New subject: Constant Expressions (was: Complex_Arctan)

7 Feb 2002


      Please see below ...
Joe.  ( Feeling a bit better :-)
...
-----Original Message-----
From:	Emil Jerabek [SMTP:ejer5183@artax.karlin.mff.cuni.cz]
Sent:	Wednesday, February 06, 2002 4:39 AM
To:	gpc@gnu.de
Subject:	Re: Complex_Arctan
Frank Heckenbach wrote:
...
Emil Jerabek wrote:
[...]
...
...
Your change doesn't solve this, it just reduces the threshold
to EpsReal^2 > 4 / MaxReal. (BTW, the original implementation in
math.pas also needs
...
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a similar condition.)
Frank, do you _really_ think there are machines with such a small
MaxReal?
...
It doesn't really matter what I think, does it? ;-) Actually, I
haven't looked at any non-IEEE FP implementation in detail.
I'm applying Joe's patch (since it's certainly a little
improvement). If you have more ideas for improvement, let me know.
Frank
So, here's another version :-) It should never overflow, AFAICS.

const Ln2 = 0.6931471805599453094172321214581765680755;
var FunnyNumber: Real;
function Complex_ArcTan (z: Complex): Complex;
var x, y, a, b, c, d: Real;
begin
  x := Abs (Re (z));
  y := Abs (Im (z));
  b := 1 - y;
  d := (1 + y) / 2;
  if b < 0 then d := -d;
  b := Abs (b);
  if x >= b then begin
    c := b / x;
    d := d * c - x / 2;
    if x <= FunnyNumber then
      b := Ln2 - Ln (4 * x * Sqrt (1 + Sqr (c)) / (1 + y)) / 2
    else
      b := Log1P (2 * (y / x) / (b * c / 2 + x / 2)) / 4;
    c := 1;
  end else begin
    c := x / b;
    a := x * c / 2;
    d := d - a;
    if b <= FunnyNumber then
      b := Ln2 - Ln (4 * b * Sqrt (1 + Sqr (c)) / (1 + y)) / 2
    else
      b := Log1P (2 * (y / b) / (b / 2 + a)) / 4
  end;
  d := ArcTan2 (c, d) / 2;
  if Re (z) < 0 then d := -d;
  if Im (z) < 0 then b := -b;
  Complex_ArcTan := Cmplx (d, b)
end;
to begin do
  begin
    FunnyNumber := 2.1 / Sqrt (MaxReal);
  end;

I encountered a compiler bug during testing :-( 
Strictly speaking, the Standard doesn't allow function calls in constant
expressions,
but I expect the compiler would reject it if it were not supposed to work.
[Joe da Silva]
Well, actually the ISO-10206 standard (since the topic was complex
    maths, that is the applicable standard) *does* allow function calls
    in constant expressions. See sections 6.3.1, 6.3.2, 6.8.2.     :-)
There are some restrictions of course, mainly to ensure that the
    constant expression is indeed constant. AFAIK, only pre-defined
    functions are allowed (although I haven't looked too closely at
that).
Example from section 6.3.2 :
const
    ...............
       pi = 4 * arctan(1);
    ...............
...
[pas]% cat exprconst.pas
program Foo (Output);
const
  Bar = 1 / Sqrt (16);
------ snip ------

Re: Constant Expressions (was: Complex_Arctan)