1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
|
------------------------------------------------------------------------------
-- --
-- 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) 1998-2014, 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. --
-- --
------------------------------------------------------------------------------
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;
|
