[splittingTimes]

That's why when I did numerical simulation of electron Dynamics in semiconductors during my phD we never used straight SI units (m, s, kg, etc), but instead expressed all physical natural constants in nm, fs, eV, etc. That way all relevant constants had numerical values between 1 and 10 which stabilized the simulations a lot.

[petschge]

Expressing all quantities as small multiples of suitable reference quantities is quite common. The resulting equations after simplifications are often referred to as dimensionless (because the reference quantity has absorbed the units and only a small dimensionless number remains).

There is two upsides to this: 1.) All quantities are order one, safely away from underflow and infinity. 2.) The equations are generally simpler, which was important when the many limitation in simulation codes was flops, not memory bandwidth.

There are also important downsides: 1.) You have to manipulate the equations before you start coding. Porting features from a code that used another normalization is quite error-prone. 2.) All quantities are order one, which makes it hard to detect if you accidentally use and electric field instead of a magnetic field. All you see is a number of roughly the correct magnitude. 3.) Comparison between codes is more difficult, in particular if the use different normalization, because you have to convert from "code units" to actual SI units.

[mschuetz]

File formats for georeferenced data also regularly store coordinates as 32 bit ints, with a scaling argument that can be used to convert the integer into a double precision number with a specific unit. E.g. your scene unit is meters and you want to store with a precision of millimeters? Multiply double precision coordinates by 1000, convert the result to an int32, save it to file. When you load the coordinate afterwards, divide it by 1000 and you you get back your double precision coordinate.

[swimfar]

I had never heard of this idea before. Do you have any references to this technique? Also, depending on what equations you're using, aren't you constrained to using a consistent system of units?

[splittingTimes]

Never anything official. It was just how it was done in the group and wider scientific community. Some googling pulled that up if it helps

https://books.google.de/books?id=uzJbyD6_3DAC&pg=PA4

And yes you have to be consistent and stay in that unit system. The other remaining two we used plain Kelvin for temperature and for the charge Coulomb was measures in electron charge. That's it.

[Scaevolus]

That shouldn't matter for the factors you're describing. The number of floats between 1 and 2 is the same as the number of floats between 2^-30 and 2^-31. Until you hit denormal numbers and start overflowing or underflowing your exponent, it doesn't have any effect on precision.

If you have a process that converts floats into other formats with more restricted exponents, like human readable strings, it might matter.