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------------------------------------------------------------------------------
-- --
-- GNAT RUN-TIME COMPONENTS --
-- --
-- ADA.NUMERICS.GENERIC_ELEMENTARY_FUNCTIONS --
-- --
-- S p e c --
-- --
-- Copyright (C) 2012-2024, Free Software Foundation, Inc. --
-- --
-- This specification is derived from the Ada Reference Manual for use with --
-- GNAT. The copyright notice above, and the license provisions that follow --
-- apply solely to the Post aspects that have been added to the spec. --
-- --
-- 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. --
-- --
------------------------------------------------------------------------------
generic
type Float_Type is digits <>;
package Ada.Numerics.Generic_Elementary_Functions with
SPARK_Mode => On,
Always_Terminates
is
pragma Pure;
-- Preconditions in this unit are meant for analysis only, not for run-time
-- checking, so that the expected exceptions are raised when calling
-- Assert. This is enforced by setting the corresponding assertion policy
-- to Ignore. This is done in the generic spec so that it applies to all
-- instances.
pragma Assertion_Policy (Pre => Ignore);
function Sqrt (X : Float_Type'Base) return Float_Type'Base with
Pre => X >= 0.0,
Post => Sqrt'Result >= 0.0
and then (if X = 0.0 then Sqrt'Result = 0.0)
and then (if X = 1.0 then Sqrt'Result = 1.0)
-- Finally if X is positive, the result of Sqrt is positive (because
-- the sqrt of numbers greater than 1 is greater than or equal to 1,
-- and the sqrt of numbers less than 1 is greater than the argument).
-- This property is useful in particular for static analysis. The
-- property that X is positive is not expressed as (X > 0.0), as
-- the value X may be held in registers that have larger range and
-- precision on some architecture (for example, on x86 using x387
-- FPU, as opposed to SSE2). So, it might be possible for X to be
-- 2.0**(-5000) or so, which could cause the number to compare as
-- greater than 0, but Sqrt would still return a zero result.
-- Note: we use the comparison with Succ (0.0) here because this is
-- more amenable to CodePeer analysis than the use of 'Machine.
and then (if X >= Float_Type'Succ (0.0) then Sqrt'Result > 0.0);
function Log (X : Float_Type'Base) return Float_Type'Base with
Pre => X > 0.0,
Post => (if X = 1.0 then Log'Result = 0.0);
function Log (X, Base : Float_Type'Base) return Float_Type'Base with
Pre => X > 0.0 and Base > 0.0 and Base /= 1.0,
Post => (if X = 1.0 then Log'Result = 0.0);
function Exp (X : Float_Type'Base) return Float_Type'Base with
Post => (if X = 0.0 then Exp'Result = 1.0);
function "**" (Left, Right : Float_Type'Base) return Float_Type'Base with
Pre => (if Left = 0.0 then Right > 0.0) and Left >= 0.0,
Post => "**"'Result >= 0.0
and then (if Right = 0.0 then "**"'Result = 1.0)
and then (if Right = 1.0 then "**"'Result = Left)
and then (if Left = 1.0 then "**"'Result = 1.0)
and then (if Left = 0.0 then "**"'Result = 0.0);
function Sin (X : Float_Type'Base) return Float_Type'Base with
Inline,
Post => Sin'Result in -1.0 .. 1.0
and then (if X = 0.0 then Sin'Result = 0.0);
function Sin (X, Cycle : Float_Type'Base) return Float_Type'Base with
Pre => Cycle > 0.0,
Post => Sin'Result in -1.0 .. 1.0
and then (if X = 0.0 then Sin'Result = 0.0);
function Cos (X : Float_Type'Base) return Float_Type'Base with
Inline,
