Which function is more accurate?

[blalien]

Homework Statement

You have a number system in base 10 with a precision of 5 digits. Which function is more accurate: x^2-y^2 or (x-y)(x+y)?

Homework Equations

None really.

The Attempt at a Solution

My intuition would tell me that (x-y)(x+y) is more accurate, since multiplication is less accurate than addition or subtraction (unless the two numbers are very close). But look at x = 1.0000, y = 0.00001:

Actual value: 0.999999999

x^2 = 1, y^2 = 1e-10 ~ 0
x^2-y^2=1

x-y = 0.99999
x+y = 1.00001 ~ 1
(x-y)(x+y) = 0.99999

So x^2-y^2 is more accurate, at least in this situation. I have no idea why, though.

 

[story645]

So x^2-y^2 is more accurate, at least in this situation. I have no idea why, though.

Which class is this for? If it's a programming class/computer architecture class/etc. the answer probably has to do with floating point errors and accumulated rounding errors.

 

[Mene]

it is important to understand that accuracy and precision are two different concepts. Accuracy refers to how close a measurement or calculation is to the true or expected value, while precision refers to how consistent or repeatable the measurement or calculation is. In this case, both functions have the same level of precision since they have the same number of significant digits (5). However, accuracy can vary depending on the values of x and y.

In the given example, x^2-y^2 may be more accurate because it avoids the potential for rounding errors that can occur when multiplying two numbers with different magnitudes. This is because the difference between x^2 and y^2 is smaller than the difference between x and y. On the other hand, (x-y)(x+y) involves both multiplication and addition, which can introduce more opportunities for rounding errors.

However, this does not mean that x^2-y^2 is always more accurate than (x-y)(x+y). In different scenarios, the accuracy may vary depending on the values of x and y. it is important to carefully consider the potential sources of error and choose the most appropriate function based on the specific situation at hand.