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Topic: Fast exponent and logarithm, given initial estimate
Replies: 29   Last Post: Nov 8, 2004 2:31 AM

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 George Russell Posts: 157 Registered: 12/6/04
Re: Fast exponent and logarithm, given initial estimate
Posted: Oct 19, 2004 5:32 AM

Glen Low wrote (snipped):
&gt; 1. Get a good exponentiation algorithm going. Probably split integer
&gt; and fraction parts, then apply a squared Taylor expansion on the
&gt; fractional part. For 24 bits of accuracy, probably end up with about 7
&gt; reduction and other esoteric algorithms, but I fail to see how they
&gt; would reduce the number of multiplies -- anyone care to explain?)

Once you have take the first few terms of a power series to make a polynomial,
there are a number of tricks you can use to evaluate that polynomial with
fewer multiplications, which are outlined in Knuth's Art of Computer Programming,
section 4.6.4. For example a polynomial over the reals of degree 4 can be
evaluated with 3 multiplications and 5 additions, of degree 5, with 4 multiplications
and 5 additions, of degree 6, with 4 multiplications and 7 additions. There is a general
method for evaluating a real polynomial of degree n&gt;=3 with (floor (n/2) + 2 )

These methods require some precomputation to generate the coefficients to be used,
but since the coefficients are constant in your case, that should not be a problem.

Date Subject Author
10/18/04 Glen Low
10/18/04 Jeremy Watts
10/19/04 Peter Spellucci
10/19/04 Glen Low
10/18/04 bv
10/19/04 Glen Low
10/19/04 George Russell
10/19/04 Glen Low
10/20/04 George Russell
10/20/04 Glen Low
10/21/04 Christer Ericson
10/21/04 Glen Low
10/22/04 Christer Ericson
10/19/04 Martin Brown
10/19/04 Glen Low
10/19/04 Richard Mathar
10/19/04 Glen Low
10/20/04 Gert Van den Eynde
10/20/04 Glen Low
10/20/04 Richard Mathar
10/21/04 Gert Van den Eynde
10/21/04 bv
10/22/04 Glen Low
10/22/04 Peter Spellucci
10/22/04 Glen Low
10/23/04 bv
10/24/04 Gert Van den Eynde
10/25/04 Peter Spellucci
10/20/04 Gert Van den Eynde
11/8/04 Glen Low