Post => Cos'Result in -1.0 .. 1.0
and then (if X = 0.0 then Cos'Result = 1.0);
function Cos (X, Cycle : Float_Type'Base) return Float_Type'Base with
Pre => Cycle > 0.0,
Post => Cos'Result in -1.0 .. 1.0
and then (if X = 0.0 then Cos'Result = 1.0);
function Tan (X : Float_Type'Base) return Float_Type'Base with
Post => (if X = 0.0 then Tan'Result = 0.0);
function Tan (X, Cycle : Float_Type'Base) return Float_Type'Base with
Pre => Cycle > 0.0
and then abs Float_Type'Base'Remainder (X, Cycle) /= 0.25 * Cycle,
Post => (if X = 0.0 then Tan'Result = 0.0);
function Cot (X : Float_Type'Base) return Float_Type'Base with
Pre => X /= 0.0;
function Cot (X, Cycle : Float_Type'Base) return Float_Type'Base with
Pre => Cycle > 0.0
and then X /= 0.0
and then Float_Type'Base'Remainder (X, Cycle) /= 0.0
and then abs Float_Type'Base'Remainder (X, Cycle) /= 0.5 * Cycle;
function Arcsin (X : Float_Type'Base) return Float_Type'Base with
Pre => abs X <= 1.0,
Post => (if X = 0.0 then Arcsin'Result = 0.0);
function Arcsin (X, Cycle : Float_Type'Base) return Float_Type'Base with
Pre => Cycle > 0.0 and abs X <= 1.0,
Post => (if X = 0.0 then Arcsin'Result = 0.0);
function Arccos (X : Float_Type'Base) return Float_Type'Base with
Pre => abs X <= 1.0,
Post => (if X = 1.0 then Arccos'Result = 0.0);
function Arccos (X, Cycle : Float_Type'Base) return Float_Type'Base with
Pre => Cycle > 0.0 and abs X <= 1.0,
Post => (if X = 1.0 then Arccos'Result = 0.0);
function Arctan
(Y : Float_Type'Base;
X : Float_Type'Base := 1.0) return Float_Type'Base
with
Pre => X /= 0.0 or Y /= 0.0,
Post => (if X > 0.0 and then Y = 0.0 then Arctan'Result = 0.0);
function Arctan
(Y : Float_Type'Base;
X : Float_Type'Base := 1.0;
Cycle : Float_Type'Base) return Float_Type'Base
with
Pre => Cycle > 0.0 and (X /= 0.0 or Y /= 0.0),
Post => (if X > 0.0 and then Y = 0.0 then Arctan'Result = 0.0);
function Arccot
(X : Float_Type'Base;
Y : Float_Type'Base := 1.0) return Float_Type'Base
with
Pre => X /= 0.0 or Y /= 0.0,
Post => (if X > 0.0 and then Y = 0.0 then Arccot'Result = 0.0);
function Arccot
(X : Float_Type'Base;
Y : Float_Type'Base := 1.0;
Cycle : Float_Type'Base) return Float_Type'Base
with
Pre => Cycle > 0.0 and (X /= 0.0 or Y /= 0.0),
Post => (if X > 0.0 and then Y = 0.0 then Arccot'Result = 0.0);
function Sinh (X : Float_Type'Base) return Float_Type'Base with
Post => (if X = 0.0 then Sinh'Result = 0.0);
function Cosh (X : Float_Type'Base) return Float_Type'Base with
Post => Cosh'Result >= 1.0
and then (if X = 0.0 then Cosh'Result = 1.0);
function Tanh (X : Float_Type'Base) return Float_Type'Base with
Post => Tanh'Result in -1.0 .. 1.0
and then (if X = 0.0 then Tanh'Result = 0.0);
function Coth (X : Float_Type'Base) return Float_Type'Base with
Pre => X /= 0.0,
Post => abs Coth'Result >= 1.0;
function Arcsinh (X : Float_Type'Base) return Float_Type'Base with
Post => (if X = 0.0 then Arcsinh'Result = 0.0);
function Arccosh (X : Float_Type'Base) return Float_Type'Base with
Pre => X >= 1.0,
Post => Arccosh'Result >= 0.0
and then (if X = 1.0 then Arccosh'Result = 0.0);
function Arctanh (X : Float_Type'Base) return Float_Type'Base with
Pre => abs X < 1.0,
Post => (if X = 0.0 then Arctanh'Result = 0.0);
function Arccoth (X : Float_Type'Base) return Float_Type'Base with
Pre => abs X > 1.0;
end Ada.Numerics.Generic_Elementary_Functions;
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