1 ------------------------------------------------------------------------------
3 -- GNAT RUNTIME COMPONENTS --
5 -- ADA.NUMERICS.GENERIC_COMPLEX_ELEMENTARY_FUNCTIONS --
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24 -- As a special exception, if other files instantiate generics from this --
25 -- unit, or you link this unit with other files to produce an executable, --
26 -- this unit does not by itself cause the resulting executable to be --
27 -- covered by the GNU General Public License. This exception does not --
28 -- however invalidate any other reasons why the executable file might be --
29 -- covered by the GNU Public License. --
31 -- GNAT was originally developed by the GNAT team at New York University. --
32 -- It is now maintained by Ada Core Technologies Inc (http://www.gnat.com). --
34 ------------------------------------------------------------------------------
36 with Ada.Numerics.Generic_Elementary_Functions;
38 package body Ada.Numerics.Generic_Complex_Elementary_Functions is
40 package Elementary_Functions is new
41 Ada.Numerics.Generic_Elementary_Functions (Real'Base);
42 use Elementary_Functions;
44 PI : constant := 3.14159_26535_89793_23846_26433_83279_50288_41971;
45 PI_2 : constant := PI / 2.0;
46 Sqrt_Two : constant := 1.41421_35623_73095_04880_16887_24209_69807_85696;
47 Log_Two : constant := 0.69314_71805_59945_30941_72321_21458_17656_80755;
49 subtype T is Real'Base;
51 Epsilon : constant T := 2.0 ** (1 - T'Model_Mantissa);
52 Square_Root_Epsilon : constant T := Sqrt_Two ** (1 - T'Model_Mantissa);
53 Inv_Square_Root_Epsilon : constant T := Sqrt_Two ** (T'Model_Mantissa - 1);
54 Root_Root_Epsilon : constant T := Sqrt_Two **
55 ((1 - T'Model_Mantissa) / 2);
56 Log_Inverse_Epsilon_2 : constant T := T (T'Model_Mantissa - 1) / 2.0;
58 Complex_Zero : constant Complex := (0.0, 0.0);
59 Complex_One : constant Complex := (1.0, 0.0);
60 Complex_I : constant Complex := (0.0, 1.0);
61 Half_Pi : constant Complex := (PI_2, 0.0);
67 function "**" (Left : Complex; Right : Complex) return Complex is
70 and then Im (Right) = 0.0
71 and then Re (Left) = 0.0
72 and then Im (Left) = 0.0
77 and then Im (Left) = 0.0
78 and then Re (Right) < 0.0
80 raise Constraint_Error;
82 elsif Re (Left) = 0.0 and then Im (Left) = 0.0 then
85 elsif Right = (0.0, 0.0) then
88 elsif Re (Right) = 0.0 and then Im (Right) = 0.0 then
91 elsif Re (Right) = 1.0 and then Im (Right) = 0.0 then
95 return Exp (Right * Log (Left));
99 function "**" (Left : Real'Base; Right : Complex) return Complex is
101 if Re (Right) = 0.0 and then Im (Right) = 0.0 and then Left = 0.0 then
102 raise Argument_Error;
104 elsif Left = 0.0 and then Re (Right) < 0.0 then
105 raise Constraint_Error;
107 elsif Left = 0.0 then
108 return Compose_From_Cartesian (Left, 0.0);
110 elsif Re (Right) = 0.0 and then Im (Right) = 0.0 then
113 elsif Re (Right) = 1.0 and then Im (Right) = 0.0 then
114 return Compose_From_Cartesian (Left, 0.0);
117 return Exp (Log (Left) * Right);
121 function "**" (Left : Complex; Right : Real'Base) return Complex is
124 and then Re (Left) = 0.0
125 and then Im (Left) = 0.0
