Maximizing Multiplier Based on Precision in Python

Question

Given the following equation:

x / y = z

Where the precision of x is 5 and the precision of z is 8, given the following values of x, y:

from decimal import Decimal

x = Decimal('15.00000')
y = Decimal('81.63')
x_prec = 5
z_prec = 8


z_unrounded = x / y # 0.1837559720690922454979786843...
z = round(z_unrounded, z_prec) # 0.18375597

the value of 0.18375597 for z is produced, which when multiplied by y gives a value of 14.9999998311.

Given the precision of x, we can "max" the value of z to 0.18375603, since:

z_max = Decimal('0.18375603')
x_unrounded = y * z_max  # 15.0000047289
x = round(x_unrounded, x_prec) # 15.00000

I know that I can find this z_max value in a loop:

extra_z = Decimal('0.0')
min_z = Decimal('0.00000001')

while True:
    next_z = z + extra_z + min_z
    if round(y * next_z, x_prec) > x:
        break
                
    extra_z += min_z

extra_z # 0.00000006
max_z = z + extra_z # 0.18375603

But using this sort of method feels wrong, or at least that I’m missing some key insight about precision and Decimal values.

Is there a clearer or more algebraic way of calculating max_z?

Asked By: Joshua Lusk

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Source

Answer 1

You are talking about significant figures. Wiki has a lot of info on it.

If x=1.0 and y=1.0, which have both 2 significant numbers, and you want to find the maximum allowable range for z, then checking the limits for the number x will suffice. In this case the range is from 0.95 to 1.049999. These two values are the limits if x_prec = 2 and will both round to 1.0.

Answered By: Bart

Answer 2

Generally speaking, the precision of a product (or quotient) of two numbers is the minimum precision of the two factors (or of dividend and divisor).

To a first approximation (pun intended), check out what Wikipedia says about significant figures.
For a fully general answer, look into the mathematical field of error analysis.

In your particular case, the precision of x is 5, so any product/division involving x can’t have higher precision than that. The number 0.18375 does seem to have a precision of 5.

Being a calculated quantity, the precision of z of course depends on that of y as well; it would be min(x_prec, y_prec), even if z nominally has a precision of 8.

I agree with the other answerer that, once you know the allowable range for each of x and y, you can just calculate the corresponding range for z as zMin = xMin/yMax and zMax = xMax/yMin (easy in this case). For a fully general answer, error analysis is your friend.

Answered By: Lover of Structure

Maximizing Multiplier Based on Precision in Python

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