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------------------------------------------------------------------------------
-- --
-- GNAT RUN-TIME COMPONENTS --
-- --
-- ADA.NUMERICS.GENERIC_COMPLEX_ELEMENTARY_FUNCTIONS --
-- --
-- B o d y --
-- --
-- Copyright (C) 1992-2021, 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 Ada.Numerics.Generic_Elementary_Functions;
package body Ada.Numerics.Generic_Complex_Elementary_Functions is
package Elementary_Functions is new
Ada.Numerics.Generic_Elementary_Functions (Real'Base);
use Elementary_Functions;
PI : constant := 3.14159_26535_89793_23846_26433_83279_50288_41971;
PI_2 : constant := PI / 2.0;
Sqrt_Two : constant := 1.41421_35623_73095_04880_16887_24209_69807_85696;
Log_Two : constant := 0.69314_71805_59945_30941_72321_21458_17656_80755;
subtype T is Real'Base;
Epsilon : constant T := 2.0 ** (1 - T'Model_Mantissa);
Square_Root_Epsilon : constant T := Sqrt_Two ** (1 - T'Model_Mantissa);
Inv_Square_Root_Epsilon : constant T := Sqrt_Two ** (T'Model_Mantissa - 1);
Root_Root_Epsilon : constant T := Sqrt_Two **
((1 - T'Model_Mantissa) / 2);
Log_Inverse_Epsilon_2 : constant T := T (T'Model_Mantissa - 1) / 2.0;
Complex_Zero : constant Complex := (0.0, 0.0);
Complex_One : constant Complex := (1.0, 0.0);
Complex_I : constant Complex := (0.0, 1.0);
Half_Pi : constant Complex := (PI_2, 0.0);
--------
-- ** --
--------
function "**" (Left : Complex; Right : Complex) return Complex is
begin
if Re (Right) = 0.0
and then Im (Right) = 0.0
and then Re (Left) = 0.0
and then Im (Left) = 0.0
then
raise Argument_Error;
elsif Re (Left) = 0.0
and then Im (Left) = 0.0
and then Re (Right) < 0.0
then
raise Constraint_Error;
elsif Re (Left) = 0.0 and then Im (Left) = 0.0 then
return Left;
elsif Right = (0.0, 0.0) then
return Complex_One;
elsif Re (Right) = 0.0 and then Im (Right) = 0.0 then
return 1.0 + Right;
elsif Re (Right) = 1.0 and then Im (Right) = 0.0 then
return Left;
else
return Exp (Right * Log (Left));
end if;
end "**";
function "**" (Left : Real'Base; Right : Complex) return Complex is
begin
if Re (Right) = 0.0 and then Im (Right) = 0.0 and then Left = 0.0 then
raise Argument_Error;
elsif Left = 0.0 and then Re (Right) < 0.0 then
raise Constraint_Error;
elsif Left = 0.0 then
return Compose_From_Cartesian (Left, 0.0);
elsif Re (Right) = 0.0 and then Im (Right) = 0.0 then
return Complex_One;
elsif Re (Right) = 1.0 and then Im (Right) = 0.0 then
return Compose_From_Cartesian (Left, 0.0);
else
return Exp (Log (Left) * Right);
end if;
end "**";
function "**" (Left : Complex; Right : Real'Base) return Complex is
begin
if Right = 0.0
and then Re (Left) = 0.0
and then Im (Left) = 0.0
then
raise Argument_Error;
elsif Re (Left) = 0.0
and then Im (Left) = 0.0
and then Right < 0.0
then
raise Constraint_Error;
elsif Re (Left) = 0.0 and then Im (Left) = 0.0 then
return Left;
elsif Right = 0.0 then
return Complex_One;
elsif Right = 1.0 then
return Left;
else
return Exp (Right * Log (Left));
end if;
end "**";
------------
-- Arccos --
------------
function Arccos (X : Complex) return Complex is
Result : Complex;
begin
if X = Complex_One then
return Complex_Zero;
elsif abs Re (X) < Square_Root_Epsilon and then
abs Im (X) < Square_Root_Epsilon
then
