And then she writes about logic cells, hardware pipeline architectures, RTL (register-transfer level), ASIC floorplans, and tapeout. Building hardware is hard, as iteration cycles are long and fabricating stuff costs real money. That was quite a journey.
Her "about" page adds more context that helps everything make sense:
> Professionally, I have a Masters in Electrical and Computer Engineering, have previously worked as a CPU designer at ARM and an FPGA engineer at Optiver. -- https://essenceia.github.io/about/
Consider me educated on the mantissa. That's a nifty pedantry.
Typo in the c++?
> cout << "x =/= x: " << ((x=x) ? "true" : "false") << endl;
Should be x != x?
For the leading 0 counter, I've found it's even better for the tool if I use a for loop. Keeps things scalable and it's even less code. I'm not understanding this takeaway though
> Sometimes a good design is about more than just performance.
The good design (unless the author's note that it's easier to read and maintain makes it worse design?) was better performing. So sometimes a good design creates performance.
Likewise for pipelining: it would have been interesting to know if the tools can reorder operations if you give them some pipeline registers to play with. In Xilinx land it'll move your logic around and utilize the flops as it sees fit.
Let me offer a nitpck: in the "Gradual underflow" section it says this about subnormal numbers:
Bonus: we have now acquired extra armour against a division by zero:
if ( x != y ) z = 1.0 / ( x - y );
Think about it this way: the smallest subnormal double is on the order of 10^-324, but the largest double is on the order of 10^308. If `x - y` is smaller than 10^-308, `1.0 / (x - y)` will be larger than 10^308, which can't be represented and must overflow to infinity.
This C program demonstrates this:
#include <stdio.h>
#include <float.h>
#include <math.h>
// return a (subnormal) number that results in zero when divided by 2:
double calc_tiny(void)
{
double x = DBL_MIN; // the normal number closest to 0.0
while (1) {
double next = x / 2.0;
if (next == 0.0) {
return x;
}
x = next;
}
}
int main(void)
{
double y = calc_tiny();
double x = 2.0 * y;
if (x != y) {
double z = 1.0 / (x - y);
printf("division is safe: %g\n", z);
} else {
printf("refusing to divide by zero\n");
}
}
- It’s not just ‘old’ FPUs that handle them verrry slowly. Benchmark this aspect of your target processors if this detail matters.
- Most modern processors provide a “flush to zero” denormal handling mode (and sometimes you can separately specify “denormals are zero”, relevant when e.g. loading old data). However, math libraries usually haven’t been written with this mode in mind, so you need to be careful with this.
It has many many warts, and many design choices were made given the constraints of hardware of that time, not by considerations in terms of a standard.
https://people.eecs.berkeley.edu/~wkahan/ieee754status/754st...
But decimal floats fall short in subtle ways. Here is the simplest example - sales tax. In Ontario it's 13%. If you buy two items for $0.98 each, the tax on each is $0.1274. There is no legal, interoperable mechanism to charge the customer a fractional number of cents, so you just can't do that. If you are in charge of producing an invoice, you have to decide where to perform the rounding(s). You can round the tax on each item, which is $0.13 each, so the total is ($0.98 + $0.13) × 2 = $2.22. Or you can add up all the pre-tax items ($1.98) and calculate the tax ($0.2548) and round that ($0.25), which brings the total to $0.98×2 + $0.25 = $2.21, a different amount. Not only do you have to decide where to perform rounding(s), you also have to keep track of how many extra decimal places you need. Massachusetts's sales tax is 6.25%, so that's two more decimal places. If you have discounts like "25% off", now you have another phenomenon that can introduce extra decimal places.
If you do any kind of interest calculation, you will necessarily have decimal places exploding. The simplest example is to take $100 at 10% annual interest compounded annually, which will give you $110, $121, $133.1, $146.41, $161.051, $177.1561, etc., and you will need to round eventually. Or another example is, 10% annual interest, but computed daily (so 10%/365 per day) and added to the account at the end of the month - not only is 10%/365 inexact in decimal arithmetic, but also many decimal places will be generated in the tiny interest calculations per day.
If you do anything that philosophically uses "real numbers", then decimal FP has zero advantages compared to binary FP. If you use pow(), exp(), cos(), sin(), etc. for engineering calculations, continuous interest, physics modeling, describing objects in 3D scene, etc., there will necessarily be all sorts of rational, irrational, and transcendental numbers flying around, and they will have to be approximated in one way or another.
Overall, yes, results need to be rounded, but it's pretty much financial software 101 not to use floats.
immediately followed by
"Time to run some tests!"
Prompted a grin. Then however "If we wanted to test this using directed testing we would need to test for all 2^32 input combinations, which sounds like a terrible idea …
… and exactly what I am going to do "
Wait, exhaustively testing float number space ... I read about that before. Might have been https://randomascii.wordpress.com/2014/01/27/theres-only-fou... covered also here https://news.ycombinator.com/item?id=34726919
Just keeping track of the shifts during a chain of multiplications and additions can really ruin your day. And the good code will look exactly the same as the bad code. I'm doing something like that right now and have moved from doubles to fixed point 64 bit ints (32.32), it works, but it took me much longer than I thought it would (phase angle estimator for SDR output).
Recently wrote a chapter in tiny-vllm course about floats in context of LLM inference, much shorter and not that deep as this one, for anyone interested in topic you might like it too https://github.com/jmaczan/tiny-vllm?tab=readme-ov-file#how-...
A really cool thing to see would be the newer block scaled fp4/fp8 data types. Maybe for their next asic they can try it haha - https://arxiv.org/abs/2310.10537