Stuart Allman wrote:

>
> Give this one a whirl and give me some feedback.
>
> (-1/sqrt(x+0.02) + 0.5) / ln(10)
>
> It's a little less accurate around .5, but that's OK.  Take a look at the
> plots of log10 and this approximation before you criticize too heavily.
>


A few months ago I picked up a book called "Dead Reckoning: Calculating
Without Instruments", by Ronald W. Doerfler. I recalled seeing something
about logarithm calculations and decided to look it up again last night. I came
across formula very similar to Stuart's:

           6*(x - 1)             1
log(x) ~ ------------------ * ------
          x + 1 + 4*sqrt(x)    ln(10)

This is called Borchardt's Algorithm. Doerfler discusses briefly how it is
derived. He also says that there is an extrapolation method called Richardson's
extrapolation that can be used to get a more accurate expression. However,
for .7 < x < 1.3, this approximation is accurate to 7 significant figures. It is
still quite accurate for the 0 < x < .5 range Stuart originally requested:

      x      log(x)    Borchardt's Approximation     error
--------------------------------------------------------------
      .1     -1           -0.99166                     0.83%
      .2     -0.69897     -0.69746                     0.22%
      .3     -0.52288     -0.52251                     0.071%
      .4     -0.39794     -0.39785                     0.023%
      .5     -0.30103     -0.30101                     0.000%
     1.5      0.17609      0.17609                     0.000%
     2.0      0.30103      0.30101                     0.000%
    10.0      1            0.99166                    -0.83%


Maybe Przemek will be kind enough to plot another one of those fancy ASCII
 plots.


Incidently, Doerfler's book has a bunch of interesting stuff (that is, if you
posess that certain geek mentality). He discusses many techniques for fast
arithmetic, roots, powers, logarithms, inverse logs, trig functions etc. His
objective is more towards mental computation and not computer algorithms.
 However,
often times the most optimum technique for one is also the most optimum for the
other.

Scott