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Re: Fast exponent and logarithm, given initial estimate
Posted:
Oct 20, 2004 11:51 PM
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> For degree 6, you can evaluate a real polynomial in x using 4 multiplications and
> 7 additions with the code
>
> z = (x + a0)*x + a1
> w = (x + a2)*z + a3
> the value of the polynomial = ((w + z + a4)*w + a5) * a6
>
> You need to solve a cubic equation to work out the constants a0-a6 given the initial
> coefficients; however there will always be a real solution.
>
> You mentioned fused multiply-adds. This algorithm requires
> 4 multiplies/fused multiply-adds and just 4 additions. I don't know if you can
> get the number of additions down even further.
On second thought, it would probably be slower on an Altivec (or other
modern pipelined) machine. Here's why:
A 6-degree polynomial using the standard Horner's rule takes exactly 5
fused multiply-adds to run. On a G5, that would take 5 x 8 cyles = 40
cyles, since each fma depends on the last.
A 6-degree polynomial using the Knuth method above uses 4 fused
multiply-adds and 4 additions. Unfortunately each addition also takes
8 cycles, so it comes out to around 8 x 8 cyles = 64 cycles, minus one
or two since the initial x + a0 doesn't depend on x + a2 and can be
pipelined. Unless some of the coefficients are zero or the same, or I
can do more of the operations in parallel... I have to a bit more
analysis on that.
Cheers,
Glen Low, Pixelglow Software
www.pixelglow.com
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