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------------------------------------------------------------------------------
-- --
-- GNAT RUN-TIME COMPONENTS --
-- --
-- A D A . N U M E R I C S . A U X --
-- --
-- B o d y --
-- (Machine Version for x86) --
-- --
-- Copyright (C) 1998-2014, 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. --
-- --
------------------------------------------------------------------------------
with System.Machine_Code; use System.Machine_Code;
package body Ada.Numerics.Aux is
NL : constant String := ASCII.LF & ASCII.HT;
-----------------------
-- Local subprograms --
-----------------------
function Is_Nan (X : Double) return Boolean;
-- Return True iff X is a IEEE NaN value
function Logarithmic_Pow (X, Y : Double) return Double;
-- Implementation of X**Y using Exp and Log functions (binary base)
-- to calculate the exponentiation. This is used by Pow for values
-- for values of Y in the open interval (-0.25, 0.25)
procedure Reduce (X : in out Double; Q : out Natural);
-- Implements reduction of X by Pi/2. Q is the quadrant of the final
-- result in the range 0 .. 3. The absolute value of X is at most Pi.
pragma Inline (Is_Nan);
pragma Inline (Reduce);
--------------------------------
-- Basic Elementary Functions --
--------------------------------
-- This section implements a few elementary functions that are used to
-- build the more complex ones. This ordering enables better inlining.
----------
-- Atan --
----------
function Atan (X : Double) return Double is
Result : Double;
begin
Asm (Template =>
"fld1" & NL
& "fpatan",
Outputs => Double'Asm_Output ("=t", Result),
Inputs => Double'Asm_Input ("0", X));
-- The result value is NaN iff input was invalid
if not (Result = Result) then
raise Argument_Error;
end if;
return Result;
end Atan;
---------
-- Exp --
---------
function Exp (X : Double) return Double is
Result : Double;
begin
Asm (Template =>
"fldl2e " & NL
& "fmulp %%st, %%st(1)" & NL -- X * log2 (E)
& "fld %%st(0) " & NL
& "frndint " & NL -- Integer (X * Log2 (E))
& "fsubr %%st, %%st(1)" & NL -- Fraction (X * Log2 (E))
& "fxch " & NL
& "f2xm1 " & NL -- 2**(...) - 1
& "fld1 " & NL
& "faddp %%st, %%st(1)" & NL -- 2**(Fraction (X * Log2 (E)))
& "fscale " & NL -- E ** X
& "fstp %%st(1) ",
Outputs => Double'Asm_Output ("=t", Result),
Inputs => Double'Asm_Input ("0", X));
return Result;
end Exp;
------------
-- Is_Nan --
------------
function Is_Nan (X : Double) return Boolean is
begin
-- The IEEE NaN values are the only ones that do not equal themselves
return not (X = X);
end Is_Nan;
---------
-- Log --
---------
function Log (X : Double) return Double is
Result : Double;
begin
Asm (Template =>
"fldln2 " & NL
& "fxch " & NL
& "fyl2x " & NL,
Outputs => Double'Asm_Output ("=t", Result),
Inputs => Double'Asm_Input ("0", X));
return Result;
end Log;
------------
-- Reduce --
------------
procedure Reduce (X : in out Double; Q : out Natural) is
Half_Pi : constant := Pi / 2.0;
Two_Over_Pi : constant := 2.0 / Pi;
HM : constant := Integer'Min (Double'Machine_Mantissa / 2, Natural'Size);
M : constant Double := 0.5 + 2.0**(1 - HM); -- Splitting constant
P1 : constant Double := Double'Leading_Part (Half_Pi, HM);
P2 : constant Double := Double'Leading_Part (Half_Pi - P1, HM);
P3 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2, HM);
P4 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2 - P3, HM);
P5 : constant Double := Double'Leading_Part (Half_Pi - P1 - P2 - P3
- P4, HM);
P6 : constant Double := Double'Model (Half_Pi - P1 - P2 - P3 - P4 - P5);
K : Double := X * Two_Over_Pi;
begin
-- For X < 2.0**32, all products below are computed exactly.
-- Due to cancellation effects all subtractions are exact as well.
-- As no double extended floating-point number has more than 75
-- zeros after the binary point, the result will be the correctly
-- rounded result of X - K * (Pi / 2.0).
