I wrote this in response to repeated questions about it. Comments/criticisms?

www.digitalmars.com/d/cppcomplex.html

"Walter" <walter@digitalmars.com> wrote in message news:bier5s$etm$1@digitaldaemon.com...
> I wrote this in response to repeated questions about it. Comments/criticisms?
>
> www.digitalmars.com/d/cppcomplex.html
>
>

> ireal a, b, c;
> c = a + b;
> In C++, it is two adds, as the real parts get added too.

explain, please

> Multiply is worse, as 4 multiplies and two adds are done instead of one
multiply.

explain, please

> Divide is the worst - D has one divide, whereas C++ implements complex
division with typically one comparison, 3 divides, 3 multiplies and 3 additions.

explain, please



The rest of it seems quite compelling, as long as it's true, for which I have every faith. :)

Sorry, Walter, had to do it:  ;)

Syntactical Aesthetics:

Mostly irrelevant.  Anyone using this in reality would make a typedef for std::complex<long double> instead of spelling it out constantly.  First step in writing portable programs is centralizing declarations.  In reality it wouldn't be so bad for the C++ programmer.  I know;  I've done this sort of thing for years in C++, with templates and without.  Certainly looks more like real math written the D way, but that method of defining values is not indefinitely extendable nor user extendable.

Besides you don't mention how to construct an imaginary long double literal, or imaginary float literal using a suffix.  Does that work?

Efficiency:

So, you finally see the logic behind my requests for builtin small fixed-size vector and quaternion types.

In theory, it should be possible to design a type system whereby you could return a type appropriate to the specific values of the inputs, thus optimizing away unnecessary calculations.  This is painstaking and tedious in C++;  you have to use overloading *heavily*, and manually specify many many combinations, then face "ambiguous conversion" errors.  Perhaps OCaml has a better way to say these sort of things?

We currently have to "go to the metal" to get performance out of these sort of values.  You know as well as I do that compilers can't optimize flow through assembly very well.  It's often a lose-lose situation.

Division behavior:  be careful here, as optimizing divides can leave theoreticians and serious number crunchers upset about precision loss. Personally I'm all for speed however, and turning divides into multiply by reciprocals is generally good enough for my purposes.

Surely you could optimize a simple user-defined struct in this manner?  If part of it is always a certain value, and the compiler knows this and can verify it through computational flow, it can disregard results of operations that are never used, and discard operations that are unnecessary.  If this is implemented by the optimizer in a general way, all programs may benefit. Giving us a single type that does it just teases us.  ;)

Semantics:

I think this guy is crazy.  Bone up on Geometric Algebra;  all the functionality is in the operations, not the data storage format.  If C++ is broken, fine, fix it.  But we all know that 0 + 1i = i and 1 + 0i = 1, so results of operations on the two values (1 + 0i  vs.  1) should be indistinguishable.  Storage format really doesn't matter;  it's all about what happens during the operations and how you define said operations (multiply, log, sqrt) and primitive values (the real numbers).

If there are sign issues, it's either a problem in the implementation of the operations, or of the input values, or of the algorithm.  I'm not sure what he was trying to do but I'm pretty confident it was either the first or the last of those three.  Probably the first.

In conclusion, I don't think an "imaginary" type is strictly necessary for correctness, nor do I believe that the implementation should favor two such "axes" and ignore the plethora of other possible axes (you might think of these as "units" or as separate types if you wish).  If you build in imaginary, pretty soon people will be asking for (or kludgily building using the available language tools) "x" "y" and "z" axis types, "length" and "time" units, ad infinitum.

Sean

"Walter" <walter@digitalmars.com> wrote in message news:bier5s$etm$1@digitaldaemon.com...
> I wrote this in response to repeated questions about it. Comments/criticisms?
>
> www.digitalmars.com/d/cppcomplex.html

>
> > ireal a, b, c;
> > c = a + b;
> > In C++, it is two adds, as the real parts get added too.
>
> explain, please
>
> > Multiply is worse, as 4 multiplies and two adds are done instead of one
> multiply.
>
> explain, please
>
> > Divide is the worst - D has one divide, whereas C++ implements complex
> division with typically one comparison, 3 divides, 3 multiplies and 3 additions.
>
> explain, please

Ok, check out the new page! www.digitalmars.com/d/cppcomplex.html

> The rest of it seems quite compelling, as long as it's true, for which I have every faith. :)

If you don't believe me, check out Prof. Kahan's referenced papers!

> Besides you don't mention how to construct an imaginary long double literal,
> or imaginary float literal using a suffix.  Does that work?

Good question.

