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
33package 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
183end Ada.Numerics.Aux;
184