An unfortunate reality of trying to represent continuous real numbers in a fixed space (e.g. with a limited number of bits) is that this comes with an inevitable loss of both precision and accuracy. Although floating point arithmetic standards – like the commonly used IEEE 754 – seek to minimize this error, it’s inevitable that across the range of a floating point variable loss of precision occurs. This is what [exozy] demonstrates, by showing just how big the error can get when performing a simple division of the exponential of an input value by the original value. This results in an amazing error of over 10%, which leads to the question of how to best fix this.
Obviously, if you have the option, you can simply increase the precision of the floating point variable, from 32-bit to 64- or even 256-bit, but this only gets you so far. The solution which [exozy] shows here involves using redundant computation by inverting the result of ex. In a demonstration using Python code (which uses IEEE 754 double precision internally), this almost eradicates the error. Other than proving that floating point arithmetic is cursed, this also raises the question of why this works.
Simplified numerator and denominator with floating point delta. (Credit: exozy)
For the explanation some basic understanding of calculus is rather helpful, as [exozy] looks at the original function (f(x)) and the version with the logarithm added (g(x)). With an assumption made about t ..
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