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248 | ------------------------------------------------------------------------------
-- --
-- GNAT COMPILER COMPONENTS --
-- --
-- S Y S T E M . L I B M --
-- --
-- S p e c --
-- --
-- Copyright (C) 2014-2023, 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. --
-- --
------------------------------------------------------------------------------
-- This is the Ada Cert Math specific version of s-libm.ads
with Ada.Numerics;
package System.Libm is
pragma Pure;
Ln_2 : constant := 0.69314_71805_59945_30941_72321_21458_17656_80755;
Ln_3 : constant := 1.09861_22886_68109_69139_52452_36922_52570_46475;
Half_Ln_2 : constant := Ln_2 / 2.0;
Inv_Ln_2 : constant := 1.0 / Ln_2;
Half_Pi : constant := Ada.Numerics.Pi / 2.0;
Third_Pi : constant := Ada.Numerics.Pi / 3.0;
Quarter_Pi : constant := Ada.Numerics.Pi / 4.0;
Sixth_Pi : constant := Ada.Numerics.Pi / 6.0;
Max_Red_Trig_Arg : constant := 0.26 * Ada.Numerics.Pi;
-- This constant representing the maximum absolute value of the reduced
-- argument of the approximation functions for Sin, Cos and Tan. A value
-- slightly larger than Pi / 4 is used. The reduction doesn't reduce to
-- an interval strictly within +-Pi / 4, as that would complicate the
-- reduction code.
One_Over_Pi : constant := 1.0 / Ada.Numerics.Pi;
Two_Over_Pi : constant := 2.0 / Ada.Numerics.Pi;
Sqrt_2 : constant := 1.41421_35623_73095_04880_16887_24209_69807_85696;
Sqrt_3 : constant := 1.73205_08075_68877_29352_74463_41505_87236_69428;
Root16_Half : constant := 0.95760_32806_98573_64693_63056_35147_91544;
-- Sixteenth root of 0.5
Sqrt_Half : constant := 0.70710_67811_86547_52440_08443_62105;
subtype Quadrant is Integer range 0 .. 3;
generic
type T is digits <>;
with function Sqrt (X : T) return T is <>;
with function Approx_Asin (X : T) return T is <>;
function Generic_Acos (X : T) return T;
generic
type T is digits <>;
with function Approx_Atan (X : T) return T is <>;
with function Infinity return T is <>;
function Generic_Atan2 (Y, X : T) return T;
generic
type T is digits <>;
procedure Generic_Pow_Special_Cases
(Left : T;
Right : T;
Is_Special : out Boolean;
Negate : out Boolean;
Result : out T);
generic
type T is private;
Mantissa : Positive;
with function "-" (X : T) return T is <>;
with function "+" (X, Y : T) return T is <>;
with function "-" (X, Y : T) return T is <>;
with function "*" (X, Y : T) return T is <>;
with function "/" (X, Y : T) return T is <>;
with function "<=" (X, Y : T) return Boolean is <>;
with function "abs" (X : T) return T is <>;
with function Exact (X : Long_Long_Float) return T is <>;
with function Maximum_Relative_Error (X : T) return Float is <>;
with function Sqrt (X : T) return T is <>;
package Generic_Approximations is
Epsilon : constant Float := 2.0**(1 - Mantissa);
-- The approximations in this package will work well for single
-- precision floating point types.
function Approx_Asin (X : T) return T
with Pre => abs X <= Exact (0.25),
Post => abs Approx_Asin'Result <= Exact (Half_Pi);
-- @llr Approx_Asin
-- The Approx_Asin function shallapproximate the mathematical function
-- arcsin (sqrt (x)) / sqrt (x) - 1.0 on -0.25 .. 0.25.
--
-- Ada accuracy requirements:
-- The approximation MRE shall be XXX T'Model_Epsilon.
