[Pixman] PDF radial gradients

[Bill Spitzak]

Andrea Canciani wrote:

> - wip/radial-float2 computes the discriminant in double procision
> (instead of 128
> bit fixed point) and should be both fast (almost two times faster than
> wip/radial)
> and accurate. Does it look good?
> I'm a little worried about the int64->double conversions in this
> branch. Are there
> architectures where it might be a problem?
> 
> - in wip/radial-float I tried to keep all the variables of the inner
> loop in floating point
> to see how much this affects the performance. I get small speed
> improvements, but
> I don't think they justify this branch (the accuracy guarantees in
> this case are very
> loose since errors accumulate quadratically with the number of iterations).

I have found that "A*x+B" is almost as fast as "x+B" on floating point 
and avoids any problems with inaccuracy of repeated adding a value. The 
FPU has an instruction to do this operation. My understanding is the 
addition is the slow part and it can merge part of it into the 
multiplication.

I am unsure if this will help for higher-order functions, but it seems 
that C*(A*x+B)+D has two of these operations and thus will be have the 
same relative speed to numerical integration.

In any case if you are getting twice the performance by switching to 
float, I suspect that a numerically stable floating point version will 
still be a win.