> Semantics:
> 
> I think this guy is crazy.  Bone up on Geometric Algebra;  all the
> functionality is in the operations, not the data storage format.  If C++ is
> broken, fine, fix it.  But we all know that 0 + 1i = i and 1 + 0i = 1, so
> results of operations on the two values (1 + 0i  vs.  1) should be
> indistinguishable.  Storage format really doesn't matter;  it's all about
> what happens during the operations and how you define said operations
> (multiply, log, sqrt) and primitive values (the real numbers).

Earlier, I did argue in favor of distinct imaginary variables, but, in
truth, I can't argue yes or no with my level of knowledge.  This I do know:

1) Being able to express imaginary literals is refreshingly easy in D.

2) C++'s method of adding an imaginary to an expression: + complex<long double>(0,7) is painfully laborious.  And so it would be for any other mathematical type that might be implemented as a template. I wouldn't like to see this happen in D, not, at least, with complex numbers.

3) D's way of being able to declare a imaginary variable is just kind of neat because even if I can express a complex imaginary as:

	cdouble = 2i; // can I do this?
or	idouble = 2i; // the same as above but imaginary only

instead of being forced:

	cdouble = 0 + 2i;

it's still "feels good" to separate the two parts (real and imaginary) into variables in some situations. It may also be a readability thing. Since you have two parts of a complex number, it's "kind of nice" to be able to specify that part as a variable. I'm afraid this is the weak extent of my argument.

> If there are sign issues, it's either a problem in the implementation of the
> operations, or of the input values, or of the algorithm.  I'm not sure what
> he was trying to do but I'm pretty confident it was either the first or the
> last of those three.  Probably the first.

I tried understanding the professor's complaints, but the points escaped me (since I don't think of or deal with complex numbers on that level to realize such problems).  Implementation details are far beyond me at this point.

> In conclusion, I don't think an "imaginary" type is strictly necessary for
> correctness, nor do I believe that the implementation should favor two such
> "axes" and ignore the plethora of other possible axes (you might think of
> these as "units" or as separate types if you wish).  If you build in
> imaginary, pretty soon people will be asking for (or kludgily building using
> the available language tools) "x" "y" and "z" axis types, "length" and
> "time" units, ad infinitum.

I still don't understand this. Complex numbers are not the same thing as
vectors, no matter how similar in some ways.  They are a special set of  numbers, nonetheless, and common enough that most languages like python, Fortran, and others have them implemented (argumentum ad populorum :P).  Vectors don't fit that category.  So far as I can see, vectors may as well be implemented as they always have been, which allows maximum flexibility to the programmer. Complex are fairly static in their definition (don't have varying number of dimensions) so it's a safe implementation. But I'm really not one to speak, ha!

Oh, by the way, Walter, can we have a set 2D, 3D, and 4D matrix variables too in D, just like the new OpenGl shader langauge? ;-D

So fun, so fun!

John

"Sean L. Palmer" <palmer.sean@verizon.net> wrote in message news:bihj7g$1kri$1@digitaldaemon.com...
> Sorry, Walter, had to do it:  ;)
>
> Syntactical Aesthetics:
>
> Mostly irrelevant.  Anyone using this in reality would make a typedef for std::complex<long double> instead of spelling it out constantly.

I think they are poor *standard* types if the first thing the programmer will want to do is typedef them into being something more palatable.

> First step
> in writing portable programs is centralizing declarations.  In reality it
> wouldn't be so bad for the C++ programmer.  I know;  I've done this sort
of
> thing for years in C++, with templates and without.

There's no easy way to get around the lack of imaginary literal syntax.

> Certainly looks more
> like real math written the D way, but that method of defining values is
not
> indefinitely extendable nor user extendable.

Int's aren't user-extendible, either <g>.

> Besides you don't mention how to construct an imaginary long double
literal,
> or imaginary float literal using a suffix.  Does that work?

Yes:
    ireal x = 5.0 + 14.567Li

> Efficiency:
>
> So, you finally see the logic behind my requests for builtin small fixed-size vector and quaternion types.
>
> In theory, it should be possible to design a type system whereby you could return a type appropriate to the specific values of the inputs, thus optimizing away unnecessary calculations.  This is painstaking and tedious in C++;  you have to use overloading *heavily*, and manually specify many many combinations, then face "ambiguous conversion" errors.  Perhaps OCaml has a better way to say these sort of things?
>
> We currently have to "go to the metal" to get performance out of these
sort
> of values.  You know as well as I do that compilers can't optimize flow through assembly very well.  It's often a lose-lose situation.
>
> Division behavior:  be careful here, as optimizing divides can leave theoreticians and serious number crunchers upset about precision loss. Personally I'm all for speed however, and turning divides into multiply by reciprocals is generally good enough for my purposes.

