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How Should You Write a Fast Integer Overflow Check?

Detecting integer overflow in languages that have wraparound semantics (or, worse, undefined behavior on overflow, as in C/C++) is a pain. The issue is that programming languages do not provide access to the hardware overflow flag that is set as a side effect of most ALU instructions. With the condition codes hidden, overflow checks become not only convoluted but also error-prone. An obvious solution — writing overflow checks in assembly — is undesirable for portability reasons and also it is not as performant as we would hope: assembly has opaque semantics to the higher-level language and the overflow checks cannot be removed in cases where the overflow can be shown to not happen.

In this post we’ll look at various implementations for this addition function:

 * perform two's complement addition on the first two arguments;
 * put the result in the location pointed to by the third argument;
 * return 1 if overflow occurred, 0 otherwise
int checked_add(int_t, int_t, int_t *);

This operation closely mimics a hardware add instruction. Here’s an optimal (at least in the number of instructions) implementation of it for the 64-bit case for x86-64 using the GCC/Linux calling convention:

        xor     %eax, %eax
        addq    %rsi, %rdi
        movq    %rdi, (%rdx)
        seto    %al

The problem is that compilers aren’t very good at taking an implementation of checked_add() in C/C++ and turning it into this code.

This is perhaps the simplest and most intuitive way to check for overflow:

int checked_add_1(int_t a, int_t b, int_t *rp) {
  wideint_t lr = (wideint_t)a + (wideint_t)b;
  *rp = lr;
  return lr > MAX || lr < MIN;

The only problem here is that we require a wider integer type than the one being added. When targeting x86-64, both GCC and Clang support 128-bit integers:

typedef int64_t int_t;
typedef __int128 wideint_t;
#define MAX INT64_MAX
#define MIN INT64_MIN

On the other hand, neither compiler supports 128-bit integers when generating code for a 32-bit target.

If we can’t or don’t want to use the wider type, we can let a narrow addition wrap around and then compare the signs of the operands and result:

int checked_add_2(int_t a, int_t b, int_t *rp) {
  int_t r = (uint_t)a + (uint_t)b;
  *rp = r;
    (a < 0 && b < 0 && r >= 0) ||
    (a >= 0 && b >= 0 && r < 0);

This code is the equivalent bitwise version:

#define BITS (8*sizeof(int_t))

int checked_add_3(int_t a, int_t b, int_t *rp) {
  uint_t ur = (uint_t)a + (uint_t)b;
  uint_t sr = ur >> (BITS-1);
  uint_t sa = (uint_t)a >> (BITS-1);
  uint_t sb = (uint_t)b >> (BITS-1);
  *rp = ur;
    (sa && sb && !sr) ||
    (!sa && !sb && sr);

What if we don’t have a wider type and also we don’t want to use unsigned math? Then we call this function:

int checked_add_4(int_t a, int_t b, int_t *rp) {
  if (b > 0 && a > MAX - b) {
    *rp =  (a + MIN) + (b + MIN);
    return 1;
  if (b < 0 && a < MIN - b) {
    *rp =  (a - MIN) + (b - MIN);
    return 1;
  *rp = a + b;
  return 0;

It looks odd but I believe it to be correct. You can find all of these functions (plus a crappy test harness) in this file and x86-64 assembly code for all widths in this file. Let me know if you have a checked add idiom that’s different from the ones here.

As a crude measure of code quality, let’s look at the number of instructions that each compiler emits, taking the smallest number of instructions across -O0, -O1, -O2, -Os, and -O3. Here I’m targeting x86-64 on Linux and using the latest releases of the compilers: Clang 3.4 and GCC 4.9.0. The results from compilers built from SVN the other day are not noticeably different.

8 bits 16 bits 32 bits 64 bits
GCC checked_add_1() 9 9 11 18
GCC checked_add_2() 14 21 21 21
GCC checked_add_3() 18 18 18 18
GCC checked_add_4() 12 12 16 23
Clang checked_add_1() 5 5 5 14
Clang checked_add_2() 16 16 14 14
Clang checked_add_3() 20 23 14 14
Clang checked_add_4() 18 18 18 22
optimal 5 5 5 5

The only real success story here is Clang and checked_add_1() for 8, 16, and 32-bit adds. The optimization responsible for this win can be found in a function called ProcessUGT_ADDCST_ADD() at line 1901 of this file from the instruction combiner pass. Its job — as you might guess — is to pattern-match on code equivalent to (but not identical to — some other passes have already transformed the code somewhat) checked_add_1(), replacing these operations with an intrinsic function that does a narrower add and separately returns the overflow bit. This intrinsic maps straightforwardly onto a machine add and an access to the hardware overflow flag, permitting optimal code generation.

