Whenever I see these tricks(see also: the quake 3 fast inverse sqrt) involving using, not casting, but using integers as floats directly and floats as integers, I wonder if there is a way to do it without the jank.
Because what do you really want? some sort of exponent or exponent math right, some variant of the log function should work. is the problem is all the log functions are gated behind the function call interface. where as the subtract function is less heavy being behind the operator interface. or are they trying to use a floating point accelerated log aka floating point subtract?
Yes of course there’s a more graceful approach. You can use an actual divide instruction or routine instead of the bit-casting subtraction trick.
It’s not about interfaces. The point is that a single subtract is simpler math, and (probably) faster than a divide, and it gets you surprisingly close to the right answer. The reason it works is subtracting two exponents is a subtraction in log space which is equivalent to a divide in linear space.
People don’t use this trick in practice that much if at all, especially these days with GPUs that do reciprocals and square roots in hardware, it’s just an interesting thing to know about that helps solidify our understanding of floating point numbers and logarithms.
For more general algorithms along these lines, see exercises 25–28 in section 1.2.2 of Knuth[1] (and note that the printed solution to exercise 28 in early printings has an error[2]).
[1] D.E. Knuth, The art of computer programming. Vol. 1, third edition, Addison-Wesley, Reading, MA, 1997.
That knuth check story was pretty great. it reminded me of when I had to write pow() from scratch many years ago. However mine was not optimized in any way, and was a very sluggish naive implementation of an algorithm I found in the nist mathematical function library.
I was young and worked night shifts feeding tapes to a ibm mainframe, in the down time I would amuse myself by reading the ibm manuals laying around and writing scripts on the operator console, some useful and some not so much.
One of my more useless scripts was a sort terminal characters drawing routine, I wanted to draw lines at a specific angle which would require sin(), cos(), however, the scripting language used had no included trig functions. So I looked it up. and smuggled in a printout of the nist reference sin function. At which point I found, to my dismay, there also was no floating point power function. So the next night I smuggled in the printout for that one as well. And now, armed with a highly questionable implementation of sin(), cos(), pow() I could finally draw the hands for my stupid analog clock screensaver on the 3270 terminal. Honestly probably the highlight of my time there, but I was too scared to let anyone know about that script, didn't want them to think I had too much downtime.
If you're not constrained to software solutions you have a whole world of opportunities. E.g. if it's a graphics or neural net pipeline you can pour tricks like this (or better) onto it. If it's a CPU then you can add special instructions that do exponent manipulation and the likes.
Because what do you really want? some sort of exponent or exponent math right, some variant of the log function should work. is the problem is all the log functions are gated behind the function call interface. where as the subtract function is less heavy being behind the operator interface. or are they trying to use a floating point accelerated log aka floating point subtract?