127 raise Argument_Error;
129 elsif Re (Left) = 0.0
130 and then Im (Left) = 0.0
133 raise Constraint_Error;
135 elsif Re (Left) = 0.0 and then Im (Left) = 0.0 then
138 elsif Right = 0.0 then
141 elsif Right = 1.0 then
145 return Exp (Right * Log (Left));
153 function Arccos (X : Complex) return Complex is
157 if X = Complex_One then
160 elsif abs Re (X) < Square_Root_Epsilon and then
161 abs Im (X) < Square_Root_Epsilon
165 elsif abs Re (X) > Inv_Square_Root_Epsilon or else
166 abs Im (X) > Inv_Square_Root_Epsilon
168 return -2.0 * Complex_I * Log (Sqrt ((1.0 + X) / 2.0) +
169 Complex_I * Sqrt ((1.0 - X) / 2.0));
172 Result := -Complex_I * Log (X + Complex_I * Sqrt (1.0 - X * X));
175 and then abs Re (X) <= 1.00
177 Set_Im (Result, Im (X));
187 function Arccosh (X : Complex) return Complex is
191 if X = Complex_One then
194 elsif abs Re (X) < Square_Root_Epsilon and then
195 abs Im (X) < Square_Root_Epsilon
197 Result := Compose_From_Cartesian (-Im (X), -PI_2 + Re (X));
199 elsif abs Re (X) > Inv_Square_Root_Epsilon or else
200 abs Im (X) > Inv_Square_Root_Epsilon
202 Result := Log_Two + Log (X);
205 Result := 2.0 * Log (Sqrt ((1.0 + X) / 2.0) +
206 Sqrt ((X - 1.0) / 2.0));
209 if Re (Result) <= 0.0 then
220 function Arccot (X : Complex) return Complex is
224 if abs Re (X) < Square_Root_Epsilon and then
225 abs Im (X) < Square_Root_Epsilon
229 elsif abs Re (X) > 1.0 / Epsilon or else
230 abs Im (X) > 1.0 / Epsilon
232 Xt := Complex_One / X;
235 Set_Re (Xt, PI - Re (Xt));
242 Xt := Complex_I * Log ((X - Complex_I) / (X + Complex_I)) / 2.0;
244 if Re (Xt) < 0.0 then
255 function Arccoth (X : Complex) return Complex is
259 if X = (0.0, 0.0) then
260 return Compose_From_Cartesian (0.0, PI_2);
262 elsif abs Re (X) < Square_Root_Epsilon
263 and then abs Im (X) < Square_Root_Epsilon
265 return PI_2 * Complex_I + X;
267 elsif abs Re (X) > 1.0 / Epsilon or else
268 abs Im (X) > 1.0 / Epsilon
273 return PI * Complex_I;
276 elsif Im (X) = 0.0 and then Re (X) = 1.0 then
277 raise Constraint_Error;
279 elsif Im (X) = 0.0 and then Re (X) = -1.0 then
280 raise Constraint_Error;
284 R := Log ((1.0 + X) / (X - 1.0)) / 2.0;
287 when Constraint_Error =>
288 R := (Log (1.0 + X) - Log (X - 1.0)) / 2.0;
292 Set_Im (R, PI + Im (R));
306 function Arcsin (X : Complex) return Complex is
310 if abs Re (X) < Square_Root_Epsilon and then
311 abs Im (X) < Square_Root_Epsilon
315 elsif abs Re (X) > Inv_Square_Root_Epsilon or else
316 abs Im (X) > Inv_Square_Root_Epsilon
318 Result := -Complex_I * (Log (Complex_I * X) + Log (2.0 * Complex_I));
320 if Im (Result) > PI_2 then
321 Set_Im (Result, PI - Im (X));
323 elsif Im (Result) < -PI_2 then
324 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 : Real'Base := Exp (Re (X));
487 return Compose_From_Cartesian (EXP_RE_X * Cos (Im (X)),
488 EXP_RE_X * Sin (Im (X)));
492 function Exp (X : Imaginary) return Complex is
493 ImX : Real'Base := Im (X);
496 return Compose_From_Cartesian (Cos (ImX), Sin (ImX));
503 function Log (X : Complex) return Complex is
509 if Re (X) = 0.0 and then Im (X) = 0.0 then
510 raise Constraint_Error;
