Wow, it’s been a long time since I posted anything here. If you’ve read the last post on this blog, you’ll know that I was working on Herbie-like improvement of looping programs. Well, it’s been a long journey since then, and the unexpected result is a new tool, Herbgrind, to help programmers weed out dubious floating point code in large numerical programs.
So how did I get there from my Herbie-loops work? Let me tell you a story.
Herbie has always been a very benchmark driven project. From day one we had 28 main benchmarks that followed us throughout the project, taken from Richard Hamming’s Numerical Methods for Scientists and Engineers. Early on we built a website to report Herbie’s results on these benchmarks, and we started running on them regularly even earlier. This benchmark-based approach served us well; it kept us from spending time on features that weren’t helpful, and kept us honest about Herbie’s abilities.
Because of our benchmarks, we never lacked for new ideas to add to Herbie’s search. All we ever had to do was pick a benchmark that Herbie wasn’t succeeding at, and ask ourselves how we would solve the problem by hand. If we could encode that intuition into a search process, we had ourselves a new search component. All of Herbie’s systems came about through this process, from simplification, to local error, to recursive rewriting: we saw a concrete problem in one of our benchmarks, and we tried to solve it in general.
So when it came time to work on loops, the first thing to do was gather the benchmarks. Unfortunately, we couldn’t find many looping examples in books like Numerical Methods for Scientists and Engineers. In fact, we couldn’t really find them anywhere. Now, this probably speaks as much to our unfamiliarity with the numerical methods literature as anything, but the problem remained. We had no benchmarks, and without benchmarks, we couldn’t develop Herbie-loops the way we’d developed Herbie.
The first thing I did was to try to write my own benchmarks. Sure, they wouldn’t be as authentic as ones in the wild, but it’s a start. I wrote some loops that calculate the mean of a list, and its variance using two different formulas. I wrote a simple partial differential equation solver, and gave it some differential equations to solve. Then I implemented a hacky version a Kahan Summation to improve them, and let it run.
It worked well. Too well. Pretty much every benchmark had its error completely improved using this simple trick. It turns out, all the benchmarks I came up with had at their heart, a sum of a bunch of similar elements. I racked my brain for more complicated examples, but I couldn’t find any. I knew that big numerical programs had challenging loops at their core, that needed more than my one simple trick to improve them. But I needed to find them.
I immediately hit a problem. I didn’t understand the source code of this software, even on a surface level. If I was going to find the numerical issues in it, I would need tools. FpDebug was a step in the right direction, but it wasn’t quite what I needed. While it tells you when floating point error occurs, it doesn’t let you know where the problematic values came from.
Luckily, building tools to help people with floating point is exactly my area of expertise. And so, work on Herbgrind began.
Herbgrind is a dynamic analysis to help people track down dubious floating point code in their programs. Because it works with Valgrind at the binary level, it works on a variety of architectures and source languages. All you need to do is run it on your binary with debug information, on some input data that you think is representative. When it finds floating point error that hurts your program, it’ll let you know, and try to tell you what computation caused it. The output looks like this:
Result in main at diff-roots.c:21 (address 1) 47.252184 bits average error 64.000000 bits max error Aggregated over 1000 instances Influenced by erroneous expression: (FPCore (x) (- (sqrt (+ 1.000000 x)) (sqrt x))) in main at diff-roots.c:20 (address 400A08) 47.252184 bits average error 64.000000 bits max error 47.250599 bits average local error 64.000000 bits max local error Aggregated over 1000 instances
So that’s Herbgrind, in a nutshell. In the next few weeks, I’m going to try to explain how Herbgrind does what it does, and what the process of developing it was like.