return Half_Pi - X;
elsif abs Re (X) > Inv_Square_Root_Epsilon or else
abs Im (X) > Inv_Square_Root_Epsilon
then
return -2.0 * Complex_I * Log (Sqrt ((1.0 + X) / 2.0) +
Complex_I * Sqrt ((1.0 - X) / 2.0));
end if;
Result := -Complex_I * Log (X + Complex_I * Sqrt (1.0 - X * X));
if Im (X) = 0.0
and then abs Re (X) <= 1.00
then
Set_Im (Result, Im (X));
end if;
return Result;
end Arccos;
-------------
-- Arccosh --
-------------
function Arccosh (X : Complex) return Complex is
Result : Complex;
begin
if X = Complex_One then
return Complex_Zero;
elsif abs Re (X) < Square_Root_Epsilon and then
abs Im (X) < Square_Root_Epsilon
then
Result := Compose_From_Cartesian (-Im (X), -PI_2 + Re (X));
elsif abs Re (X) > Inv_Square_Root_Epsilon or else
abs Im (X) > Inv_Square_Root_Epsilon
then
Result := Log_Two + Log (X);
else
Result := 2.0 * Log (Sqrt ((1.0 + X) / 2.0) +
Sqrt ((X - 1.0) / 2.0));
end if;
if Re (Result) <= 0.0 then
Result := -Result;
end if;
return Result;
end Arccosh;
------------
-- Arccot --
------------
function Arccot (X : Complex) return Complex is
Xt : Complex;
begin
if abs Re (X) < Square_Root_Epsilon and then
abs Im (X) < Square_Root_Epsilon
then
return Half_Pi - X;
elsif abs Re (X) > 1.0 / Epsilon or else
abs Im (X) > 1.0 / Epsilon
then
Xt := Complex_One / X;
if Re (X) < 0.0 then
Set_Re (Xt, PI - Re (Xt));
return Xt;
else
return Xt;
end if;
end if;
Xt := Complex_I * Log ((X - Complex_I) / (X + Complex_I)) / 2.0;
if Re (Xt) < 0.0 then
Xt := PI + Xt;
end if;
return Xt;
end Arccot;
--------------
-- Arccoth --
--------------
function Arccoth (X : Complex) return Complex is
R : Complex;
begin
if X = (0.0, 0.0) then
return Compose_From_Cartesian (0.0, PI_2);
elsif abs Re (X) < Square_Root_Epsilon
and then abs Im (X) < Square_Root_Epsilon
then
return PI_2 * Complex_I + X;
elsif abs Re (X) > 1.0 / Epsilon or else
abs Im (X) > 1.0 / Epsilon
then
if Im (X) > 0.0 then
return (0.0, 0.0);
else
return PI * Complex_I;
end if;
elsif Im (X) = 0.0 and then Re (X) = 1.0 then
raise Constraint_Error;
elsif Im (X) = 0.0 and then Re (X) = -1.0 then
raise Constraint_Error;
end if;
begin
R := Log ((1.0 + X) / (X - 1.0)) / 2.0;
exception
when Constraint_Error =>
R := (Log (1.0 + X) - Log (X - 1.0)) / 2.0;
end;
if Im (R) < 0.0 then
Set_Im (R, PI + Im (R));
end if;
if Re (X) = 0.0 then
Set_Re (R, Re (X));
end if;
return R;
end Arccoth;
------------
-- Arcsin --
------------
function Arcsin (X : Complex) return Complex is
Result : Complex;
begin
-- For very small argument, sin (x) = x
if abs Re (X) < Square_Root_Epsilon and then
abs Im (X) < Square_Root_Epsilon
then
return X;
elsif abs Re (X) > Inv_Square_Root_Epsilon or else
abs Im (X) > Inv_Square_Root_Epsilon
then
Result := -Complex_I * (Log (Complex_I * X) + Log (2.0 * Complex_I));
if Im (Result) > PI_2 then
Set_Im (Result, PI - Im (X));
elsif Im (Result) < -PI_2 then
Set_Im (Result, -(PI + Im (X)));
end if;
return Result;
end if;
Result := -Complex_I * Log (Complex_I * X + Sqrt (1.0 - X * X));
if Re (X) = 0.0 then
Set_Re (Result, Re (X));
elsif Im (X) = 0.0
and then abs Re (X) <= 1.00
then
Set_Im (Result, Im (X));
end if;
return Result;
end Arcsin;
-------------
-- Arcsinh --
-------------
function Arcsinh (X : Complex) return Complex is
Result : Complex;