while abs K >= 2.0**HM loop
K := K * M - (K * M - K);
X := (((((X - K * P1) - K * P2) - K * P3)
- K * P4) - K * P5) - K * P6;
K := X * Two_Over_Pi;
end loop;
if K /= K then
-- K is not a number, because X was not finite
raise Constraint_Error;
end if;
K := Double'Rounding (K);
Q := Integer (K) mod 4;
X := (((((X - K * P1) - K * P2) - K * P3)
- K * P4) - K * P5) - K * P6;
end Reduce;
----------
-- Sqrt --
----------
function Sqrt (X : Double) return Double is
Result : Double;
begin
if X < 0.0 then
raise Argument_Error;
end if;
Asm (Template => "fsqrt",
Outputs => Double'Asm_Output ("=t", Result),
Inputs => Double'Asm_Input ("0", X));
return Result;
end Sqrt;
--------------------------------
-- Other Elementary Functions --
--------------------------------
-- These are built using the previously implemented basic functions
----------
-- Acos --
----------
function Acos (X : Double) return Double is
Result : Double;
begin
Result := 2.0 * Atan (Sqrt ((1.0 - X) / (1.0 + X)));
-- The result value is NaN iff input was invalid
if Is_Nan (Result) then
raise Argument_Error;
end if;
return Result;
end Acos;
----------
-- Asin --
----------
function Asin (X : Double) return Double is
Result : Double;
begin
Result := Atan (X / Sqrt ((1.0 - X) * (1.0 + X)));
-- The result value is NaN iff input was invalid
if Is_Nan (Result) then
raise Argument_Error;
end if;
return Result;
end Asin;
---------
-- Cos --
---------
function Cos (X : Double) return Double is
Reduced_X : Double := abs X;
Result : Double;
Quadrant : Natural range 0 .. 3;
begin
if Reduced_X > Pi / 4.0 then
Reduce (Reduced_X, Quadrant);
case Quadrant is
when 0 =>
Asm (Template => "fcos",
Outputs => Double'Asm_Output ("=t", Result),
Inputs => Double'Asm_Input ("0", Reduced_X));
when 1 =>
Asm (Template => "fsin",
Outputs => Double'Asm_Output ("=t", Result),
Inputs => Double'Asm_Input ("0", -Reduced_X));
when 2 =>
Asm (Template => "fcos ; fchs",
Outputs => Double'Asm_Output ("=t", Result),
Inputs => Double'Asm_Input ("0", Reduced_X));
when 3 =>
Asm (Template => "fsin",
Outputs => Double'Asm_Output ("=t", Result),
Inputs => Double'Asm_Input ("0", Reduced_X));
end case;
else
Asm (Template => "fcos",
Outputs => Double'Asm_Output ("=t", Result),
Inputs => Double'Asm_Input ("0", Reduced_X));
end if;
return Result;
end Cos;
---------------------
-- Logarithmic_Pow --
---------------------
function Logarithmic_Pow (X, Y : Double) return Double is
Result : Double;
begin
Asm (Template => "" -- X : Y
& "fyl2x " & NL -- Y * Log2 (X)
& "fld %%st(0) " & NL -- Y * Log2 (X) : Y * Log2 (X)
& "frndint " & NL -- Int (...) : Y * Log2 (X)
& "fsubr %%st, %%st(1)" & NL -- Int (...) : Fract (...)
& "fxch " & NL -- Fract (...) : Int (...)
& "f2xm1 " & NL -- 2**Fract (...) - 1 : Int (...)
& "fld1 " & NL -- 1 : 2**Fract (...) - 1 : Int (...)
& "faddp %%st, %%st(1)" & NL -- 2**Fract (...) : Int (...)
& "fscale ", -- 2**(Fract (...) + Int (...))
Outputs => Double'Asm_Output ("=t", Result),
Inputs =>
(Double'Asm_Input ("0", X),
Double'Asm_Input ("u", Y)));
return Result;
end Logarithmic_Pow;
---------
-- Pow --
---------
function Pow (X, Y : Double) return Double is
type Mantissa_Type is mod 2**Double'Machine_Mantissa;
-- Modular type that can hold all bits of the mantissa of Double
-- For negative exponents, do divide at the end of the processing
Negative_Y : constant Boolean := Y < 0.0;
Abs_Y : constant Double := abs Y;
-- During this function the following invariant is kept:
-- X ** (abs Y) = Base**(Exp_High + Exp_Mid + Exp_Low) * Factor
Base : Double := X;
Exp_High : Double := Double'Floor (Abs_Y);
Exp_Mid : Double;
Exp_Low : Double;
Exp_Int : Mantissa_Type;
Factor : Double := 1.0;
begin
-- Select algorithm for calculating Pow (integer cases fall through)
if Exp_High >= 2.0**Double'Machine_Mantissa then
-- In case of Y that is IEEE infinity, just raise constraint error
if Exp_High > Double'Safe_Last then
raise Constraint_Error;
end if;
-- Large values of Y are even integers and will stay integer
-- after division by two.
loop
-- Exp_Mid and Exp_Low are zero, so
-- X**(abs Y) = Base ** Exp_High = (Base**2) ** (Exp_High / 2)
Exp_High := Exp_High / 2.0;
Base := Base * Base;
exit when Exp_High < 2.0**Double'Machine_Mantissa;
end loop;
elsif Exp_High /= Abs_Y then
Exp_Low := Abs_Y - Exp_High;
Factor := 1.0;
if Exp_Low /= 0.0 then
-- Exp_Low now is in interval (0.0, 1.0)
-- Exp_Mid := Double'Floor (Exp_Low * 4.0) / 4.0;
Exp_Mid := 0.0;
Exp_Low := Exp_Low - Exp_Mid;
if Exp_Low >= 0.5 then
Factor := Sqrt (X);
Exp_Low := Exp_Low - 0.5; -- exact
if Exp_Low >= 0.25 then
Factor := Factor * Sqrt (Factor);
Exp_Low := Exp_Low - 0.25; -- exact
end if;
elsif Exp_Low >= 0.25 then
Factor := Sqrt (Sqrt (X));
Exp_Low := Exp_Low - 0.25; -- exact
end if;
-- Exp_Low now is in interval (0.0, 0.25)
-- This means it is safe to call Logarithmic_Pow
-- for the remaining part.