512 elsif abs (1.0 - Re (X)) < Root_Root_Epsilon
513 and then abs Im (X) < Root_Root_Epsilon
516 Set_Re (Z, Re (Z) - 1.0);
518 return (1.0 - (1.0 / 2.0 -
519 (1.0 / 3.0 - (1.0 / 4.0) * Z) * Z) * Z) * Z;
523 ReX := Log (Modulus (X));
526 when Constraint_Error =>
527 ReX := Log (Modulus (X / 2.0)) - Log_Two;
530 ImX := Arctan (Im (X), Re (X));
533 ImX := ImX - 2.0 * PI;
536 return Compose_From_Cartesian (ReX, ImX);
543 function Sin (X : Complex) return Complex is
545 if abs Re (X) < Square_Root_Epsilon and then
546 abs Im (X) < Square_Root_Epsilon then
551 Compose_From_Cartesian
552 (Sin (Re (X)) * Cosh (Im (X)),
553 Cos (Re (X)) * Sinh (Im (X)));
560 function Sinh (X : Complex) return Complex is
562 if abs Re (X) < Square_Root_Epsilon and then
563 abs Im (X) < Square_Root_Epsilon
568 return Compose_From_Cartesian (Sinh (Re (X)) * Cos (Im (X)),
569 Cosh (Re (X)) * Sin (Im (X)));
577 function Sqrt (X : Complex) return Complex is
578 ReX : constant Real'Base := Re (X);
579 ImX : constant Real'Base := Im (X);
580 XR : constant Real'Base := abs Re (X);
581 YR : constant Real'Base := abs Im (X);
587 -- Deal with pure real case, see (RM G.1.2(39))
592 Compose_From_Cartesian
600 Compose_From_Cartesian
601 (0.0, Real'Copy_Sign (Sqrt (-ReX), ImX));
605 R_X := Sqrt (YR / 2.0);
608 return Compose_From_Cartesian (R_X, R_X);
610 return Compose_From_Cartesian (R_X, -R_X);
614 R := Sqrt (XR ** 2 + YR ** 2);
616 -- If the square of the modulus overflows, try rescaling the
617 -- real and imaginary parts. We cannot depend on an exception
618 -- being raised on all targets.
620 if R > Real'Base'Last then
621 raise Constraint_Error;
624 -- We are solving the system
626 -- XR = R_X ** 2 - Y_R ** 2 (1)
627 -- YR = 2.0 * R_X * R_Y (2)
629 -- The symmetric solution involves square roots for both R_X and
630 -- R_Y, but it is more accurate to use the square root with the
631 -- larger argument for either R_X or R_Y, and equation (2) for the
635 R_Y := Sqrt (0.5 * (R - ReX));
636 R_X := YR / (2.0 * R_Y);
639 R_X := Sqrt (0.5 * (R + ReX));
640 R_Y := YR / (2.0 * R_X);
644 if Im (X) < 0.0 then -- halve angle, Sqrt of magnitude
647 return Compose_From_Cartesian (R_X, R_Y);
650 when Constraint_Error =>
652 -- Rescale and try again.
654 R := Modulus (Compose_From_Cartesian (Re (X / 4.0), Im (X / 4.0)));
655 R_X := 2.0 * Sqrt (0.5 * R + 0.5 * Re (X / 4.0));
656 R_Y := 2.0 * Sqrt (0.5 * R - 0.5 * Re (X / 4.0));
658 if Im (X) < 0.0 then -- halve angle, Sqrt of magnitude
662 return Compose_From_Cartesian (R_X, R_Y);
669 function Tan (X : Complex) return Complex is
671 if abs Re (X) < Square_Root_Epsilon and then
672 abs Im (X) < Square_Root_Epsilon
676 elsif Im (X) > Log_Inverse_Epsilon_2 then
679 elsif Im (X) < -Log_Inverse_Epsilon_2 then
683 return Sin (X) / Cos (X);
691 function Tanh (X : Complex) return Complex is
693 if abs Re (X) < Square_Root_Epsilon and then
694 abs Im (X) < Square_Root_Epsilon
698 elsif Re (X) > Log_Inverse_Epsilon_2 then
701 elsif Re (X) < -Log_Inverse_Epsilon_2 then
705 return Sinh (X) / Cosh (X);
709 end Ada.Numerics.Generic_Complex_Elementary_Functions;