begin
if abs Re (X) < Square_Root_Epsilon and then
abs Im (X) < Square_Root_Epsilon
then
return X;
elsif abs Re (X) > Inv_Square_Root_Epsilon or else
abs Im (X) > Inv_Square_Root_Epsilon
then
Result := Log_Two + Log (X); -- may have wrong sign
if (Re (X) < 0.0 and then Re (Result) > 0.0)
or else (Re (X) > 0.0 and then Re (Result) < 0.0)
then
Set_Re (Result, -Re (Result));
end if;
return Result;
end if;
Result := Log (X + Sqrt (1.0 + X * X));
if Re (X) = 0.0 then
Set_Re (Result, Re (X));
elsif Im (X) = 0.0 then
Set_Im (Result, Im (X));
end if;
return Result;
end Arcsinh;
------------
-- Arctan --
------------
function Arctan (X : Complex) return Complex is
begin
if abs Re (X) < Square_Root_Epsilon and then
abs Im (X) < Square_Root_Epsilon
then
return X;
else
return -Complex_I * (Log (1.0 + Complex_I * X)
- Log (1.0 - Complex_I * X)) / 2.0;
end if;
end Arctan;
-------------
-- Arctanh --
-------------
function Arctanh (X : Complex) return Complex is
begin
if abs Re (X) < Square_Root_Epsilon and then
abs Im (X) < Square_Root_Epsilon
then
return X;
else
return (Log (1.0 + X) - Log (1.0 - X)) / 2.0;
end if;
end Arctanh;
---------
-- Cos --
---------
function Cos (X : Complex) return Complex is
begin
return
Compose_From_Cartesian
(Cos (Re (X)) * Cosh (Im (X)),
-(Sin (Re (X)) * Sinh (Im (X))));
end Cos;
----------
-- Cosh --
----------
function Cosh (X : Complex) return Complex is
begin
return
Compose_From_Cartesian
(Cosh (Re (X)) * Cos (Im (X)),
Sinh (Re (X)) * Sin (Im (X)));
end Cosh;
---------
-- Cot --
---------
function Cot (X : Complex) return Complex is
begin
if abs Re (X) < Square_Root_Epsilon and then
abs Im (X) < Square_Root_Epsilon
then
return Complex_One / X;
elsif Im (X) > Log_Inverse_Epsilon_2 then
return -Complex_I;
elsif Im (X) < -Log_Inverse_Epsilon_2 then
return Complex_I;
end if;
return Cos (X) / Sin (X);
end Cot;
----------
-- Coth --
----------
function Coth (X : Complex) return Complex is
begin
if abs Re (X) < Square_Root_Epsilon and then
abs Im (X) < Square_Root_Epsilon
then
return Complex_One / X;
elsif Re (X) > Log_Inverse_Epsilon_2 then
return Complex_One;
elsif Re (X) < -Log_Inverse_Epsilon_2 then
return -Complex_One;
else
return Cosh (X) / Sinh (X);
end if;
end Coth;
---------
-- Exp --
---------
function Exp (X : Complex) return Complex is
ImX : constant Real'Base := Im (X);
EXP_RE_X : constant Real'Base := Exp (Re (X));
begin
return Compose_From_Cartesian (EXP_RE_X * Cos (ImX),
EXP_RE_X * Sin (ImX));
end Exp;
function Exp (X : Imaginary) return Complex is
ImX : constant Real'Base := Im (X);
begin
return Compose_From_Cartesian (Cos (ImX), Sin (ImX));
end Exp;
---------
-- Log --
---------
function Log (X : Complex) return Complex is
ReX : Real'Base;
ImX : Real'Base;
Z : Complex;
begin
if Re (X) = 0.0 and then Im (X) = 0.0 then
raise Constraint_Error;
elsif abs (1.0 - Re (X)) < Root_Root_Epsilon
and then abs Im (X) < Root_Root_Epsilon
then
Z := X;
Set_Re (Z, Re (Z) - 1.0);
return (1.0 - (1.0 / 2.0 -
(1.0 / 3.0 - (1.0 / 4.0) * Z) * Z) * Z) * Z;
end if;
begin
ReX := Log (Modulus (X));
exception
when Constraint_Error =>
ReX := Log (Modulus (X / 2.0)) - Log_Two;
end;
ImX := Arctan (Im (X), Re (X));
if ImX > PI then
ImX := ImX - 2.0 * PI;
end if;
