Systematically avoiding arbitarily large exponents in calculations with small answers

In summary, avoiding arbitrarily large exponents can improve the accuracy and efficiency of calculations by preventing overflow errors and loss of precision. This can be achieved by breaking down calculations into smaller steps or using scientific notation. It is important to avoid arbitrarily large exponents in scientific calculations to ensure reliable and accurate results. Consistently avoiding arbitrarily large exponents can be achieved by setting clear guidelines and protocols for handling calculations with small answers.
  • #1
kmarinas86
979
1
[tex]y=\frac{100}{100}\frac{\frac{(((100/100)^2-100*100)e^{x/(100*100)}+(100*100))e^{-x/(100*100)})-(100/100)^2e^{-x*100/100}}{100/100-100*100}}{1}[/tex]

http://www.quickmath.com/msolver//graphs/2011-12-06/e5/61/e2/e561e26ac85a4b1da0176565a4827772-2.png?t=1323201139

Code:
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704	#NUM!

How do I avoid arbitrarily large exponents in calculations with such small answers? Is there a way of getting rid of them?
 
Last edited by a moderator:
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  • #2


You need to clarify your expression. You have several terms (100/100 = 1) which are confusing. Other terms are 100*100 = 10000. Why the product?
 
  • #3


Modified:

kmarinas86 said:
[tex]y=\frac{\left((1-10000)e^{x/10000}+10000)e^{-x/10000}\right)-e^{-x}}{1-10000}[/tex]

http://www.quickmath.com/msolver//graphs/2011-12-06/e5/61/e2/e561e26ac85a4b1da0176565a4827772-2.png?t=1323201139

Code:
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704	#NUM!

How do I avoid arbitrarily large exponents in calculations with such small answers? Is there a way of getting rid of them?
 
Last edited by a moderator:
  • #4


You could start by using the approximation eu = 1 + u + u2/2 for small u.
 
  • #5


kmarinas86,

The parentheses aren't balanced in your modified version.
 
  • #6


kmarinas86 said:
[tex]y=\frac{100}{100}\frac{\frac{(((100/100)^2-100*100)e^{x/(100*100)}+(100*100))e^{-x/(100*100)})-(100/100)^2e^{-x*100/100}}{100/100-100*100}}{1}[/tex]

http://www.quickmath.com/msolver//graphs/2011-12-06/e5/61/e2/e561e26ac85a4b1da0176565a4827772-2.png?t=1323201139

Code:
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703	0.067792636
704	#NUM!

How do I avoid arbitrarily large exponents in calculations with such small answers? Is there a way of getting rid of them?

It appears that this was easier than I thought:

[tex]y=\frac{100}{100}\frac{\frac{((((100/100)^2-100*100)e^{x/(100*100)}+(100*100))e^{-x/(100*100)})-(100/100)^2e^{-x*100/100}}{100/100-100*100}}{1}[/tex]

[tex]y=\frac{100}{100}\frac{((((100/100)^2-100*100)e^{x/(100*100)}+(100*100))e^{-x/(100*100)})-(100/100)^2e^{-x*100/100}}{100/100-100*100}[/tex]

[tex]y=\frac{(((1-10000)e^{x/10000}+10000)e^{-x/10000})-e^{-x}}{1-10000}[/tex]

[tex]y=\frac{(((1-10000)e^{x/10000}e^{-x/10000})+10000e^{-x/10000}))-e^{-x}}{1-10000}[/tex]

[tex]1=e^{x}e^{-x}[/tex]

[tex]y=\frac{1-10000+10000e^{-x/10000}-e^{-x}}{1-10000}[/tex]

I figured this out a few days ago, but I hadn't decided to post it until now.
 
Last edited by a moderator:

1. How does avoiding arbitrarily large exponents affect the accuracy of calculations?

Avoiding arbitrarily large exponents can improve the accuracy of calculations by preventing overflow errors and loss of precision. This is especially important when working with small numbers, as even slight rounding errors can significantly impact the final result.

2. What techniques can be used to systematically avoid arbitrary large exponents?

One technique is to break down the calculation into smaller steps, using intermediate results that are within a manageable range. Another approach is to use scientific notation, which allows for a more compact representation of numbers with large or small exponents.

3. Why is it important to avoid arbitrarily large exponents in scientific calculations?

In scientific calculations, accuracy is crucial. Using arbitrarily large exponents can lead to errors and inaccuracies, which can impact the validity and reliability of research findings. It is important to use precise and consistent methods to ensure accurate results.

4. Can avoiding arbitrarily large exponents also affect the efficiency of calculations?

Yes, avoiding arbitrarily large exponents can also improve the efficiency of calculations. By breaking down the calculation into smaller steps, the overall computational load is reduced, making it faster and more efficient. Additionally, using scientific notation can also save time and memory space.

5. How can one ensure that arbitrarily large exponents are consistently avoided in calculations?

One way to ensure consistent avoidance of arbitrarily large exponents is to establish clear guidelines and protocols for handling calculations with small answers. This can include using scientific notation, setting limits for exponent values, and regularly checking for and correcting any potential errors.

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