------------------------------------------------------------------------------ -- -- -- GNAT RUN-TIME COMPONENTS -- -- -- -- ADA.NUMERICS.GENERIC_ELEMENTARY_FUNCTIONS -- -- -- -- S p e c -- -- -- -- Copyright (C) 2012-2018, 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 -- -- . -- -- -- -- 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 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 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 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;