Question
Answer and Explanation
In the context of computing, especially within areas involving floating-point arithmetic like computer graphics and scientific simulations, the term "Float32 EPS" refers to the machine epsilon for single-precision floating-point numbers (specifically, the 32-bit IEEE 754 floating-point format). So, What exactly is EPS?
EPS, short for Epsilon, is defined as the smallest representable positive number that, when added to 1, yields a result that is different from 1 within the constraints of floating-point precision. Put more technically, it's the distance between 1 and the next largest representable floating-point number. Because computers represent floating-point numbers with finite precision, not all numbers can be stored exactly, leading to small rounding errors. EPS is an inherent limit on how finely you can differentiate two similar floating point values in your system.
For a float32 number (also known as "single precision"), which occupies 32 bits of memory, EPS is approximately equal to 1.1920929 x 10-7 or more precisely 2-23. It represents the smallest increment that you can effectively add to the value one without rounding to the exact value. That value depends on the used architecture/programming language.
Why is this Important?
1. Precision Awareness: Understanding EPS is crucial in numerical calculations and simulations. You can't depend on every float being represented perfectly. Instead of direct comparison of equality of two float values ( floatA == floatB) , comparing if they are “near” to each other is necessary: `if( abs(floatA - floatB) < float32Epsilon factor)`
where `factor` is small integer.
2. Error Handling: In mathematical calculations where cumulative errors are likely, using EPS can assist to check the tolerance.
3. Algorithm Design: Some algorithms involve working with infinitesimals, which aren't real numbers on the computer. Knowing the system's smallest epsilon aids in effective management of such calculations. If a number gets smaller than `epsilon` it may considered equal zero, within the computation/system
How to get/calculate it? Some libraries provide epsilon constants, if they are not readily available in a given programing language you can implement such routine:
For example, a rough approximation in JavaScript to get float32Epsilon:
function getFloat32Epsilon(){
let epsilon = 1.0;
let nextEpsilon = 1.0;
while((1.0 + nextEpsilon) > 1.0) {
epsilon = nextEpsilon;
nextEpsilon /= 2.0;
}
return epsilon;
}
const float32Epsilon = getFloat32Epsilon();
console.log("Float32 EPS is:", float32Epsilon); // Expected output will be roughly 1.1920929e-7, check it to make sure.
This algorithm reduces the current tested value of the epsilon until, while adding to one ,the outcome remains 1 and will provide correct machine precision value of epsilon.
Keep in mind `float32` numbers behave in some unique, unusual sometimes, and counter-intuitive way, for example consider floatA = 0.1 + 0.2 the outcome of such addition may be a bit off of expected `0.3`. Awareness of Float32 EPS help with understating the limitations of these number in computing world and to adjust accordingly when designing such calculations.