There are many ways to express complex division, some certainly better than others, but they all involve a lot more than a single divide, which was all I was trying to show.

> Surely you could optimize a simple user-defined struct in this manner?  If part of it is always a certain value, and the compiler knows this and can verify it through computational flow, it can disregard results of
operations
> that are never used, and discard operations that are unnecessary.  If this is implemented by the optimizer in a general way, all programs may
benefit.
> Giving us a single type that does it just teases us.  ;)

Yet the compiler does not know this. Consider passing a value to a function. The function cannot know that the real part is always 0.

> Semantics:
>
> I think this guy is crazy.  Bone up on Geometric Algebra;  all the functionality is in the operations, not the data storage format.  If C++
is
> broken, fine, fix it.  But we all know that 0 + 1i = i and 1 + 0i = 1, so results of operations on the two values (1 + 0i  vs.  1) should be indistinguishable.

They aren't with IEEE arithmetic because of the sign of 0.

> Storage format really doesn't matter;  it's all about
> what happens during the operations and how you define said operations
> (multiply, log, sqrt) and primitive values (the real numbers).

Prof. Kahan isn't crazy, in fact, he is the prime mover behind IEEE 754 arithmetic. There really isn't anyone who knows floating point better than him. The trouble comes not from the mathematics, but from imperfections in how floating point is implemented. In dealing with complex numbers, you have a thing called "branch cuts" which wind up relying on the sign of 0. Carrying along an extra 0 for the real part winds up getting the signs wrong.

> If there are sign issues, it's either a problem in the implementation of
the
> operations, or of the input values, or of the algorithm.  I'm not sure
what
> he was trying to do but I'm pretty confident it was either the first or
the
> last of those three.  Probably the first.
> In conclusion, I don't think an "imaginary" type is strictly necessary for
> correctness, nor do I believe that the implementation should favor two
such
> "axes" and ignore the plethora of other possible axes (you might think of these as "units" or as separate types if you wish).  If you build in imaginary, pretty soon people will be asking for (or kludgily building
using
> the available language tools) "x" "y" and "z" axis types, "length" and "time" units, ad infinitum.

It's too bad that the analysis of what exactly is going wrong isn't on his web page, but in his book.

> It's too bad that the analysis of what exactly is going wrong isn't on his
> web page, but in his book.
> 
> 

It is an interesting analysis that seems to warrant a real study.  I've heard you mention the professor before, but never really understood the topics importance.  What's the name of the book again?

Later,

John

"Walter" <walter@digitalmars.com> wrote in message news:bihjbf$1kv6$1@digitaldaemon.com...
>
> >
> > > ireal a, b, c;
> > > c = a + b;
> > > In C++, it is two adds, as the real parts get added too.
> >
> > explain, please
> >
> > > Multiply is worse, as 4 multiplies and two adds are done instead of
one
> > multiply.
> >
> > explain, please
> >
> > > Divide is the worst - D has one divide, whereas C++ implements complex
> > division with typically one comparison, 3 divides, 3 multiplies and 3 additions.
> >
> > explain, please
>
> Ok, check out the new page! www.digitalmars.com/d/cppcomplex.html

Get it posted up on c.l.c.m!

> > The rest of it seems quite compelling, as long as it's true, for which I have every faith. :)
>
> If you don't believe me, check out Prof. Kahan's referenced papers!

Twas never in doubt, E.C.W. (esteemed compiler-walter)

"John Reimer" <jjreimer@telus.net> wrote in message news:bihsoe$24tq$1@digitaldaemon.com...
> > It's too bad that the analysis of what exactly is going wrong isn't on
his
> > web page, but in his book.
> It is an interesting analysis that seems to warrant a real study.  I've heard you mention the professor before, but never really understood the topics importance.  What's the name of the book again?

" Branch Cuts for Complex Elementary Functions, or Much Ado About Nothing's Sign Bit " by W. Kahan, ch.

7 in The State of the Art in Numerical Analysis ( 1987 ) ed. by M. Powell and A. Iserles for Oxford U.P.

> 
> " Branch Cuts for Complex Elementary Functions, or Much Ado About Nothing's
> Sign Bit " by W. Kahan, ch.
> 
> 7 in The State of the Art in Numerical Analysis ( 1987 ) ed. by M. Powell
> and A. Iserles for Oxford U.P.
> 
> 

Thanks,  I see now that it is one of the references listed in the references!  Oops! :-P

So much for my observation skills...