1 ------------------------------------------------------------------------------
3 -- GNAT RUN-TIME COMPONENTS --
5 -- ADA.NUMERICS.GENERIC_COMPLEX_ELEMENTARY_FUNCTIONS --
9 -- Copyright (C) 1992-2009, Free Software Foundation, Inc. --
11 -- GNAT is free software; you can redistribute it and/or modify it under --
12 -- terms of the GNU General Public License as published by the Free Soft- --
13 -- ware Foundation; either version 3, or (at your option) any later ver- --
14 -- sion. GNAT is distributed in the hope that it will be useful, but WITH- --
15 -- OUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY --
16 -- or FITNESS FOR A PARTICULAR PURPOSE. --
18 -- As a special exception under Section 7 of GPL version 3, you are granted --
19 -- additional permissions described in the GCC Runtime Library Exception, --
20 -- version 3.1, as published by the Free Software Foundation. --
22 -- You should have received a copy of the GNU General Public License and --
23 -- a copy of the GCC Runtime Library Exception along with this program; --
24 -- see the files COPYING3 and COPYING.RUNTIME respectively. If not, see --
25 -- <http://www.gnu.org/licenses/>. --
27 -- GNAT was originally developed by the GNAT team at New York University. --
28 -- Extensive contributions were provided by Ada Core Technologies Inc. --
30 ------------------------------------------------------------------------------
32 with Ada.Numerics.Generic_Elementary_Functions;
34 package body Ada.Numerics.Generic_Complex_Elementary_Functions is
36 package Elementary_Functions is new
37 Ada.Numerics.Generic_Elementary_Functions (Real'Base);
38 use Elementary_Functions;
40 PI : constant := 3.14159_26535_89793_23846_26433_83279_50288_41971;
41 PI_2 : constant := PI / 2.0;
42 Sqrt_Two : constant := 1.41421_35623_73095_04880_16887_24209_69807_85696;
43 Log_Two : constant := 0.69314_71805_59945_30941_72321_21458_17656_80755;
45 subtype T is Real'Base;
47 Epsilon : constant T := 2.0 ** (1 - T'Model_Mantissa);
48 Square_Root_Epsilon : constant T := Sqrt_Two ** (1 - T'Model_Mantissa);
49 Inv_Square_Root_Epsilon : constant T := Sqrt_Two ** (T'Model_Mantissa - 1);
50 Root_Root_Epsilon : constant T := Sqrt_Two **
51 ((1 - T'Model_Mantissa) / 2);
52 Log_Inverse_Epsilon_2 : constant T := T (T'Model_Mantissa - 1) / 2.0;
54 Complex_Zero : constant Complex := (0.0, 0.0);
55 Complex_One : constant Complex := (1.0, 0.0);
56 Complex_I : constant Complex := (0.0, 1.0);
57 Half_Pi : constant Complex := (PI_2, 0.0);
63 function "**" (Left : Complex; Right : Complex) return Complex is
66 and then Im (Right) = 0.0
67 and then Re (Left) = 0.0
68 and then Im (Left) = 0.0
73 and then Im (Left) = 0.0
74 and then Re (Right) < 0.0
76 raise Constraint_Error;
78 elsif Re (Left) = 0.0 and then Im (Left) = 0.0 then
81 elsif Right = (0.0, 0.0) then
84 elsif Re (Right) = 0.0 and then Im (Right) = 0.0 then
87 elsif Re (Right) = 1.0 and then Im (Right) = 0.0 then
91 return Exp (Right * Log (Left));
