Welcome back to my series of posts on Herbgrind, a dynamic analysis tool that finds floating point issues in compiled programs. The purpose of this series is to explain how Herbgrind works and the principles behind its design. To do that, we’re starting with a simple hypothetical computer called the “Float Machine”, which is just enough to capture the interesting floating point behavior of a real computer, like the one that’s probably sitting on your desk. In the beginning of the series I described the basics of the float machine. Then, we extended it with a “Real” machine for figuring out the correct output of the program. In the previous post, I showed how we can “localize” errors to a particular operation to find the source of error.

The only problem is, one operation alone can’t explain the true cause of floating point error. Each operation on its own is about as accurate as it can be in IEEE754 floating point. It’s only when operations compose that non-trivial error starts to appear. So while the Localizing Machine is good for figuring out where the error first appears, it doesn’t tell the whole story.

You can see this in this simple example:

$(x + 1) - x$

As I mentioned in earlier posts, this computation should always return $1$, but for large values of $x$ it returns $0$ instead. This is because floating point can only represent numbers accurately at one magnitude at a time. So when you add the small value $1$ to a large value of $x$, it gets rounded off, and you just get $x$. But this isn’t bad floating point error yet, because that’s still the closest floating point number to the correct answer, so it’s as “correct” as it can be. However, once you subtract $x$ back out, you get $0$ instead of $1$, and that’s not the most correct answer.

Pointing the Localizing Machine at this thing will tell you that the subtraction is the problem. After all, this is where the value first becomes different from the closest floating point number to the correct answer. But the subtraction alone isn’t enough to express the problem: given the inputs $x$ and $x$, zero is the correct answer. It’s only because we know that the first $x$ is wrong in a very specific way that we know that the subtraction causes a problem.

In general, to figure out enough about the error to fix it, you also need to look at the operations that come before the error appears.

One way to do this is to just look at the source code. Once you know where the error appears, you sometimes only need to look a few lines back to see the relevant operations. Unfortunately this isn’t always the case. Sometimes the operations that matter are scattered across the program, in different functions and even different modules.

The final machine that makes up Herbgrind is called the Symbolic Machine, and is intended to bring together the computation that produced a value so that it can be reported to the user. This way when you find local error, you’ll also know how that value was computed, and therefore which operations need to be fixed to solve the problem.

## How the Symbolic Machine Works

Given the above example, what we want the symbolic machine to do is produce the expression “$\texttt{(x + 1) - x}$”. Since the addition and subtraction could be in different modules though, we can’t trust the code to easily tell us this. Instead, we rely on the values in the program to tell us how they were built.

The Symbolic Machine takes programs, and runs them “symbolically”. When the float machine sees a program that computes $1 + 2$, it produces $3$. But when the symbolic machine sees that same program, it produces an object representing the computation abstractly, in this case “$\texttt{1 + 2}$”.

This is especially powerful when the values flow through the program in interesting ways. Even if this $3$ gets passed around in the program, and put in a hash table by one piece of code and taken out by another, it still has the “symbolic” value “$\texttt{1 + 2}$”. Then, if someone adds $4$ to that $3$, it’ll produce the expression “$\texttt{(1 + 2) + 4}$”, while the client program produces $7$.

So the Symbolic Machine has “symbolic memory” which it uses to track the float memory in the Float Machine. It also has a symbolic processor which looks at each operation in the program, and does the correct “symbolic” operation.

This dynamic approach to finding expressions means that we will never miss part of the computation, because branches and crazy control flow don’t present a problem. Unfortunately it makes it harder for us to find some information about the program that would be obvious in source code, like which values are a constant and which values are a variable. Luckily, we have a few tricks up our sleeves to get that information back.

## Constant or Variable?

When we’re trying to improve a computation, it’s usually pretty important to know which values are “variables” and which values are “constants”. You can do a lot more transformation of the expression when you know that a particular value is going to be always the same value. But try those transformations on a value that changes, and you’ll be in for a bad time!

Unfortunately, as long as we’re being completely dynamic, it’s nearly impossible to figure out which is which perfectly. However, some good guesses can go a long way. For instance: most parts of the program that are standalone “procedures” with “inputs” that make variables, are going to be run more than once. When this happens, the symbolic machine will see the same code fragment run with some of its values different (the variables), and some the same (the constants).

While so far we’ve only been talking about traces of values, that is the history that a particular value went through, now we care about something else: “code fragments”. This is a semi-static (the opposite of dynamic) notion, so it’s worth considering how the two mix. While with our “dynamic” techniques we stored information at each value, now we want to also know something about procedures that are run more than once, so we’re going to consider a different kind of storage. We’ll add another kind of memory to our symbolic machine. But instead of this one having one memory location for each location in the Float Machine’s float memory, it’ll have one memory location for every program instruction.

This new program-based memory will store, for each operation in the program, every trace that was produced at it. Therefore, when we evaluate $(1 + 2) + 3$, we won’t just store $"\texttt{(1 + 2) + 3}"$ in the symbolic memory where the value $3$ goes in float memory, we’ll also store it as one of the traces cooresponding to the $+$ operation.

With each operation holding a set of traces, we can start to infer which values are constant, and which ones vary. Whenever two traces disagree on a value, or even a whole branch of the tree, we’ll call that part a variable. And anything that doesn’t change between traces we’ll call a constant.