Across both compilers, checked_add_1() gives the best results. Since it is also perhaps the most difficult check to mess up, that’s the one I recommend using, and also it has fairly straightforward equivalents for subtract, multiply, and divide. The only issue is that a double-width operation cannot be used to perform safe addition of 64-bit quantities on a 32-bit platform, since neither Clang nor GCC supports 128-bit math on that platform.

Ideally, authors of safe math libraries would spend some time creating code that can be optimized effectively by both GCC and LLVM, and then we could all just use that library. So far there’s one such example: Xi Wang’s libo, which avoids leaning so hard on the optimizer by directly invoking the llvm.x.with.overflow intrinsics. Unfortunately there’s no equivalent mechanism in GCC (that I know of, at least) for guaranteeing fast overflow checks.

Testing with Small Capacities

The goal of testing is to expose bugs. To find a bug we must drive the software under test into a part of its state space where the bug becomes evident, for example by causing a crash. Many bugs lurk in dark corners of the state space that are ordinarily difficult to reach. This piece is about a nice technique for making those dark corners easier to reach using either random and non-random testing. Although the method is one that I have used before, I hadn’t really thought about it explicitly until a conversation with Colin Runciman the other day brought it out into the open.

The issue is the small scope hypothesis, which states that most bugs can be triggered using small test cases. Here we’re talking about a class of bug that falls outside of this hypothesis: these bugs are triggered when some implementation limit is exceeded. For example, if we have a bounded buffer with length 10,000, and this buffer contains a bug in its handling of the buffer-full case, then it is clear that no test case can trigger the bug unless it results in more than 10,000 insertions into the buffer. Similarly, a bug in an operating system’s swapping code will not be hit until we put memory pressure on the machine — this will be difficult if we’re testing on a machine with lots of RAM. A bug in a compiler’s register spill logic will not be hit until we run out of registers, which may happen only rarely on a RISC with a large register file.

The testing technique, then, is simply to recognize capacities in the system under test and to artificially reduce them in order to make capacity bugs much easier to trigger. I think that probalby every accomplished programmer has used this technique, perhaps without even thinking about it: a simple example is testing the code for an auto-resizing hash table by making its initial capacity very small. This is just good sense.

Of course, while some capacities appear as explicit constants in the code that can be easily changed, others, such as the number of registers available to a compiler, will tend to be encoded in a more implicit fashion, and will therefore be more difficult to find and change. Also, the fact that a constant is hard-coded into a program doesn’t mean this constant can be freely changed: in some cases the constant is fake (how many programs that reference the CHAR_BIT constant can really tolerate values other than 8?) and in others the program is only designed to operate for a certain range of choices. To continue with the compiler example, the calling convention (and other factors) will almost certainly cause failures when the number of registers falls below a certain number. Similarly, a pipe or a thread pool can easily cause the system to deadlock when it is instantiated with a capacity that is too small. These things are design constraints, not the implementation errors that we are looking for.

In summary, if you have a system that offers a choice of capacity for an internal resource, it is often very useful to reduce this capacity for testing purposes, even if this results in poor throughput. This effect is particularly pronounced during fuzzing, where the random walks through the state space that are induced by random test cases will tend to hover around mean values with high probability.

Find the Integer Bug

Not all of the functions below are correct. The first person to leave a comment containing a minimal fix to a legitimate bug will get a small prize. I’m posting this not because the bug is particularly subtle or interesting but rather because I wrote this code for a piece about integer overflow and thought — wrongly, as usual — that I could get this stuff right the first time.

By “legitimate bug” I mean a bug that would cause the function to execute undefined behavior or return an incorrect result using GCC or Clang on Linux on x86 or x64 — I’m not interested in unexpected implementation-defined integer truncation behaviors, patches for failures under K&R C on a VAX-11, or style problems. For convenience, here are GCC’s implementation-defined behaviors for integers. I don’t know that Clang has a section in the manual about this but in general we expect its integers to behave like GCC’s.

