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) 1998-2014, 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 package body Ada.Numerics.Aux is
  34
  35    -----------------------
  36    -- Local subprograms --
  37    -----------------------
  38
  39    procedure Reduce (X : in out Double; Q : out Natural);
  40    --  Implements reduction of X by Pi/2. Q is the quadrant of the final
  41    --  result in the range 0 .. 3. The absolute value of X is at most Pi/4.
  42
  43    --  The following three functions implement Chebishev approximations
  44    --  of the trigonometric functions in their reduced domain.
  45    --  These approximations have been computed using Maple.
  46
  47    function Sine_Approx (X : Double) return Double;
  48    function Cosine_Approx (X : Double) return Double;
  49
  50    pragma Inline (Reduce);
  51    pragma Inline (Sine_Approx);
  52    pragma Inline (Cosine_Approx);
  53
  54    function Cosine_Approx (X : Double) return Double is
  55       XX : constant Double := X * X;
  56    begin
  57       return (((((16#8.DC57FBD05F640#E-08 * XX
  58               - 16#4.9F7D00BF25D80#E-06) * XX
  59               + 16#1.A019F7FDEFCC2#E-04) * XX
  60               - 16#5.B05B058F18B20#E-03) * XX
  61               + 16#A.AAAAAAAA73FA8#E-02) * XX
  62               - 16#7.FFFFFFFFFFDE4#E-01) * XX
  63               - 16#3.655E64869ECCE#E-14 + 1.0;
  64    end Cosine_Approx;
  65
  66    function Sine_Approx (X : Double) return Double is
  67       XX : constant Double := X * X;
  68    begin
  69       return (((((16#A.EA2D4ABE41808#E-09 * XX
  70               - 16#6.B974C10F9D078#E-07) * XX
  71               + 16#2.E3BC673425B0E#E-05) * XX
  72               - 16#D.00D00CCA7AF00#E-04) * XX
  73               + 16#2.222222221B190#E-02) * XX
  74               - 16#2.AAAAAAAAAAA44#E-01) * (XX * X) + X;
  75    end Sine_Approx;
  76
  77    ------------
  78    -- Reduce --
  79    ------------
  80
  81    procedure Reduce (X : in out Double; Q : out Natural) is
  82       Half_Pi     : constant := Pi / 2.0;
  83       Two_Over_Pi : constant := 2.0 / Pi;
  84
  85       HM : constant := Integer'Min (Double'Machine_Mantissa / 2, Natural'Size);
  86       M  : constant Double := 0.5 + 2.0**(1 - HM); -- Splitting constant
  87       P1 : constant Double := Double'Leading_Part (Half_Pi, HM);
  88       P2 : constant Double := Double'Leading_Part (Half_Pi - P1, HM);
  89       P3 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2, HM);
  90       P4 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2 - P3, HM);
  91       P5 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2 - P3
  92                                                                  - P4, HM);
  93       P6 : constant Double := Double'Model (Half_Pi - P1 - P2 - P3 - P4 - P5);
  94       K  : Double;
  95
  96    begin
  97       --  For X < 2.0**HM, all products below are computed exactly.
  98       --  Due to cancellation effects all subtractions are exact as well.
  99       --  As no double extended floating-point number has more than 75
 100       --  zeros after the binary point, the result will be the correctly
 101       --  rounded result of X - K * (Pi / 2.0).
 102
 103       K := X * Two_Over_Pi;
 104       while abs K >= 2.0 ** HM loop
 105          K := K * M - (K * M - K);
 106          X :=
 107            (((((X - K * P1) - K * P2) - K * P3) - K * P4) - K * P5) - K * P6;
 108          K := X * Two_Over_Pi;
 109       end loop;
 110
 111       --  If K is not a number (because X was not finite) raise exception
 112
 113       if K /= K then
 114          raise Constraint_Error;
 115       end if;
 116
 117       K := Double'Rounding (K);
 118       Q := Integer (K) mod 4;
 119       X := (((((X - K * P1) - K * P2) - K * P3)
 120                   - K * P4) - K * P5) - K * P6;
 121    end Reduce;
 122
 123    ---------
 124    -- Cos --
 125    ---------
 126
 127    function Cos (X : Double) return Double is
 128       Reduced_X : Double := abs X;
 129       Quadrant  : Natural range 0 .. 3;
 130
 131    begin
 132       if Reduced_X > Pi / 4.0 then
 133          Reduce (Reduced_X, Quadrant);
 134
 135          case Quadrant is
 136             when 0 =>
 137                return Cosine_Approx (Reduced_X);
 138
 139             when 1 =>
 140                return Sine_Approx (-Reduced_X);
 141
 142             when 2 =>
 143                return -Cosine_Approx (Reduced_X);
 144
 145             when 3 =>
 146                return Sine_Approx (Reduced_X);
 147          end case;
 148       end if;
 149
 150       return Cosine_Approx (Reduced_X);
 151    end Cos;
 152
 153    ---------
 154    -- Sin --
 155    ---------
 156
 157    function Sin (X : Double) return Double is
 158       Reduced_X : Double := X;
 159       Quadrant  : Natural range 0 .. 3;
 160
 161    begin
 162       if abs X > Pi / 4.0 then
 163          Reduce (Reduced_X, Quadrant);
 164
 165          case Quadrant is
 166             when 0 =>
 167                return Sine_Approx (Reduced_X);
 168
 169             when 1 =>
 170                return Cosine_Approx (Reduced_X);
 171
 172             when 2 =>
 173                return Sine_Approx (-Reduced_X);
 174
 175             when 3 =>
 176                return -Cosine_Approx (Reduced_X);
 177          end case;
 178       end if;
 179
 180       return Sine_Approx (Reduced_X);
 181    end Sin;
 182
 183 end Ada.Numerics.Aux;