95 function "**" (Left : Real'Base; Right : Complex) return Complex is
97 if Re (Right) = 0.0 and then Im (Right) = 0.0 and then Left = 0.0 then
100 elsif Left = 0.0 and then Re (Right) < 0.0 then
101 raise Constraint_Error;
103 elsif Left = 0.0 then
104 return Compose_From_Cartesian (Left, 0.0);
106 elsif Re (Right) = 0.0 and then Im (Right) = 0.0 then
109 elsif Re (Right) = 1.0 and then Im (Right) = 0.0 then
110 return Compose_From_Cartesian (Left, 0.0);
113 return Exp (Log (Left) * Right);
117 function "**" (Left : Complex; Right : Real'Base) return Complex is
120 and then Re (Left) = 0.0
121 and then Im (Left) = 0.0
123 raise Argument_Error;
125 elsif Re (Left) = 0.0
126 and then Im (Left) = 0.0
129 raise Constraint_Error;
131 elsif Re (Left) = 0.0 and then Im (Left) = 0.0 then
134 elsif Right = 0.0 then
137 elsif Right = 1.0 then
141 return Exp (Right * Log (Left));
149 function Arccos (X : Complex) return Complex is
153 if X = Complex_One then
156 elsif abs Re (X) < Square_Root_Epsilon and then
157 abs Im (X) < Square_Root_Epsilon
161 elsif abs Re (X) > Inv_Square_Root_Epsilon or else
162 abs Im (X) > Inv_Square_Root_Epsilon
164 return -2.0 * Complex_I * Log (Sqrt ((1.0 + X) / 2.0) +
165 Complex_I * Sqrt ((1.0 - X) / 2.0));
168 Result := -Complex_I * Log (X + Complex_I * Sqrt (1.0 - X * X));
171 and then abs Re (X) <= 1.00
173 Set_Im (Result, Im (X));
183 function Arccosh (X : Complex) return Complex is
187 if X = Complex_One then
190 elsif abs Re (X) < Square_Root_Epsilon and then
191 abs Im (X) < Square_Root_Epsilon
193 Result := Compose_From_Cartesian (-Im (X), -PI_2 + Re (X));
195 elsif abs Re (X) > Inv_Square_Root_Epsilon or else
196 abs Im (X) > Inv_Square_Root_Epsilon
198 Result := Log_Two + Log (X);
201 Result := 2.0 * Log (Sqrt ((1.0 + X) / 2.0) +
202 Sqrt ((X - 1.0) / 2.0));
205 if Re (Result) <= 0.0 then
216 function Arccot (X : Complex) return Complex is
220 if abs Re (X) < Square_Root_Epsilon and then
221 abs Im (X) < Square_Root_Epsilon
225 elsif abs Re (X) > 1.0 / Epsilon or else
226 abs Im (X) > 1.0 / Epsilon
228 Xt := Complex_One / X;
231 Set_Re (Xt, PI - Re (Xt));
238 Xt := Complex_I * Log ((X - Complex_I) / (X + Complex_I)) / 2.0;
240 if Re (Xt) < 0.0 then
251 function Arccoth (X : Complex) return Complex is
255 if X = (0.0, 0.0) then
256 return Compose_From_Cartesian (0.0, PI_2);
258 elsif abs Re (X) < Square_Root_Epsilon
259 and then abs Im (X) < Square_Root_Epsilon
261 return PI_2 * Complex_I + X;
263 elsif abs Re (X) > 1.0 / Epsilon or else
264 abs Im (X) > 1.0 / Epsilon
269 return PI * Complex_I;
272 elsif Im (X) = 0.0 and then Re (X) = 1.0 then
273 raise Constraint_Error;
275 elsif Im (X) = 0.0 and then Re (X) = -1.0 then
276 raise Constraint_Error;
280 R := Log ((1.0 + X) / (X - 1.0)) / 2.0;
283 when Constraint_Error =>
284 R := (Log (1.0 + X) - Log (X - 1.0)) / 2.0;
288 Set_Im (R, PI + Im (R));
302 function Arcsin (X : Complex) return Complex is
306 -- For very small argument, sin (x) = x
