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   1 ------------------------------------------------------------------------------
   2 --                                                                          --
   3 --                         GNAT RUN-TIME COMPONENTS                         --
   4 --                                                                          --
   5 --                ADA.NUMERICS.GENERIC_ELEMENTARY_FUNCTIONS                 --
   6 --                                                                          --
   7 --                                 S p e c                                  --
   8 --                                                                          --
   9 --          Copyright (C) 2012-2015, 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
  36 generic
  37    type Float_Type is digits <>;
  38
  39 package Ada.Numerics.Generic_Elementary_Functions is
  40    pragma Pure;
  41
  42    function Sqrt (X : Float_Type'Base) return Float_Type'Base with
  43      Post => Sqrt'Result >= 0.0
  44        and then (if X = 0.0 then Sqrt'Result = 0.0)
  45        and then (if X = 1.0 then Sqrt'Result = 1.0)
  46
  47        --  Finally if X is positive, the result of Sqrt is positive (because
  48        --  the sqrt of numbers greater than 1 is greater than or equal to 1,
  49        --  and the sqrt of numbers less than 1 is greater than the argument).
  50
  51        --  This property is useful in particular for static analysis. The
  52        --  property that X is positive is not expressed as (X > 0.0), as
  53        --  the value X may be held in registers that have larger range and
  54        --  precision on some architecture (for example, on x86 using x387
  55        --  FPU, as opposed to SSE2). So, it might be possible for X to be
  56        --  2.0**(-5000) or so, which could cause the number to compare as
  57        --  greater than 0, but Sqrt would still return a zero result.
  58
  59        --  Note: we use the comparison with Succ (0.0) here because this is
  60        --  more amenable to CodePeer analysis than the use of 'Machine.
  61
  62        and then (if X >= Float_Type'Succ (0.0) then Sqrt'Result > 0.0);
  63
  64    function Log (X : Float_Type'Base) return Float_Type'Base with
  65      Post => (if X = 1.0 then Log'Result = 0.0);
  66
  67    function Log (X, Base : Float_Type'Base) return Float_Type'Base with
  68      Post => (if X = 1.0 then Log'Result = 0.0);
  69
  70    function Exp (X : Float_Type'Base) return Float_Type'Base with
  71      Post => (if X = 0.0 then Exp'Result = 1.0);
  72
  73    function "**" (Left, Right : Float_Type'Base) return Float_Type'Base with
  74      Post => "**"'Result >= 0.0
  75        and then (if Right = 0.0 then "**"'Result = 1.0)
  76        and then (if Right = 1.0 then "**"'Result = Left)
  77        and then (if Left  = 1.0 then "**"'Result = 1.0)
  78        and then (if Left  = 0.0 then "**"'Result = 0.0);
  79
  80    function Sin (X : Float_Type'Base) return Float_Type'Base with
  81      Post => Sin'Result in -1.0 .. 1.0
  82        and then (if X = 0.0 then Sin'Result = 0.0);
  83
  84    function Sin (X, Cycle : Float_Type'Base) return Float_Type'Base with
  85      Post => Sin'Result in -1.0 .. 1.0
  86        and then (if X = 0.0 then Sin'Result = 0.0);
  87
  88    function Cos (X : Float_Type'Base) return Float_Type'Base with
  89      Post => Cos'Result in -1.0 .. 1.0
  90        and then (if X = 0.0 then Cos'Result = 1.0);
  91
  92    function Cos (X, Cycle : Float_Type'Base) return Float_Type'Base with
  93      Post => Cos'Result in -1.0 .. 1.0
  94        and then (if X = 0.0 then Cos'Result = 1.0);
  95
  96    function Tan (X : Float_Type'Base) return Float_Type'Base with
  97      Post => (if X = 0.0 then Tan'Result = 0.0);
  98
  99    function Tan (X, Cycle : Float_Type'Base) return Float_Type'Base with
 100      Post => (if X = 0.0 then Tan'Result = 0.0);
 101
 102    function Cot (X : Float_Type'Base) return Float_Type'Base;
 103
 104    function Cot (X, Cycle : Float_Type'Base) return Float_Type'Base;
 105
 106    function Arcsin (X : Float_Type'Base) return Float_Type'Base with
 107      Post => (if X = 0.0 then Arcsin'Result = 0.0);
 108
 109    function Arcsin (X, Cycle : Float_Type'Base) return Float_Type'Base with
 110      Post => (if X = 0.0 then Arcsin'Result = 0.0);
 111
 112    function Arccos (X : Float_Type'Base) return Float_Type'Base with
 113      Post => (if X = 1.0 then Arccos'Result = 0.0);
 114
 115    function Arccos (X, Cycle : Float_Type'Base) return Float_Type'Base with
 116      Post => (if X = 1.0 then Arccos'Result = 0.0);
 117
 118    function Arctan
 119      (Y : Float_Type'Base;
 120       X : Float_Type'Base := 1.0) return Float_Type'Base
 121    with
 122      Post => (if X > 0.0 and then Y = 0.0 then Arctan'Result = 0.0);
 123
 124    function Arctan
 125      (Y     : Float_Type'Base;
 126       X     : Float_Type'Base := 1.0;
 127       Cycle : Float_Type'Base) return Float_Type'Base
 128    with
 129      Post => (if X > 0.0 and then Y = 0.0 then Arctan'Result = 0.0);
 130
 131    function Arccot
 132      (X   : Float_Type'Base;
 133       Y   : Float_Type'Base := 1.0) return Float_Type'Base
 134    with
 135      Post => (if X > 0.0 and then Y = 0.0 then Arccot'Result = 0.0);
 136
 137    function Arccot
 138      (X     : Float_Type'Base;
 139       Y     : Float_Type'Base := 1.0;
 140       Cycle : Float_Type'Base) return Float_Type'Base
 141    with
 142      Post => (if X > 0.0 and then Y = 0.0 then Arccot'Result = 0.0);
 143
 144    function Sinh (X : Float_Type'Base) return Float_Type'Base with
 145      Post => (if X = 0.0 then Sinh'Result = 0.0);
 146
 147    function Cosh (X : Float_Type'Base) return Float_Type'Base with
 148      Post => Cosh'Result >= 1.0
 149        and then (if X = 0.0 then Cosh'Result = 1.0);
 150
 151    function Tanh (X : Float_Type'Base) return Float_Type'Base with
 152      Post => Tanh'Result in -1.0 .. 1.0
 153        and then (if X = 0.0 then Tanh'Result = 0.0);
 154
 155    function Coth (X : Float_Type'Base) return Float_Type'Base with
 156      Post => abs Coth'Result >= 1.0;
 157
 158    function Arcsinh (X : Float_Type'Base) return Float_Type'Base with
 159      Post => (if X = 0.0 then Arcsinh'Result = 0.0);
 160
 161    function Arccosh (X : Float_Type'Base) return Float_Type'Base with
 162      Post => Arccosh'Result >= 0.0
 163        and then (if X = 1.0 then Arccosh'Result = 0.0);
 164
 165    function Arctanh (X : Float_Type'Base) return Float_Type'Base with
 166      Post => (if X = 0.0 then Arctanh'Result = 0.0);
 167
 168    function Arccoth (X : Float_Type'Base) return Float_Type'Base;
 169
 170 end Ada.Numerics.Generic_Elementary_Functions;