exponential algorithm

kc6zut wrote:
> I need an algorithm for computing the exponential of a real number using
> only elementary operations (addition, subtraction, multiplication and/or
> division).  I have a PLC (Programmable Logic Controller - used in
> industrial controls) as the processor.  It has no built-in math functions
> other that the above. I need to convert a voltage from a pressure
> transducer to a displayed value.  The transducer output is scaled to
> produce X volts per decade of pressure e.g. 1 volt is 1.6e-10 Torr, 1.6
> volts is 1.6e-9 Torr.  The conversion formula is; pressure =
> 10^(1.667*Voltage - 11.46).  I only need 2 - 3 significant figures for the
> display.  I've tried using a Taylor series but even with 5 terms it is only
> good over a small range.  The available memory would only support a small
> look up table.  Any other ideas?  Any references?  

Calculate pow(10,x) as pow(2,x). The integer part of X is just the 
number of arithmetic shifts. The fractional part of X is going to be in 
the range from 1 to 2. In this range, pow(2,x) function can be easily 
approximated to desired accuracy by a polynomial of the order ~2 or ~3. 
Or by piecewise linear interpolation.

Vladimir Vassilevsky
DSP and Mixed Signal Design Consultant
http://www.abvolt.com

kc6zut <mmiller@ushio.com> wrote:
> I need an algorithm for computing the exponential of a real number using
> only elementary operations (addition, subtraction, multiplication and/or
> division).  I have a PLC (Programmable Logic Controller - used in
> industrial controls) as the processor.  It has no built-in math functions
> other that the above. I need to convert a voltage from a pressure
> transducer to a displayed value.  The transducer output is scaled to
> produce X volts per decade of pressure e.g. 1 volt is 1.6e-10 Torr, 1.6
> volts is 1.6e-9 Torr.  The conversion formula is; pressure =
> 10^(1.667*Voltage - 11.46).  I only need 2 - 3 significant figures for the
> display.  I've tried using a Taylor series but even with 5 terms it is only
> good over a small range.  The available memory would only support a small
> look up table.  Any other ideas?  Any references?  

When the input is floating point, the usual system is to 
multiply by the appropriate constant to get in in the from 2**x.
The integer part goes to the exponent of the result.  Then a 
taylor series in frac(x)-1 shouldn't take many terms.  For small x, 
exp(x) is about 1+x.

Otherwise, for a scaled fixed point exp see Knuth's 
"Metafont: The Program".  

-- glen

Scott Hemphill <hemphill@hemphills.net> wrote:
> "kc6zut" <mmiller@ushio.com> writes:
(snip)
 
> Over what range do you want the approximation to be good?  Are you
> displaying the result in exponential form?
 
> If you are displaying the result in exponential form, you will need to
> pick off the decade part of the answer first.  

Oh, yes, I forgot to say that.  On machines with binary floating
point, one scales such that it is the form 2**x, but for decimal
then scale to 10**x.  The integer part is the exponent for the
(decimal) result, the fractional part goes to the polynomial
approximation.  

> Then you can use a
> Chebyshev polynomial approximation on the remainder.  Otherwise, if
> you are converting to fixed point, you can use Chebyshev polynomials
> on the entire quantity.  Chebyshev polynomials will minimize the
> maximum absolute error for a given order of polynomial.

-- glen