For example, let’s say we see a $+$ operation with the traces:

\begin{align} \texttt{1 +}&\texttt{ 4} \\ \texttt{1 +}&\texttt{ 3} \\ \texttt{1 +}&\texttt{ (2 + 5)}\\ \end{align}

Since the 1 on the left hand side of the plus always stays the same, we’ll say it’s a constant. Since the right hand side changes, and is even sometimes another operation, we’ll call it a variable. The resulting inferred expression is:

$\texttt{1 + x}$.

In practice finding this solution algorithmically can be a bit tricky, but we’ll discuss this when we talk about bringing Herbgrind to a real machine, not the abstract Float Machine.

## Variable Matching

In the previous example we gave the variable the name “$x$”. We could have given it a different name, like “$y$” or “$cow$”, but “$x$” is as good a name as any. When there’s only one variable, the name we give it doesn’t matter. When multiple variables enter the picture though, we have to be a little more careful, because we can have different variables that have the same name, and that means something special.

For example, in the expression:

$(x + 1) - x$

We have the same variable, $x$, in two places. Rather than these both being independent variables, this means that they can vary in value, but they are always the same as each other. They could both be $10000000$, or they could both be $3.5$, but they have to match.

To reproduce this equation with our dynamic system, we’ll need to figure out which variables are actually the same variable. Once again, we’ll look at the traces to determine this. To find out whether a value was a constant, we would see if it never changed across all the traces. Similarly, to find out whether two parts of the traces are actually the same variable, we’ll see if their values match across all traces.

For instance, if we see the traces:

\begin{align} \texttt{(1 + 2) - 1} \\ \texttt{(2 + 2) - 2} \\ \texttt{(3 + 4) - 3} \\ \end{align}

We’ll produce the expression:

$\texttt{(x + y) - x}$

In this expression, the left side of the $+$, and the right side of the $-$ are given the same variable name, $x$. The right side of the $+$ is given a different variable name, $y$. This is because, while the first and third number vary across traces, in every trace they have the same value as each other; in the first they are both $1$, in the second they are both $2$, and in the third they are both $3$.

Even though in the second trace the second number matches the other, we don’t give it the same name. Because it differs in the other traces, we consider it a different variable. Herbgrind only treats two variables as the same if they match in every trace.

## The Rules of the Game

To make this process of inferring operation trees more concrete, let’s give it a name, and some rules. We’ll call this “generalizing the traces” into a “symbolic expression”. Symbolic, because while the traces just involve numbers and operations, when we generalize them we get variables also, which are “symbols” representing more than one value.

The rules of this process are all about not being too specific, and not being too general. If the expression is too specific, then it won’t properly be able to represent the different traces. If it’s too general, than it won’t give you enough information.

What does it mean to be too specific? Well, a variable like $x$, $y$, or $cow$, can represent anything. A number, an operation, whatever. A variable by itself is never too specific. On the opposite end, an operation, or a constant, represents only a single thing: either running that operation, or that constant. We’re also going to do something kind of weird here, and say that an operation on constants can be “represented” by its result. For instance, the trace:

$\texttt{(1 + 2) - 3}$

can be represented as “$\texttt{(1 + 2) - 3}$”, or “$\texttt{(x + y) - z}$”, or “$\texttt{3 - 3}$”, or “$\texttt{0}$”, or even “$\texttt{x}$”. That’s a lot of options! If instead we had two traces:

\begin{align} \texttt{(1 + 2) - 3}\\ \texttt{(1 + 2) - 4} \end{align}

Then we wouldn’t be able to use “$\texttt{(1 + 2) - 3}$”, “$\texttt{3 - 3}$”, or “$\texttt{0}$”, because they are now too specific; they don’t encompass all of the traces they are supposed to represent. But we could still use “$\texttt{(1 + 2) - z}$”, “$\texttt{(x + y) - z}$”, “$\texttt{3 - x}$”, or “$\texttt{x}$”.

With so many options, we need to figure out which one is best. The easiest one to pick is always $x$, because it can represent anything. However, it doesn’t tell the user very much, and it’s not useful for rewriting the code to be more accurate. The more specific of a trace we can give, the more chance we have of helping the user. So we’ll always look for the most specific trace that is still general enough to represent all the traces that we care about.

In the case of:

\begin{align} \texttt{(1 + 2) - 3} \\ \texttt{(1 + 2) - 4} \end{align}

the most specific trace that is general enough is:

$\texttt{(1 + 2) - x}$

Because in both traces the left hand side of the subtraction is “$\texttt{(1 + 2)}$”, it’s not too specific to use that for the generalized symbolic expression. And, it’s the most specific trace possible, since it’s completely concrete (just numbers and operations) and doesn’t pre-compute any operations.

On the right hand side of the subtraction, we have a variable $x$, because the value in that position is different in the two different traces. We couldn’t make it a concrete value like $3$ or $4$, because that would be too specific, and wouldn’t represent both traces.

Wooh, okay. That’s the Symbolic Machine, the last machine in Herbgrind. Now that we’ve got all the machines in one place, we can do some pretty cool things. We can find that there is error in the output of a program, track it down to a particular part of the computation, and represent that computation as a symbolic expression to the user. From there, it’s up to the user to fix the computation, and put the fixed version back in the program.

What we’ve defined up until now has been on an abstract machine, called the Float Machine. While the Float Machine has many of the things a normal computer does, there are still a few differences that make implementing Herbgrind a bit tricky. In future posts, I’ll talk about these differences, and what Herbgrind does to work on normal computers.

Still though, what we’ve defined so far is most of the work. I hope it’s been useful to anyone out there trying to track down and understand floating point error!