Factor := Factor * Logarithmic_Pow (X, Exp_Low);
end if;
elsif X = 0.0 then
return 0.0;
end if;
-- Exp_High is non-zero integer smaller than 2**Double'Machine_Mantissa
Exp_Int := Mantissa_Type (Exp_High);
-- Standard way for processing integer powers > 0
while Exp_Int > 1 loop
if (Exp_Int and 1) = 1 then
-- Base**Y = Base**(Exp_Int - 1) * Exp_Int for Exp_Int > 0
Factor := Factor * Base;
end if;
-- Exp_Int is even and Exp_Int > 0, so
-- Base**Y = (Base**2)**(Exp_Int / 2)
Base := Base * Base;
Exp_Int := Exp_Int / 2;
end loop;
-- Exp_Int = 1 or Exp_Int = 0
if Exp_Int = 1 then
Factor := Base * Factor;
end if;
if Negative_Y then
Factor := 1.0 / Factor;
end if;
return Factor;
end Pow;
---------
-- Sin --
---------
function Sin (X : Double) return Double is
Reduced_X : Double := X;
Result : Double;
Quadrant : Natural range 0 .. 3;
begin
if abs X > Pi / 4.0 then
Reduce (Reduced_X, Quadrant);
case Quadrant is
when 0 =>
Asm (Template => "fsin",
Outputs => Double'Asm_Output ("=t", Result),
Inputs => Double'Asm_Input ("0", Reduced_X));
when 1 =>
Asm (Template => "fcos",
Outputs => Double'Asm_Output ("=t", Result),
Inputs => Double'Asm_Input ("0", Reduced_X));
when 2 =>
Asm (Template => "fsin",
Outputs => Double'Asm_Output ("=t", Result),
Inputs => Double'Asm_Input ("0", -Reduced_X));
when 3 =>
Asm (Template => "fcos ; fchs",
Outputs => Double'Asm_Output ("=t", Result),
Inputs => Double'Asm_Input ("0", Reduced_X));
end case;
else
Asm (Template => "fsin",
Outputs => Double'Asm_Output ("=t", Result),
Inputs => Double'Asm_Input ("0", Reduced_X));
end if;
return Result;
end Sin;
---------
-- Tan --
---------
function Tan (X : Double) return Double is
Reduced_X : Double := X;
Result : Double;
Quadrant : Natural range 0 .. 3;
begin
if abs X > Pi / 4.0 then
Reduce (Reduced_X, Quadrant);
if Quadrant mod 2 = 0 then
Asm (Template => "fptan" & NL
& "ffree %%st(0)" & NL
& "fincstp",
Outputs => Double'Asm_Output ("=t", Result),
Inputs => Double'Asm_Input ("0", Reduced_X));
else
Asm (Template => "fsincos" & NL
& "fdivp %%st, %%st(1)" & NL
& "fchs",
Outputs => Double'Asm_Output ("=t", Result),
Inputs => Double'Asm_Input ("0", Reduced_X));
end if;
else
Asm (Template =>
"fptan " & NL
& "ffree %%st(0) " & NL
& "fincstp ",
Outputs => Double'Asm_Output ("=t", Result),
Inputs => Double'Asm_Input ("0", Reduced_X));
end if;
return Result;
end Tan;
----------
-- Sinh --
----------
function Sinh (X : Double) return Double is
begin
-- Mathematically Sinh (x) is defined to be (Exp (X) - Exp (-X)) / 2.0
if abs X < 25.0 then
return (Exp (X) - Exp (-X)) / 2.0;
else
return Exp (X) / 2.0;
end if;
end Sinh;
----------
-- Cosh --
----------
function Cosh (X : Double) return Double is
begin
-- Mathematically Cosh (X) is defined to be (Exp (X) + Exp (-X)) / 2.0
if abs X < 22.0 then
return (Exp (X) + Exp (-X)) / 2.0;
else
return Exp (X) / 2.0;
end if;
end Cosh;
----------
-- Tanh --
----------
function Tanh (X : Double) return Double is
begin
-- Return the Hyperbolic Tangent of x
-- x -x
-- e - e Sinh (X)
-- Tanh (X) is defined to be ----------- = --------
-- x -x Cosh (X)
-- e + e
if abs X > 23.0 then
return Double'Copy_Sign (1.0, X);
end if;
return 1.0 / (1.0 + Exp (-(2.0 * X))) - 1.0 / (1.0 + Exp (2.0 * X));
end Tanh;
end Ada.Numerics.Aux;