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) 1998-2018, 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   function Is_Nan (X : Double) return Boolean;
40   --  Return True iff X is a IEEE NaN value
41
42   procedure Reduce (X : in out Double; Q : out Natural);
43   --  Implement reduction of X by Pi/2. Q is the quadrant of the final
44   --  result in the range 0..3. The absolute value of X is at most Pi/4.
45   --  It is needed to avoid a loss of accuracy for sin near Pi and cos
46   --  near Pi/2 due to the use of an insufficiently precise value of Pi
47   --  in the range reduction.
48
49   --  The following two functions implement Chebishev approximations
50   --  of the trigonometric functions in their reduced domain.
51   --  These approximations have been computed using Maple.
52
53   function Sine_Approx (X : Double) return Double;
54   function Cosine_Approx (X : Double) return Double;
55
56   pragma Inline (Reduce);
57   pragma Inline (Sine_Approx);
58   pragma Inline (Cosine_Approx);
59
60   -------------------
61   -- Cosine_Approx --
62   -------------------
63
64   function Cosine_Approx (X : Double) return Double is
65      XX : constant Double := X * X;
66   begin
67      return (((((16#8.DC57FBD05F640#E-08 * XX
68              - 16#4.9F7D00BF25D80#E-06) * XX
69              + 16#1.A019F7FDEFCC2#E-04) * XX
70              - 16#5.B05B058F18B20#E-03) * XX
71              + 16#A.AAAAAAAA73FA8#E-02) * XX
72              - 16#7.FFFFFFFFFFDE4#E-01) * XX
73              - 16#3.655E64869ECCE#E-14 + 1.0;
74   end Cosine_Approx;
75
76   -----------------
77   -- Sine_Approx --
78   -----------------
79
80   function Sine_Approx (X : Double) return Double is
81      XX : constant Double := X * X;
82   begin
83      return (((((16#A.EA2D4ABE41808#E-09 * XX
84              - 16#6.B974C10F9D078#E-07) * XX
85              + 16#2.E3BC673425B0E#E-05) * XX
86              - 16#D.00D00CCA7AF00#E-04) * XX
87              + 16#2.222222221B190#E-02) * XX
88              - 16#2.AAAAAAAAAAA44#E-01) * (XX * X) + X;
89   end Sine_Approx;
90
91   ------------
92   -- Is_Nan --
93   ------------
94
95   function Is_Nan (X : Double) return Boolean is
96   begin
97      --  The IEEE NaN values are the only ones that do not equal themselves
98
99      return X /= X;
100   end Is_Nan;
101
102   ------------
103   -- Reduce --
104   ------------
105
106   procedure Reduce (X : in out Double; Q : out Natural) is
107      Half_Pi     : constant := Pi / 2.0;
108      Two_Over_Pi : constant := 2.0 / Pi;
109
110      HM : constant := Integer'Min (Double'Machine_Mantissa / 2, Natural'Size);
111      M  : constant Double := 0.5 + 2.0**(1 - HM); -- Splitting constant
112      P1 : constant Double := Double'Leading_Part (Half_Pi, HM);
113      P2 : constant Double := Double'Leading_Part (Half_Pi - P1, HM);
114      P3 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2, HM);
115      P4 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2 - P3, HM);
116      P5 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2 - P3
117                                                                 - P4, HM);
118      P6 : constant Double := Double'Model (Half_Pi - P1 - P2 - P3 - P4 - P5);
119      K  : Double;
120      R  : Integer;
121
122   begin
123      --  For X < 2.0**HM, all products below are computed exactly.
124      --  Due to cancellation effects all subtractions are exact as well.
125      --  As no double extended floating-point number has more than 75
126      --  zeros after the binary point, the result will be the correctly
127      --  rounded result of X - K * (Pi / 2.0).
128
129      K := X * Two_Over_Pi;
130      while abs K >= 2.0**HM loop
131         K := K * M - (K * M - K);
132         X :=
133           (((((X - K * P1) - K * P2) - K * P3) - K * P4) - K * P5) - K * P6;
134         K := X * Two_Over_Pi;
135      end loop;
136
137      --  If K is not a number (because X was not finite) raise exception
138
139      if Is_Nan (K) then
140         raise Constraint_Error;
141      end if;
142
143      --  Go through an integer temporary so as to use machine instructions
144
145      R := Integer (Double'Rounding (K));
146      Q := R mod 4;
147      K := Double (R);
148      X := (((((X - K * P1) - K * P2) - K * P3) - K * P4) - K * P5) - K * P6;
149   end Reduce;
150
151   ---------
152   -- Cos --
153   ---------
154
155   function Cos (X : Double) return Double is
156      Reduced_X : Double := abs X;
157      Quadrant  : Natural range 0 .. 3;
158
159   begin
160      if Reduced_X > Pi / 4.0 then
161         Reduce (Reduced_X, Quadrant);
162
163         case Quadrant is
164            when 0 =>
165               return Cosine_Approx (Reduced_X);
166
167            when 1 =>
168               return Sine_Approx (-Reduced_X);
169
170            when 2 =>
171               return -Cosine_Approx (Reduced_X);
172
173            when 3 =>
174               return Sine_Approx (Reduced_X);
175         end case;
176      end if;
177
178      return Cosine_Approx (Reduced_X);
179   end Cos;
180
181   ---------
182   -- Sin --
183   ---------
184
185   function Sin (X : Double) return Double is
186      Reduced_X : Double := X;
187      Quadrant  : Natural range 0 .. 3;
188
189   begin
190      if abs X > Pi / 4.0 then
191         Reduce (Reduced_X, Quadrant);
192
193         case Quadrant is
194            when 0 =>
195               return Sine_Approx (Reduced_X);
196
197            when 1 =>
198               return Cosine_Approx (Reduced_X);
199
200            when 2 =>
201               return Sine_Approx (-Reduced_X);
202
203            when 3 =>
204               return -Cosine_Approx (Reduced_X);
205         end case;
206      end if;
207
208      return Sine_Approx (Reduced_X);
209   end Sin;
210
211end Ada.Numerics.Aux;
212