308 if abs Re (X) < Square_Root_Epsilon and then
309 abs Im (X) < Square_Root_Epsilon
313 elsif abs Re (X) > Inv_Square_Root_Epsilon or else
314 abs Im (X) > Inv_Square_Root_Epsilon
316 Result := -Complex_I * (Log (Complex_I * X) + Log (2.0 * Complex_I));
318 if Im (Result) > PI_2 then
319 Set_Im (Result, PI - Im (X));
321 elsif Im (Result) < -PI_2 then
322 Set_Im (Result, -(PI + Im (X)));
328 Result := -Complex_I * Log (Complex_I * X + Sqrt (1.0 - X * X));
331 Set_Re (Result, Re (X));
334 and then abs Re (X) <= 1.00
336 Set_Im (Result, Im (X));
346 function Arcsinh (X : Complex) return Complex is
350 if abs Re (X) < Square_Root_Epsilon and then
351 abs Im (X) < Square_Root_Epsilon
355 elsif abs Re (X) > Inv_Square_Root_Epsilon or else
356 abs Im (X) > Inv_Square_Root_Epsilon
358 Result := Log_Two + Log (X); -- may have wrong sign
360 if (Re (X) < 0.0 and Re (Result) > 0.0)
361 or else (Re (X) > 0.0 and Re (Result) < 0.0)
363 Set_Re (Result, -Re (Result));
369 Result := Log (X + Sqrt (1.0 + X * X));
372 Set_Re (Result, Re (X));
373 elsif Im (X) = 0.0 then
374 Set_Im (Result, Im (X));
384 function Arctan (X : Complex) return Complex is
386 if abs Re (X) < Square_Root_Epsilon and then
387 abs Im (X) < Square_Root_Epsilon
392 return -Complex_I * (Log (1.0 + Complex_I * X)
393 - Log (1.0 - Complex_I * X)) / 2.0;
401 function Arctanh (X : Complex) return Complex is
403 if abs Re (X) < Square_Root_Epsilon and then
404 abs Im (X) < Square_Root_Epsilon
408 return (Log (1.0 + X) - Log (1.0 - X)) / 2.0;
416 function Cos (X : Complex) return Complex is
419 Compose_From_Cartesian
420 (Cos (Re (X)) * Cosh (Im (X)),
421 -(Sin (Re (X)) * Sinh (Im (X))));
428 function Cosh (X : Complex) return Complex is
431 Compose_From_Cartesian
432 (Cosh (Re (X)) * Cos (Im (X)),
433 Sinh (Re (X)) * Sin (Im (X)));
440 function Cot (X : Complex) return Complex is
442 if abs Re (X) < Square_Root_Epsilon and then
443 abs Im (X) < Square_Root_Epsilon
445 return Complex_One / X;
447 elsif Im (X) > Log_Inverse_Epsilon_2 then
450 elsif Im (X) < -Log_Inverse_Epsilon_2 then
454 return Cos (X) / Sin (X);
461 function Coth (X : Complex) return Complex is
463 if abs Re (X) < Square_Root_Epsilon and then
464 abs Im (X) < Square_Root_Epsilon
466 return Complex_One / X;
468 elsif Re (X) > Log_Inverse_Epsilon_2 then
471 elsif Re (X) < -Log_Inverse_Epsilon_2 then
475 return Cosh (X) / Sinh (X);
483 function Exp (X : Complex) return Complex is
484 EXP_RE_X : constant Real'Base := Exp (Re (X));
487 return Compose_From_Cartesian (EXP_RE_X * Cos (Im (X)),
488 EXP_RE_X * Sin (Im (X)));
491 function Exp (X : Imaginary) return Complex is
492 ImX : constant Real'Base := Im (X);
495 return Compose_From_Cartesian (Cos (ImX), Sin (ImX));
502 function Log (X : Complex) return Complex is
508 if Re (X) = 0.0 and then Im (X) = 0.0 then
509 raise Constraint_Error;
511 elsif abs (1.0 - Re (X)) < Root_Root_Epsilon
512 and then abs Im (X) < Root_Root_Epsilon
515 Set_Re (Z, Re (Z) - 1.0);