return Compose_From_Cartesian (ReX, ImX);
end Log;
---------
-- Sin --
---------
function Sin (X : Complex) return Complex is
begin
if abs Re (X) < Square_Root_Epsilon
and then
abs Im (X) < Square_Root_Epsilon
then
return X;
end if;
return
Compose_From_Cartesian
(Sin (Re (X)) * Cosh (Im (X)),
Cos (Re (X)) * Sinh (Im (X)));
end Sin;
----------
-- Sinh --
----------
function Sinh (X : Complex) return Complex is
begin
if abs Re (X) < Square_Root_Epsilon and then
abs Im (X) < Square_Root_Epsilon
then
return X;
else
return Compose_From_Cartesian (Sinh (Re (X)) * Cos (Im (X)),
Cosh (Re (X)) * Sin (Im (X)));
end if;
end Sinh;
----------
-- Sqrt --
----------
function Sqrt (X : Complex) return Complex is
ReX : constant Real'Base := Re (X);
ImX : constant Real'Base := Im (X);
XR : constant Real'Base := abs Re (X);
YR : constant Real'Base := abs Im (X);
R : Real'Base;
R_X : Real'Base;
R_Y : Real'Base;
begin
-- Deal with pure real case, see (RM G.1.2(39))
if ImX = 0.0 then
if ReX > 0.0 then
return
Compose_From_Cartesian
(Sqrt (ReX), 0.0);
elsif ReX = 0.0 then
return X;
else
return
Compose_From_Cartesian
(0.0, Real'Copy_Sign (Sqrt (-ReX), ImX));
end if;
elsif ReX = 0.0 then
R_X := Sqrt (YR / 2.0);
if ImX > 0.0 then
return Compose_From_Cartesian (R_X, R_X);
else
return Compose_From_Cartesian (R_X, -R_X);
end if;
else
R := Sqrt (XR ** 2 + YR ** 2);
-- If the square of the modulus overflows, try rescaling the
-- real and imaginary parts. We cannot depend on an exception
-- being raised on all targets.
if R > Real'Base'Last then
raise Constraint_Error;
end if;
-- We are solving the system
-- XR = R_X ** 2 - Y_R ** 2 (1)
-- YR = 2.0 * R_X * R_Y (2)
--
-- The symmetric solution involves square roots for both R_X and
-- R_Y, but it is more accurate to use the square root with the
-- larger argument for either R_X or R_Y, and equation (2) for the
-- other.
if ReX < 0.0 then
R_Y := Sqrt (0.5 * (R - ReX));
R_X := YR / (2.0 * R_Y);
else
R_X := Sqrt (0.5 * (R + ReX));
R_Y := YR / (2.0 * R_X);
end if;
end if;
if Im (X) < 0.0 then -- halve angle, Sqrt of magnitude
R_Y := -R_Y;
end if;
return Compose_From_Cartesian (R_X, R_Y);
exception
when Constraint_Error =>
-- Rescale and try again
R := Modulus (Compose_From_Cartesian (Re (X / 4.0), Im (X / 4.0)));
R_X := 2.0 * Sqrt (0.5 * R + 0.5 * Re (X / 4.0));
R_Y := 2.0 * Sqrt (0.5 * R - 0.5 * Re (X / 4.0));
if Im (X) < 0.0 then -- halve angle, Sqrt of magnitude
R_Y := -R_Y;
end if;
return Compose_From_Cartesian (R_X, R_Y);
end Sqrt;
---------
-- Tan --
---------
function Tan (X : Complex) return Complex is
begin
if abs Re (X) < Square_Root_Epsilon and then
abs Im (X) < Square_Root_Epsilon
then
return X;
elsif Im (X) > Log_Inverse_Epsilon_2 then
return Complex_I;
elsif Im (X) < -Log_Inverse_Epsilon_2 then
return -Complex_I;
else
return Sin (X) / Cos (X);
end if;
end Tan;
----------
-- Tanh --
----------
function Tanh (X : Complex) return Complex is
begin
if abs Re (X) < Square_Root_Epsilon and then
abs Im (X) < Square_Root_Epsilon
then
return X;
elsif Re (X) > Log_Inverse_Epsilon_2 then
return Complex_One;
elsif Re (X) < -Log_Inverse_Epsilon_2 then
return -Complex_One;
else
return Sinh (X) / Cosh (X);
end if;
end Tanh;
end Ada.Numerics.Generic_Complex_Elementary_Functions;