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Re: Fast exponent and logarithm, given initial estimate
Posted:
Oct 19, 2004 5:32 AM
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Glen Low wrote (snipped):
> 1. Get a good exponentiation algorithm going. Probably split integer
> and fraction parts, then apply a squared Taylor expansion on the
> fractional part. For 24 bits of accuracy, probably end up with about 7
> multiplies. (I read some stuff about binary splitting, binary
> reduction and other esoteric algorithms, but I fail to see how they
> 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>=3 with (floor (n/2) + 2 )
multiplications and n additions.
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.
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