#include <limits.h>
#include <stdint.h>

 * specification:
 * perform 32-bit two's complement addition on arguments a and b
 * store the result into *rp
 * return 1 if overflow occurred, 0 otherwise

int32_t checked_add_1(int32_t a, int32_t b, int32_t *rp) {
  int64_t lr = (int64_t)a + (int64_t)b;
  *rp = lr;
  return lr > INT32_MAX || lr < INT32_MIN;

int32_t checked_add_2(int32_t a, int32_t b, int32_t *rp) {
  int32_t r = (uint32_t)a + (uint32_t)b;
  *rp = r;
    (a < 0 && b < 0 && r > 0) ||
    (a > 0 && b > 0 && r < 0);

int32_t checked_add_3(int32_t a, int32_t b, int32_t *rp) {
  if (b > 0 && a > INT32_MAX - b) {
    *rp =  (a + INT32_MIN) + (b + INT32_MIN);
    return 1;
  if (b < 0 && a < INT32_MIN - b) {
    *rp =  (a - INT32_MIN) + (b - INT32_MIN);
    return 1;
  *rp = a + b;
  return 0;

UPDATE: The prize has been awarded, see comment 5. The prize is an Amazon gift certificate for $27.49 — the cost of the kindle edition of Hacker’s Delight.

Research Advice from Alan Adler

Although I am a happy French press user, I enjoyed reading an article about Alan Adler and the AeroPress that showed up recently on Hacker News. In particular, I love Adler’s advice to inventors:

  1. Learn all you can about the science behind your invention.
  2. Scrupulously study the existing state of your idea by looking at current products and patents.
  3. Be willing to try things even if you aren’t too confident they’ll work. Sometimes you’ll get lucky.
  4. Try to be objective about the value of your invention. People get carried away with the thrill of inventing and waste good money pursuing something that doesn’t work any better than what’s already out there.
  5. You don’t need a patent in order to sell an invention. A patent is not a business license; it’s a permission to be the sole maker of product (even this is limited to 20 years).

Now notice that (disregarding the last suggestion) we can simply replace “invention” with “research project” and Adler’s suggestions become a great set of principles for doing research. I think #4 is particularly important: lacking the feedback that people in the private sector get from product sales (or not), us academics are particularly susceptible to falling in love with pretty ideas that don’t improve anything.

A New Development for Coverity and Heartbleed

As a consequence of my last post, I spent some time on the phone Friday with Andy Chou, Coverity’s CTO and former member of Dawson Engler’s research group at Stanford, from which Coverity spun off over a decade ago. Andy confirmed that Coverity does not spot the heartbleed flaw and said that it remained stubborn even when they tweaked various analysis settings. Basically, the control flow and data flow between the socket read() from which the bad data originates and the eventual bad memcpy() is just too complicated.

Let’s be clear: it is trivial to create a static analyzer that runs fast and flags heartbleed. I can accomplish this, for example, by flagging a taint error in every line of code that is analyzed. The task that is truly difficult is to create a static analysis tool that is performant and that has a high signal to noise ratio for a broad range of analyzed programs. This is the design point that Coverity is aiming for, and while it is an excellent tool there is obviously no general-purpose silver bullet: halting problem arguments guarantee the non-existence of static analysis tools that can reliably and automatically detect even simple kinds of bugs such as divide by zero. In practice, it’s not halting problem stuff that stops analyzers but rather code that has a lot of indirection and a lot of data-dependent control flow. If you want to make a program that is robustly resistant to static analysis, implement some kind of interpreter.

Anyhow, Andy did not call me to admit defeat. Rather, he wanted to show off a new analysis that he and his engineers prototyped over the last couple of days. Their insight is that we might want to consider byte-swap operations to be sources of tainted data. Why? Because there is usually no good reason to call ntohs() or a similar function on a data item unless it came from the network or from some other outside source. This is one of those ideas that is simple, sensible, and obvious — in retrospect. Analyzing a vulnerable OpenSSL using this new technique gives this result for the heartbleed code:

Nice! As you might guess, additional locations in OpenSSL are also flagged by this analysis, but it isn’t my place to share those here.

What issues are left? Well, for one thing it may not be trivial to reliably recognize byte-swapping code; for example, some people may use operations on a char array instead of shift-and-mask. A trickier issue is reliably determining when untrusted data items should be untainted: do they need to be tested against both a lower and upper bound, or just an upper bound? What if they are tested against a really large bound? Does every bit-masking operation untaint? Etc. The Coverity people will need to work out these and other issues, such as the effect of this analysis on the overall false alarm rate. In case this hasn’t been clear, the analysis that Andy showed me is a proof-of-concept prototype and he doesn’t know when it will make it into the production system.

UPDATE: Also see Andy’s post at the Coverity blog.

