Ah very nice, I was close with using max error - 0.05051 is the same number I got. Pretty sure 0x7ef311c2 came up for me at least a few times as I was fiddling with parameters. Is this using minimum good bits as the deciding criteria, or is it the best overall number using one of the averages and also 1-3 NR steps? Did you limit the input range, or use all finite floats? Having the min/avg error in bits is nice, it’s more intuitive than relative error.
I like the FMA simulation, that’s smart; I didn’t think about it. I did my search in Python. I don’t have it in front of me right now, and off the top of my head I’m not even sure whether my NR steps are in Python precision or fp32. :P My posts in this thread were with NR turned off, I wanted to find the best raw approximation and noticed I got a different magic number when using refinement. It really is an amazing trick, right? Even knowing how it works it still looks like magic when plotting the result.
Thanks for the update!
BTW I was also fiddling with another possible trick that is specific to reciprocal. I suspect you can simply negate all the bits except the sign and get something that’s a decent starting point for Newton iters, though it’s a much worse approximation than the subtraction. So maybe (x ^ 0x7fffffff). Not sure if negating the mantissa helps or if it’s better to negate only the exponent. I haven’t had time to analyze it properly yet, and I don’t know of any cases where it would be preferred, but I still think it’s another interesting/cute observation about how fp32 bits are stored.
I like the FMA simulation, that’s smart; I didn’t think about it. I did my search in Python. I don’t have it in front of me right now, and off the top of my head I’m not even sure whether my NR steps are in Python precision or fp32. :P My posts in this thread were with NR turned off, I wanted to find the best raw approximation and noticed I got a different magic number when using refinement. It really is an amazing trick, right? Even knowing how it works it still looks like magic when plotting the result.
Thanks for the update!
BTW I was also fiddling with another possible trick that is specific to reciprocal. I suspect you can simply negate all the bits except the sign and get something that’s a decent starting point for Newton iters, though it’s a much worse approximation than the subtraction. So maybe (x ^ 0x7fffffff). Not sure if negating the mantissa helps or if it’s better to negate only the exponent. I haven’t had time to analyze it properly yet, and I don’t know of any cases where it would be preferred, but I still think it’s another interesting/cute observation about how fp32 bits are stored.