P1) - K * P2) - K * P3) - K * P4) - K * P5) - K * P6;
 110          K := X * Two_Over_Pi;
 111       end loop;
 112
 113       --  If K is not a number (because X was not finite) raise exception
 114
 115       if K /= K then
 116          raise Constraint_Error;
 117       end if;
 118
 119       K := Double'Rounding (K);
 120       Q := Integer (K) mod 4;
 121       X := (((((X - K * P1) - K * P2) - K * P3)
 122                   - K * P4) - K * P5) - K * P6;
 123    end Reduce;
 124
 125    ---------
 126    -- Cos --
 127    ---------
 128
 129    function Cos (X : Double) return Double is
 130       Reduced_X : Double := abs X;
 131       Quadrant  : Natural range 0 .. 3;
 132
 133    begin
 134       if Reduced_X > Pi / 4.0 then
 135          Reduce (Reduced_X, Quadrant);
 136
 137          case Quadrant is
 138             when 0 =>
 139                return Cosine_Approx (Reduced_X);
 140
 141             when 1 =>
 142                return Sine_Approx (-Reduced_X);
 143
 144             when 2 =>
 145                return -Cosine_Approx (Reduced_X);
 146
 147             when 3 =>
 148                return Sine_Approx (Reduced_X);
 149          end case;
 150       end if;
 151
 152       return Cosine_Approx (Reduced_X);
 153    end Cos;
 154
 155    ---------
 156    -- Sin --
 157    ---------
 158
 159    function Sin (X : Double) return Double is
 160       Reduced_X : Double := X;
 161       Quadrant  : Natural range 0 .. 3;
 162
 163    begin
 164       if abs X > Pi / 4.0 then
 165          Reduce (Reduced_X, Quadrant);
 166
 167          case Quadrant is
 168             when 0 =>
 169                return Sine_Approx (Reduced_X);
 170
 171             when 1 =>
 172                return Cosine_Approx (Reduced_X);
 173
 174             when 2 =>
 175                return Sine_Approx (-Reduced_X);
 176
 177             when 3 =>
 178                return -Cosine_Approx (Reduced_X);
 179          end case;
 180       end if;
 181
 182       return Sine_Approx (Reduced_X);
 183    end Sin;
 184
 185 end Ada.Numerics.Aux;