Heartbleed and Static Analysis

Today in my Writing Solid Code class we went through some of the 151 defects that Coverity Scan reports for OpenSSL. I can’t link to these results but take my word for it that they are a pleasure to read — the interface clearly explains each flaw and the reasoning that leads up to it, even across multiple function calls. Some of the problems were slightly alarming but we didn’t see anything that looks like the next heartbleed. Unfortunately, Coverity did not find the heartbleed bug itself. (UPDATE: See this followup post.) This is puzzling; here’s a bit of speculation about what might be going on. There are basically two ways to find heartbleed using static analysis (here I’ll assume you’re familiar with the bug; if not, this post is useful). First, a taint analysis should be able to observe that two bytes from an untrusted source find their way into the length argument of a memcpy() call. This is clearly undesirable. The Coverity documentation indicates that it taints the buffer stored to by a read() system call (unfortunately you will need to login to Coverity Scan before you can see this). So why don’t we get a defect report? One guess is that since the data buffer is behind two pointer dereferences (the access is via s->s3->, the scanner’s alias analysis loses track of what is going on. Another guess is that two bytes of tainted data are not enough to trigger an alarm. Only someone familiar with the Coverity implementation can say for sure what is going on — the tool is highly sophisticated not just in its static analysis but also in its defect ranking system.

The other kind of static analysis that would find heartbleed is one that insists that the length argument to any memcpy() call does not exceed the size of either the source or destination buffer. Frama-C in value analysis mode is a tool that can do this. It is sound, meaning that it will not stop complaining until it can prove that certain defect classes are not present, and as such it requires far more handholding than does Coverity, which is designed to unsoundly analyze huge quantities of code. To use Frama-C, we would make sure that its own header files are included instead of the system headers. In one of those files we would find a model for memcpy():

/*@ requires \valid(((char*)dest)+(0..n - 1));
  @ requires \valid_read(((char*)src)+(0..n - 1));
  @ requires \separated(((char *)dest)+(0..n-1),((char *)src)+(0..n-1));
  @ assigns ((char*)dest)[0..n - 1] \from ((char*)src)[0..n-1];
  @ assigns \result \from dest;
  @ ensures memcmp((char*)dest,(char*)src,n) == 0;
  @ ensures \result == dest;
extern void *memcpy(void *restrict dest,
                    const void *restrict src, 
                    size_t n);

The comments are consumed by Frama-C. Basically they say that src and dest are pointers to valid storage of at least the required size, that the buffers do not overlap (recall that memcpy() has undefined behavior when called with overlapping regions), that it moves data in the proper direction, that the return value is dest, and that a subsequent memcmp() of the two regions will return zero.

The Frama-C value analyzer tracks an integer variable using an interval: a representation of the smallest and largest value that the integer could contain at some program point. Upon reaching the problematic memcpy() call in t1_lib.c, the value of payload is in the interval [0..65535]. This interval comes from the n2s() macro which turns two arbitrary-valued bytes from the client into an unsigned int:

#define n2s(c,s)        ((s=(((unsigned int)(c[0]))<< 8)| \
                            (((unsigned int)(c[1]))    )),c+=2)

The dest argument of this memcpy() turns out to be large enough. However, the source buffer is way too small. Frama-C would gripe about this and it would not shut up until the bug was really fixed.

How much effort would be required to push OpenSSL through Frama-C? I don’t know, but wouldn’t plan on getting this done in a week or three. Interestingly, a company spun off by Frama-C developers has recently used the tool to create a version of PolarSSL that they promise is immune to CWEs 119, 120, 121, 122, 123, 124, 125, 126, 127, 369, 415, 416, 457, 562, and 690. I think it would be reasonable for the open source security community to start thinking more seriously about what this kind of tool can do for us.


  • In a comment below, Masklinn states that OpenSSL’s custom allocators would defeat the detection of the too-large argument to memcpy(). This is indeed a danger. To avoid it, as part of applying Frama-C to OpenSSL, the custom malloc/free functions would be marked as being malloc-like using the “allocates” and “frees” keywords supported by ACSL 1.8. Coverity lets you do the same thing, and so would any competent analyzer for C. Custom allocators are regrettably common in oldish C code.
  • I’m interested to see what other tools have to say about heartbleed. If you have a link to results, please put it in a comment.
  • I re-ran Coverity after disabling OpenSSL’s custom freelist and also hacking CRYPTO_malloc() and friends to just directly call the obvious function from the malloc family. This caused Coverity to report 173 new defects: mostly use-after-free and resource leaks. Heartbleed wasn’t in the list, however, so I stand by my guess (above) that perhaps something related to indirection caused this defect to not be ranked highly enough to be reported.
  • HN has some discussion of PolarSSL and of this blog post. Also Reddit.


