Path: utzoo!utgpu!news-server.csri.toronto.edu!cs.utexas.edu!uunet!mcsun!ukc!acorn!dseal
From: dseal@acorn.co.uk (David Seal)
Newsgroups: comp.theory
Subject: Re: High precision Mandelbrot set
Message-ID: <4611@acorn.co.uk>
Date: 10 Jan 91 10:24:23 GMT
References: <10288@jarthur.Claremont.EDU>
Sender: dseal@acorn.co.uk
Distribution: comp
Organization: Acorn Computers Ltd, Cambridge, England
Lines: 34

In article <10288@jarthur.Claremont.EDU> rspangle@jarthur.Claremont.EDU (Froot Loop) writes:

>In article <1991Jan8.142354.1732@pegasus.com> shaw@pegasus.com (Sandy Shaw) writes:
>>My question is : Is there there a way short of using an extended precision
>>math package to gain further detail upon continuing the zoom? Perhaps some 
>>sort of renormalization? Any thoughts on this question or references would
>>be greatly appreciated.
>
>...
>
>Another would be to write your own floating point routines in assembly.
>(So take a 20-byte structure, use 1 bit for sign, 15 bits for implied base-16
>exponent, and use the other 144 bits for the mantissa.  All you have to
>do are multiplies and adds, which aren't too difficult.

Writing floating point routines is somewhat overkill, in fact. Generally,
the Mandelbrot set algorithms terminate if the numbers get more than a few
binary orders of magnitude greater than 1. Also, in the recurrence z := z**2
+ c, the addition of c generally restricts the accuracy of the result to
that of c. As all numbers with magnitude less than 1/4 are in the Mandelbrot
set, the useful range of the exponent in floating point arithmetic for the
Mandelbrot set runs from -2 to a fairly small positive number.

All this means that you will get greater accuracy and simpler, faster code
if your arithmetic is implemented as fixed point routines, with a few bits
before the binary point and as many as you need after it.

It can also pay to have more than one set of fixed point routines, of
differing accuracies. Use the least accurate on the whole Mandelbrot set and
when a limited amount of zooming in has been done, then switch to higher
accuracies as the degree of zooming in increases.

David Seal
dseal@acorn.co.uk