Re: fixed point vs floating point

[William A Whitaker <whitaker@erols.com>]

Joe Gwinn wrote:
> 
snip
> 
> In one recent example, Square Root was taking 60 microseconds on a 50-MHz
> 68060; this should take no more than 2 uS.  Where did the time go? 

  Etc.  Same kind of
> story with the transcendental functions, only worse.  One needs to read
> the generated assembly code to know what a given compiler is up to.
> 
> We only need a few functions, specifically Sine, Cosine, Arc Tangent, and
> Square Root, speed is very much of the essence, and we need only 10e-4 to
> 10e-5 (16-bit) accuracy anyway.  So, we will write our own versions of
> these functions, in Ada, C, or even assembly, as determined by performance
> tests.
> 
> 
> 
> As for the polynomial approximations, the bible is to this day is
> "Approximations for Digital Computers"; Cecil Hastings, Jr.; Princeton
> University Press; 1955.  This has been a classic since its publication.
> In those days, the computers were small, so the programmers had to be very
> focused on performance.  For computation of polynomials in real systems,
> convert the published polynomials into Horner's form.  This is discussed
> in many texts on numerical methods.  In short,  a + bX + cX^2 becomes a +
> X(b+cX), which is faster to compute, and has better numerical properties
> than the standard power-series form.
> 
> Joe Gwinn

I remember working hard to get an exponential in 3.7 microseconds, using
a self-generated "Hasty Approximation" (in honor of Hastings who was the
father of them all).  This was 30 years ago.

Bill Whitaker