[SOLVED] Precison of digital computers

Issue

I read that multiplying multiple values between 0 and 1 will significantly reduce the precision of digital computers; I want to know the basis on which such postulate is based? And does it still holds for modern-day computers?

Solution

The typical IEEE-conformant representation of fractional numbers only supports a limited number of (binary) digits. So, very often, the result of some computation isn’t an exact representation of the expected mathematical value, but something close to it (rounded to the next number representable within the digits limit), meaning that there is some amount of error in most calculations.

If you do multi-step calculations, you might be lucky that the error introduced by one step is compensated by some complementary error at a later step. But that’s pure luck, and statistics teaches us that the expected error will indeed increase with every step.

If you e.g. do 1000 multiplications using the float datatype (typically achieving 6-7 significant decimal digits accuracy), I’d expect the result to be correct only up to about 5 digits, and in worst case only 3-4 digits.

There are ways to do precise calculations (at least for addition, subtraction, multiplication and division), e.g. using the ratio type in the LISP programming language, but in practice they are rarely used.

So yes, doing multi-step calculations in datatypes supporting fractional numbers quickly degrades precision, and it happens with all number ranges, not only with numbers between 0 and 1.

If this is a problem for some application, it’s a special skill to transform mathematical formulas into equivalent ones that can be computed with better precision (e.g. formulas with fewer intermediate steps).

Answered By – Ralf Kleberhoff

Answer Checked By – Terry (BugsFixing Volunteer)

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