517 return (1.0 - (1.0 / 2.0 -
518 (1.0 / 3.0 - (1.0 / 4.0) * Z) * Z) * Z) * Z;
522 ReX := Log (Modulus (X));
525 when Constraint_Error =>
526 ReX := Log (Modulus (X / 2.0)) - Log_Two;
529 ImX := Arctan (Im (X), Re (X));
532 ImX := ImX - 2.0 * PI;
535 return Compose_From_Cartesian (ReX, ImX);
542 function Sin (X : Complex) return Complex is
544 if abs Re (X) < Square_Root_Epsilon and then
545 abs Im (X) < Square_Root_Epsilon then
550 Compose_From_Cartesian
551 (Sin (Re (X)) * Cosh (Im (X)),
552 Cos (Re (X)) * Sinh (Im (X)));
559 function Sinh (X : Complex) return Complex is
561 if abs Re (X) < Square_Root_Epsilon and then
562 abs Im (X) < Square_Root_Epsilon
567 return Compose_From_Cartesian (Sinh (Re (X)) * Cos (Im (X)),
568 Cosh (Re (X)) * Sin (Im (X)));
576 function Sqrt (X : Complex) return Complex is
577 ReX : constant Real'Base := Re (X);
578 ImX : constant Real'Base := Im (X);
579 XR : constant Real'Base := abs Re (X);
580 YR : constant Real'Base := abs Im (X);
586 -- Deal with pure real case, see (RM G.1.2(39))
591 Compose_From_Cartesian
599 Compose_From_Cartesian
600 (0.0, Real'Copy_Sign (Sqrt (-ReX), ImX));
604 R_X := Sqrt (YR / 2.0);
607 return Compose_From_Cartesian (R_X, R_X);
609 return Compose_From_Cartesian (R_X, -R_X);
613 R := Sqrt (XR ** 2 + YR ** 2);
615 -- If the square of the modulus overflows, try rescaling the
616 -- real and imaginary parts. We cannot depend on an exception
617 -- being raised on all targets.
619 if R > Real'Base'Last then
620 raise Constraint_Error;
623 -- We are solving the system
625 -- XR = R_X ** 2 - Y_R ** 2 (1)
626 -- YR = 2.0 * R_X * R_Y (2)
628 -- The symmetric solution involves square roots for both R_X and
629 -- R_Y, but it is more accurate to use the square root with the
630 -- larger argument for either R_X or R_Y, and equation (2) for the
634 R_Y := Sqrt (0.5 * (R - ReX));
635 R_X := YR / (2.0 * R_Y);
638 R_X := Sqrt (0.5 * (R + ReX));
639 R_Y := YR / (2.0 * R_X);
643 if Im (X) < 0.0 then -- halve angle, Sqrt of magnitude
646 return Compose_From_Cartesian (R_X, R_Y);
649 when Constraint_Error =>
651 -- Rescale and try again
653 R := Modulus (Compose_From_Cartesian (Re (X / 4.0), Im (X / 4.0)));
654 R_X := 2.0 * Sqrt (0.5 * R + 0.5 * Re (X / 4.0));
655 R_Y := 2.0 * Sqrt (0.5 * R - 0.5 * Re (X / 4.0));
657 if Im (X) < 0.0 then -- halve angle, Sqrt of magnitude
661 return Compose_From_Cartesian (R_X, R_Y);
668 function Tan (X : Complex) return Complex is
670 if abs Re (X) < Square_Root_Epsilon and then
671 abs Im (X) < Square_Root_Epsilon
675 elsif Im (X) > Log_Inverse_Epsilon_2 then
678 elsif Im (X) < -Log_Inverse_Epsilon_2 then
682 return Sin (X) / Cos (X);
690 function Tanh (X : Complex) return Complex is
692 if abs Re (X) < Square_Root_Epsilon and then
693 abs Im (X) < Square_Root_Epsilon
697 elsif Re (X) > Log_Inverse_Epsilon_2 then
700 elsif Re (X) < -Log_Inverse_Epsilon_2 then
704 return Sinh (X) / Cosh (X);
708 end Ada.Numerics.Generic_Complex_Elementary_Functions;