First, a paper you should consider reading, if you want to understand floating point foibles better: “What Every Computer Scientist Should Know About Floating Point Arithmetic,” http://www.validlab.com/goldberg/paper.pdf And now to some meat. The following code is bare bones, and attempts to produce an IEEE-754 single precision float from an unsigned int in the range 0 … Read more
From N3337: [basic.fundamental/8]: There are three floating point types: float, double, and long double. The type double provides at least as much precision as float, and the type long double provides at least as much precision as double. The set of values of the type float is a subset of the set of values of … Read more
When you have a subnormal number with odd significand, that is, an odd multiple of 2^(-1074), you have a number whose last nonzero digit in the decimal representation is the 1074th after the decimal point. You then have around 300 zeros directly following the decimal point, so the number has around 750-770 significant decimal digits. … Read more
Update: Since I wrote the original answer below, the situation has changed slightly. CPython versions 3.11 and later now require that the platform C double follows the IEEE 754 binary64 format. This was mostly a matter of convenience for developers – it allowed us to remove special-case code that was in practice close to untestable. … Read more
It was thought to be a good idea when IEEE754 and NaN’s were developed. I have actually seen it used to store the reason why a NaN was created. Today, I wouldn’t use it in portable code for several reasons. How sure are you that this payload will survive for example an assignment? If you … Read more
Using the new ArrayBuffer access arrays, it is actually possible to retrieve the exact mantissa and exponent, by extracting them from the Uint8Array. If you need more speed, consider reusing the Float64Array. function getNumberParts(x) { var float = new Float64Array(1), bytes = new Uint8Array(float.buffer); float[0] = x; var sign = bytes[7] >> 7, exponent = … Read more
I suspect the restrictions should be taken as referring to the relation’s behavior on the values actually used as keys, not necessarily on all values of the type. Don’t have time at the moment to go through the standard looking for “smoking gun” language that refers to actual container elements rather than all values of … Read more
There are functions available for doing exactly that, but they can depend on what language you use. Two examples: if you have access to a decent C99 math library, you can use nextafter (and its float and long double variants, nextafterf and nextafterl); or the nexttoward family (which take a long double as second argument). … Read more
Just to make sure we’re on the same page, G is the most significant bit of the three, R comes next and S can be thought of as the least significant bit because its value partially represents the even less significant bits that have been truncated in the calculations. These three bits are only used … Read more
23 fraction bits (22-0) of the significand appear in the memory format but the total precision is actually 24 bits since we assume there is a leading 1. This is equivalent to log10(2^24) ≈ 7.225 decimal digits. Double-precision float has 52 bits in fraction, plus the leading 1 is 53. Therefore a double can hold … Read more