1------------------------------------------------------------------------------
2--                                                                          --
3--                         GNAT RUN-TIME COMPONENTS                         --
4--                                                                          --
5--                ADA.NUMERICS.GENERIC_ELEMENTARY_FUNCTIONS                 --
6--                                                                          --
7--                                 S p e c                                  --
8--                                                                          --
9--          Copyright (C) 2012-2018, Free Software Foundation, Inc.         --
10--                                                                          --
11-- This specification is derived from the Ada Reference Manual for use with --
12-- GNAT. The copyright notice above, and the license provisions that follow --
13-- apply solely to the Post aspects that have been added to the spec.       --
14--                                                                          --
15-- GNAT is free software;  you can  redistribute it  and/or modify it under --
16-- terms of the  GNU General Public License as published  by the Free Soft- --
17-- ware  Foundation;  either version 3,  or (at your option) any later ver- --
18-- sion.  GNAT is distributed in the hope that it will be useful, but WITH- --
19-- OUT ANY WARRANTY;  without even the  implied warranty of MERCHANTABILITY --
20-- or FITNESS FOR A PARTICULAR PURPOSE.                                     --
21--                                                                          --
22-- As a special exception under Section 7 of GPL version 3, you are granted --
23-- additional permissions described in the GCC Runtime Library Exception,   --
24-- version 3.1, as published by the Free Software Foundation.               --
25--                                                                          --
26-- You should have received a copy of the GNU General Public License and    --
27-- a copy of the GCC Runtime Library Exception along with this program;     --
28-- see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see    --
29-- <http://www.gnu.org/licenses/>.                                          --
30--                                                                          --
31-- GNAT was originally developed  by the GNAT team at  New York University. --
32-- Extensive contributions were provided by Ada Core Technologies Inc.      --
33--                                                                          --
34------------------------------------------------------------------------------
35
36generic
37   type Float_Type is digits <>;
38
39package Ada.Numerics.Generic_Elementary_Functions with
40  SPARK_Mode => On
41is
42   pragma Pure;
43
44   --  Preconditions in this unit are meant for analysis only, not for run-time
45   --  checking, so that the expected exceptions are raised when calling
46   --  Assert. This is enforced by setting the corresponding assertion policy
47   --  to Ignore. This is done in the generic spec so that it applies to all
48   --  instances.
49
50   pragma Assertion_Policy (Pre => Ignore);
51
52   function Sqrt (X : Float_Type'Base) return Float_Type'Base with
53     Pre  => X >= 0.0,
54     Post => Sqrt'Result >= 0.0
55       and then (if X = 0.0 then Sqrt'Result = 0.0)
56       and then (if X = 1.0 then Sqrt'Result = 1.0)
57
58       --  Finally if X is positive, the result of Sqrt is positive (because
59       --  the sqrt of numbers greater than 1 is greater than or equal to 1,
60       --  and the sqrt of numbers less than 1 is greater than the argument).
61
62       --  This property is useful in particular for static analysis. The
63       --  property that X is positive is not expressed as (X > 0.0), as
64       --  the value X may be held in registers that have larger range and
65       --  precision on some architecture (for example, on x86 using x387
66       --  FPU, as opposed to SSE2). So, it might be possible for X to be
67       --  2.0**(-5000) or so, which could cause the number to compare as
68       --  greater than 0, but Sqrt would still return a zero result.
69
70       --  Note: we use the comparison with Succ (0.0) here because this is
71       --  more amenable to CodePeer analysis than the use of 'Machine.
72
73       and then (if X >= Float_Type'Succ (0.0) then Sqrt'Result > 0.0);
74
75   function Log (X : Float_Type'Base) return Float_Type'Base with
76     Pre  => X > 0.0,
77     Post => (if X = 1.0 then Log'Result = 0.0);
78
79   function Log (X, Base : Float_Type'Base) return Float_Type'Base with
80     Pre  => X > 0.0 and Base > 0.0 and Base /= 1.0,
81     Post => (if X = 1.0 then Log'Result = 0.0);
82
83   function Exp (X : Float_Type'Base) return Float_Type'Base with
84     Post => (if X = 0.0 then Exp'Result = 1.0);
