Re: Handling really large numbers, dynamic ranges....

My computationally-savvy friend Ned Conklin recommends:

Don't do the brute force cubing first and then multiply by the coefficient.
Instead, rewrite

y = -7.0970E-11x3 + 2.0936E-06x2 - 2.4255E-02x + 1.6214E+02

as

y = ((-7.097E-11x + 2.0936E-06)x -2.4255E-02)x + 1.6214E+02

So it's a sequence of multiply, add, multiply, add, multiply, add ....
This way you get to apply the coefficients early on before the numbers get
too large. You only have a dynamic range of about 120 in your data, so
there ought to be a set of scale factors that let you stay within the range
of double precision neumbers while not losing significant resolution.

Cheers,
Elizabeth

At 04:09 PM 9/3/02 -0700, Gene Silvernail wrote:
>The data is 24 bits, as stated. Actually 28 bit plus sign. The lower 4
>bits I discard as sub-lsb bits. The noise as stated is +/- 10 or +/- .59
>ppm. Good resolution not over sampling generated bits.
>
>The solutions that I see are:
>
>1. Do something like below, cleaning it up with Horner's method
>2. Do it with table look up
>3. Do it externally in floating point
>
>
>This kludge handles 2nd degree polynomials but is not factored
>: >DEG-F ( d1 -- d2 )
> 2DUP
> >R >R
> 1 1000 m*/
> 2DUP D>S 1 M*/
> 2ND-COEF-X2 10000 M*/
> R> R>
> 2ND-COEF-X1 1000 M*/
> D-
> 135620000. \
> D+
> 1 10000 M*/
> ;
>
>-----Original Message-----
>From: swiftx-bounce_at_forth.com [mailto:swiftx-bounce_at_forth.com] On Behalf
>Of Elizabeth D. Rather
>Sent: Tuesday, September 03, 2002 11:54 AM
>To: Swiftx Listserve
>Subject: [swiftx] Re: Handling really large numbers, dynamic ranges....
>
>
>At 02:42 PM 9/3/02 -0400, Bill Hall wrote:
>
> >hi
> >First I must ask what is the expected domain of the resultant y ?
> >I ran 12,000,000 thru the equation you gave, and y goes way past 32
>bits,
> >in the negative direction.
> >
> >One suggestion is to rearrange the equation as follows:
> >
> >y = (((a)x + b)x + c)x +d .
> >
> >Then evaluate it as follows
> >[etc....]
> >But a is so small that you will need to scale the result UP to
> >obtain any
> >integer effect from the a term. Which leads me back to the first
> >question
> >because it is the a term that causes the result to go beyond 32
>bits
> >negative.
>
>What is the actual resolution of the data you're starting with? That
>is ultimately the limiting factor on the _real_ resolution of any result
>you get. It's a common fallacy to take, say, 12-bit data and process
>it with 32 bits of accuracy; you end up with mostly noise bits.
>
> >You may be able to pick a scaling denominator for the data and use */
>,
> >thinking of the data as a fraction of some "full scale" like 16777216.
>
>Better still, use M*/ to multiply your 32-bit intermediate result by
>a ratio of 16-bit numbers (w/ extended-precision intermediate
>product).
>
>Cheers,
>Elizabeth
>
>
>==================================================
>Elizabeth D. Rather (US & Canada) 800-55-FORTH
>FORTH Inc. +1 310-491-3356
>5155 W. Rosecrans Ave. #1018 Fax: +1 310-978-9454
>Hawthorne, CA 90250
>http://www.forth.com
>
>"Forth-based products and Services for real-time
>applications since 1973."
>==================================================
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==================================================
Elizabeth D. Rather (US & Canada) 800-55-FORTH
FORTH Inc. +1 310-491-3356
5155 W. Rosecrans Ave. #1018 Fax: +1 310-978-9454
Hawthorne, CA 90250
http://www.forth.com

"Forth-based products and Services for real-time
applications since 1973."
==================================================

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Received on Wed Sep 04 2002 - 15:38:25 PDT