Which values cannot be represented correctly by a

Question

The Double data type cannot correctly represent some base 10 values. This is because of how floating point numbers represent real numbers. What this means is that when representing monetary values, one should use the decimal value type to prevent errors. (feel free to correct errors in this preamble)

What I want to know is what are the values which present such a problem under the Double data-type under a 64 bit architecture in the standard .Net framework (C# if that makes a difference) ?

I expect the answer the be a formula or rule to find such values but I would also like some example values.

Answer 1

This question actually goes beyond any single programming language or platform. The inaccuracy is actually inherent in binary data.

Consider that with a double, each number N to the left (at 0-based index I) of the decimal point represents the value N * 2^I and every digit to the right of the decimal point represents the value N * 2^(-I).

As an example, 5.625 (base 10) would be 101.101 (base 2).

Given this calculation, and decimal value that can't be calculated as a sum of 2^(-I) for different values of I would have an incorrect value as a double.

Answer 2

You generally need to be prepared for the possibility that any value you store in a double has some small amount of error. Unless you're storing a constant value, chances are it could be something with at least some error. If it's imperative that there never be any error, and the values aren't constant, you probably shouldn't be using a floating point type.

What you probably should be asking in many cases is, "How do I deal with the minor floating point errors?" You'll want to know what types of operations can result in a lot of error, and what types don't. You'll want to ensure that comparing two values for "equality" actually just ensures they are "close enough" rather than exactly equal, etc.

Answer 3

A float is represented as s, e and m in the following formula

s * m * 2^e

This means that any number that cannot be represented using the given expression (and in the respective domains of s, e and m) cannot be represented exactly.

Basically, you can represent all numbers between 0 and 2^53 - 1 multiplied by a certain power of two (possibly a negative power).

As an example, all numbers between 0 and 2^53 - 1 can be represented multiplied with 2^0 = 1. And you can also represent all those numbers by dividing them by 2 (with a .5 fraction). And so on.

This answer does not fully cover the topic, but I hope it helps.

Answer 4

Any number which cannot be written as the sum of positive and negative powers of 2 cannot be exactly represented as a binary floating-point number.

The common IEEE formats for 32- and 64-bit representations of floating-point numbers impose further constraints; they limit the number of binary digits in both the significand and the exponent. So there are maximum and minimum representable numbers (approximately +/- 10^308 (base-10) if memory serves) and limits to the precision of a number that can be represented. This limit on the precision means that, for 64-bit numbers, the difference between the exponent of the largest power of 2 and the smallest power in a number is limited to 52, so if your number includes a term in 2^52 it can't also include a term in 2^-1.

Simple examples of numbers which cannot be exactly represented in binary floating-point numbers include 1/3, 2/3, 1/5.

Since the set of floating-point numbers (in any representation) is finite, and the set of real numbers is infinite, one algorithm to find a real number which is not exactly representable as a floating-point number is to select a real number at random. The probability that the real number is exactly representable as a floating-point number is 0.