85
86   function "**" (Left, Right : Float_Type'Base) return Float_Type'Base with
87     Pre  => (if Left = 0.0 then Right > 0.0) and Left >= 0.0,
88     Post => "**"'Result >= 0.0
89       and then (if Right = 0.0 then "**"'Result = 1.0)
90       and then (if Right = 1.0 then "**"'Result = Left)
91       and then (if Left  = 1.0 then "**"'Result = 1.0)
92       and then (if Left  = 0.0 then "**"'Result = 0.0);
93
94   function Sin (X : Float_Type'Base) return Float_Type'Base with
95     Post => Sin'Result in -1.0 .. 1.0
96       and then (if X = 0.0 then Sin'Result = 0.0);
97
98   function Sin (X, Cycle : Float_Type'Base) return Float_Type'Base with
99     Pre  => Cycle > 0.0,
100     Post => Sin'Result in -1.0 .. 1.0
101       and then (if X = 0.0 then Sin'Result = 0.0);
102
103   function Cos (X : Float_Type'Base) return Float_Type'Base with
104     Post => Cos'Result in -1.0 .. 1.0
105       and then (if X = 0.0 then Cos'Result = 1.0);
106
107   function Cos (X, Cycle : Float_Type'Base) return Float_Type'Base with
108     Pre  => Cycle > 0.0,
109     Post => Cos'Result in -1.0 .. 1.0
110       and then (if X = 0.0 then Cos'Result = 1.0);
111
112   function Tan (X : Float_Type'Base) return Float_Type'Base with
113     Post => (if X = 0.0 then Tan'Result = 0.0);
114
115   function Tan (X, Cycle : Float_Type'Base) return Float_Type'Base with
116     Pre  => Cycle > 0.0
117       and then abs Float_Type'Base'Remainder (X, Cycle) /= 0.25 * Cycle,
118     Post => (if X = 0.0 then Tan'Result = 0.0);
119
120   function Cot (X : Float_Type'Base) return Float_Type'Base with
121     Pre => X /= 0.0;
122
123   function Cot (X, Cycle : Float_Type'Base) return Float_Type'Base with
124     Pre => Cycle > 0.0
125       and then X /= 0.0
126       and then Float_Type'Base'Remainder (X, Cycle) /= 0.0
127       and then abs Float_Type'Base'Remainder (X, Cycle) = 0.5 * Cycle;
128
129   function Arcsin (X : Float_Type'Base) return Float_Type'Base with
130     Pre  => abs X <= 1.0,
131     Post => (if X = 0.0 then Arcsin'Result = 0.0);
132
133   function Arcsin (X, Cycle : Float_Type'Base) return Float_Type'Base with
134     Pre  => Cycle > 0.0 and abs X <= 1.0,
135     Post => (if X = 0.0 then Arcsin'Result = 0.0);
136
137   function Arccos (X : Float_Type'Base) return Float_Type'Base with
138     Pre  => abs X <= 1.0,
139     Post => (if X = 1.0 then Arccos'Result = 0.0);
140
141   function Arccos (X, Cycle : Float_Type'Base) return Float_Type'Base with
142     Pre  => Cycle > 0.0 and abs X <= 1.0,
143     Post => (if X = 1.0 then Arccos'Result = 0.0);
144
145   function Arctan
146     (Y : Float_Type'Base;
147      X : Float_Type'Base := 1.0) return Float_Type'Base
148   with
149     Pre  => X /= 0.0 or Y /= 0.0,
150     Post => (if X > 0.0 and then Y = 0.0 then Arctan'Result = 0.0);
151
152   function Arctan
153     (Y     : Float_Type'Base;
154      X     : Float_Type'Base := 1.0;
155      Cycle : Float_Type'Base) return Float_Type'Base
156   with
157     Pre  => Cycle > 0.0 and (X /= 0.0 or Y /= 0.0),
158     Post => (if X > 0.0 and then Y = 0.0 then Arctan'Result = 0.0);
159
160   function Arccot
161     (X   : Float_Type'Base;
162      Y   : Float_Type'Base := 1.0) return Float_Type'Base
163   with
164     Pre  => X /= 0.0 or Y /= 0.0,
165     Post => (if X > 0.0 and then Y = 0.0 then Arccot'Result = 0.0);
166
167   function Arccot
168     (X     : Float_Type'Base;
169      Y     : Float_Type'Base := 1.0;
170      Cycle : Float_Type'Base) return Float_Type'Base
171   with
172     Pre  => Cycle > 0.0 and (X /= 0.0 or Y /= 0.0),
173     Post => (if X > 0.0 and then Y = 0.0 then Arccot'Result = 0.0);
174
175   function Sinh (X : Float_Type'Base) return Float_Type'Base with
176     Post => (if X = 0.0 then Sinh'Result = 0.0);
177
178   function Cosh (X : Float_Type'Base) return Float_Type'Base with
179     Post => Cosh'Result >= 1.0
180       and then (if X = 0.0 then Cosh'Result = 1.0);
181
182   function Tanh (X : Float_Type'Base) return Float_Type'Base with
183     Post => Tanh'Result in -1.0 .. 1.0
184       and then (if X = 0.0 then Tanh'Result = 0.0);
185
186   function Coth (X : Float_Type'Base) return Float_Type'Base with
187     Pre  => X /= 0.0,
188     Post => abs Coth'Result >= 1.0;
189
190   function Arcsinh (X : Float_Type'Base) return Float_Type'Base with
191     Post => (if X = 0.0 then Arcsinh'Result = 0.0);
192
193   function Arccosh (X : Float_Type'Base) return Float_Type'Base with
194     Pre  => X >= 1.0,
195     Post => Arccosh'Result >= 0.0
196       and then (if X = 1.0 then Arccosh'Result = 0.0);
197
198   function Arctanh (X : Float_Type'Base) return Float_Type'Base with
199     Pre  => abs X /= 1.0,
200     Post => (if X = 0.0 then Arctanh'Result = 0.0);
201
202   function Arccoth (X : Float_Type'Base) return Float_Type'Base with
203     Pre => X <= 1.0 and abs X /= 1.0;
204
205end Ada.Numerics.Generic_Elementary_Functions;
206