function Approx_Atan (X : T) return T
with Pre => abs X <= Exact (Sqrt_3),
Post => abs (Approx_Atan'Result) <= Exact (Half_Pi) and then
Maximum_Relative_Error (Approx_Atan'Result) <= 2.0 * Epsilon;
-- @llr Approx_Atan
-- The Approx_Atan approximates the mathematical inverse tangent on
-- -Sqrt (3) .. Sqrt (3)
--
-- Accuracy requirements:
-- The approximation MRE is XXX T'Model_Epsilon
function Approx_Cos (X : T) return T
with Pre => abs (X) <= Exact (Max_Red_Trig_Arg),
Post => abs (Approx_Cos'Result) <= Exact (1.0)
and then Maximum_Relative_Error (Approx_Cos'Result)
<= 2.0 * Epsilon;
-- @llr Approx_Cos
-- The Approx_Cos approximates the mathematical cosine on
-- -0.26 * Pi .. 0.26 * Pi
--
-- Accuracy requirements:
-- The approximation MRE is XXX T'Model_Epsilon
function Approx_Exp (X : T) return T
with Pre => abs (X) <= Exact (Ln_2 / 2.0),
Post => Exact (0.0) <= Approx_Exp'Result and then
Maximum_Relative_Error (Approx_Exp'Result) <= 2.0 * Epsilon;
-- @llr Approx_Exp
-- The Approx_Exp function approximates the mathematical exponent on
-- -Ln (2.0) / 2.0 .. Ln (2.0) / 2.0
--
-- Accuracy requirements:
-- The approximation MRE is XXX T'Model_Epsilon
function Approx_Exp2 (X : T) return T
with Pre => abs (X) <= Exact (Ln_2 / 2.0),
Post => Exact (0.0) <= Approx_Exp2'Result and then
Maximum_Relative_Error (Approx_Exp2'Result) <= 2.0 * Epsilon;
-- @llr Approx_Exp2
-- The Approx_Exp2 function approximates the mathematical function
-- (x-> 2.0 ** x) on -Ln (2.0) / 2.0 .. Ln (2.0) / 2.0
--
-- Accuracy requirements:
-- The approximation MRE is XXX T'Model_Epsilon
function Approx_Log (X : T) return T
with
Pre => Exact (Sqrt_2 / 2.0) <= X and then
X <= Exact (Sqrt_2),
Post => Maximum_Relative_Error (Approx_Log'Result) <= 2.0 * Epsilon;
-- @llr Approx_Log
-- The Approx_Log function approximates the mathematical logarithm on
-- Sqrt (2.0) /2.0 .. Sqrt (2.0)
--
-- Accuracy requirements:
-- The approximation MRE is XXX T'Model_Epsilon
function Approx_Power_Log (X : T) return T;
-- @llr Approx_Power_Log
-- The Approx_Power_Log approximates the function
-- (x -> Log ((x-1) / (x+1), base => 2)) on the interval
-- TODO
--
-- Accuracy requirements:
-- The approximation MRE is XXX T'Model_Epsilon
function Approx_Sin (X : T) return T
with Pre => (abs X) <= Exact (Max_Red_Trig_Arg),
Post => abs Approx_Sin'Result <= Exact (1.0) and then
Maximum_Relative_Error (Approx_Sin'Result) <= 2.0 * Epsilon;
-- @llr Approx_Sin
-- The Approx_Sin function approximates the mathematical sine on
-- -0.26 * Pi .. 0.26 * Pi
--
-- Ada accuracy requirements:
-- The approximation MRE is XXX T'Model_Epsilon
function Approx_Sinh (X : T) return T
with Pre => True, -- The original X >= 0.0 and X <= 1.0, fails ???
Post => abs Approx_Sinh'Result <= Exact
((Ada.Numerics.e - 1.0 / Ada.Numerics.e) / 2.0)
and then Maximum_Relative_Error (Approx_Sinh'Result)
<= 2.0 * Epsilon;
-- @llr Approx_Sinh
-- The Approx_Sinh function approximates the mathematical hyperbolic
-- sine on XXX
--
-- Accuracy requirements:
-- The approximation MRE is XXX T'Model_Epsilon
function Approx_Tan (X : T) return T
with Pre => abs X <= Exact (Max_Red_Trig_Arg),
Post =>
Maximum_Relative_Error (Approx_Tan'Result) <= 2.0 * Epsilon;
-- @llr Approx_Tan
-- The Approx_Tan function approximates the mathematical tangent on
-- -0.26 * Pi .. 0.26 * Pi
--
-- Accuracy requirements:
-- The approximation MRE is XXX T'Model_Epsilon
function Approx_Cot (X : T) return T
with Pre => abs X <= Exact (Max_Red_Trig_Arg),
Post =>
Maximum_Relative_Error (Approx_Cot'Result) <= 2.0 * Epsilon;
-- @llr Approx_Cot
-- The Approx_Cot function approximates the mathematical cotangent on
-- -0.26 * Pi .. 0.26 * Pi
--
-- Accuracy requirements:
-- The approximation MRE is XXX T'Model_Epsilon
function Approx_Tanh (X : T) return T
with Pre => Exact (0.0) <= X and then X <= Exact (Ln_3 / 2.0),
Post => abs (Approx_Tanh'Result) <= Exact (Half_Pi)
and then Maximum_Relative_Error (Approx_Tanh'Result)
<= 2.0 * Epsilon;
-- @llr Approx_Tanh
-- The Approx_Tanh function approximates the mathematical hyperbolic
-- tangent on 0.0 .. Ln (3.0) / 2.0
--
-- Accuracy requirements:
-- The approximation MRE is XXX T'Model_Epsilon
function Asin (X : T) return T
with Pre => abs X <= Exact (1.0),
Post => Maximum_Relative_Error (Asin'Result) <= 400.0 * Epsilon;
-- @llr Asin
-- The Asin function has a maximum relative error of 2 epsilon.
end Generic_Approximations;
end System.Libm;
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