Testing The Accuracy of LN and EXP Approximations
Introduction
Today we are testing an approximation polynomial for two common scientific functions. I used a TI-84 Plus CE in testing. The values take a range from x = 0.5 to x = 11.
Approximating e^x
Approximation used:
e^x ≈ 1 + x * (1 + x/2 * (1 + x/3 * (1 + x/4 * (1 + x/5 * (1 + x/6)))))
(fractions are simplified, see source)
TI-84 Plus CE Program: EXPTEST
"EWS 2021-01-19"
Disp "e^(X) TEST","JON M. SMITH"
Disp "RANGE, MIN. 0.5"
Input "START? ",A
Input "INCREMENT? ",I
Input "NO STEPS? ",N
seq(X,X,A,A+I*N,I)→L₁
e^(L₁)→L₂
1+L₁*(1+L₁/2*(1+L₁/3*(1+L₁/4*(1+L₁/5*(1+L₁/6)))))→L₃
abs(L₂-L₃)→L₄
iPart(log(L₄))→L₅
Disp "L₁:X, L₂:EXP, L₃:APPROX","L₄:ABS ERROR","L₅:PLACES"
SetUpEditor L₁,L₂,L₃,L₄,L₅
Disp "PRESS STAT, 1. EDIT"
Test: x range: 0.5 to 11, increments of 0.5
Approximating ln x
Approximation used:
ln x ≈ y * (1 + y/2 * (1 + 2y/3 * (1 + 3y/4 * (1 + 4y/5))))
where x ≥ 0.5
and y = (x - 1)/x
(see source)
TI-84 Plus CE Program: LNTEST
"EWS 2021-01-19"
Disp "ln(X) TEST","JON M. SMITH"
Disp "RANGE, MIN. 0.5"
Input "START? ",A
Input "INCREMENT? ",I
Input "NO STEPS? ",N
seq(X,X,A,A+I*N,I)→L₁
ln(L₁)→L₂
(L₁-1)/L₁→L₆
L₆*(1+L₆/2*(1+2*L₆/3*(1+3*L₆/4*(1+4*L₆/5))))→L₃
abs(L₂-L₃)→L₄
Disp "L₁:X, L₂:LN, L₃:APPROX","L₄:ABS ERROR"
SetUpEditor L₁,L₂,L₃,L₄
Disp "PRESS STAT, 1. EDIT"
Test: x range: 0.5 to 11, increments of 0.5
Final Thoughts
The approximation for ln x hold much better than e^x as x increases. I would not recommend the following approximations for values above 5.
Source:
Smith, Jon M. Scientific Analysis on the Pocket Calculator John Wiley & Sons: New York. 1975
Eddie
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