I’m teaching a small Advanced Operating Systems course this spring. Preparing for the course over winter break, I spent some time reading various Linux subsystems such as the scheduler, and was a bit shocked at how complex it has become. I’ve been using Linux, looking at its code, and occasionally hacking it for more than 20 years, and it seems that my impression of it as a fairly simple, easy-to-follow kernel has become badly out of date. It’s not that this glorious 7991-line file is impossible to understand, but rather that it — along with the other 16,000 lines of code in the kernel/sched directory — isn’t obviously the correct thing to inflict on some undergrads who are interested enough in operating systems to take a second course.

While looking around for alternatives I tried out Xv6, which I had known about for a while but hadn’t looked at closely. Xv6 is a rewrite of v6 UNIX in modern C that runs on multicore x86 chips. It compiles in a couple of seconds and is trivial to boot up in QEMU. It took me a while to see the genius of Xv6, which is that it is simpler than I would have thought a working multicore OS with shell and filesystem could be. For example, it lacks wait queues and ready queues — in Xv6, both wakeup and scheduling are accomplished by looping over the all-process table. Similarly, there’s no malloc() in the kernel, but rather just a page allocator. The pipe implementation copies one byte at a time. Amazingly, even the bootloader is a pleasure to read. Another nice thing about Xv6 is that it comes with a short textbook that explains OS concepts in terms of their implementations in Xv6.

So what did we do with Xv6 beside read it? One exercise was to speed up its pipe implementation as much as possible while preserving UNIX semantics. This turned out to be a really nice exercise (I thought). It is fairly easy to get within a factor of two of Linux pipe throughput on a two-core VM. We implemented a ring buffer for zero-copy bulk data transfer between processes. Finally, we added priorities and ready queues.

I have no real complaints about Xv6: the code is clean and commented, the functions are small, and overall it makes a great instructional OS. The look and feel are very similar to what I remember from hacking on Minix when I took an advanced OS class around 1994. Actually I do have one small complaint, which is that in a few places liberties are taken with error checking. For example, in the allocuvm() function from vm.c, which grows a process, there is a call to mappages() that mysteriously fails to check the return value. Is there some reason that this particular call cannot fail? If so, a comment explaining the reasoning is needed. If not, error checking is required. I’m sensitive about this issue since I tell the students over and over that they cannot ignore failures in this kind of code.

Another OS that we’ve been learning from is the Windows 2000 kernel which is — surprisingly, to many people — a simple and elegant multicore OS that provides some real-world contrast to Xv6′s over-simplicity. I haven’t seen any Windows kernels later than 2000, I’d be curious to hear if they have remained so nice.

Automated Reasoning About LLVM Optimizations and Undefined Behavior

Following up on last week’s toy use of Z3 and my LLVM superoptimizer post from a few weeks ago, I wanted to try to prove things about optimizations that involve undefined behavior.

We’ll be working with the following example:

int triple (int x)
  return x + x + x;

Clang’s -O0 output for this code is cluttered but after running mem2reg we get a nice literal translation of the C code:

define i32 @triple(i32 %x) #0 {
  %1 = add nsw i32 %x, %x
  %2 = add nsw i32 %1, %x
  ret i32 %2

Next we can run instcombine, with this result:

define i32 @triple(i32 %x) #0 {
  %1 = mul i32 %x, 3
  ret i32 %1

The optimization itself (replacing x+x+x with x*3) is uninteresting and proving that it is correct is trivial in Z3 or any similar tool that supports reasoning about bitvector operations. To see this, go to the Z3 site and paste this code into the Python window on the right:

x = BitVec('x', 32)
slow = x+x+x
fast = x*3
prove (slow == fast)

The interesting aspect of this optimization is that the original C code and LLVM code are not using two’s complement addition. Rather, + in C and add nsw in LLVM are only defined when the addition doesn’t overflow. On the other hand, the multiplication emitted by instcombine is not qualified by nsw: it wraps in the expected fashion when the multiplication overflows (see the docs for more detail). My goal was to use Z3 to automatically prove that instcombine dropped the ball here: it could have emitted mul nsw which has weaker semantics and is therefore more desirable: compared to the unqualified mul it gives subsequent optimizations more freedom.

When we talk informally about compiler optimizations, we often say that the compiler should replace code with (faster or smaller) code that is functionally equivalent. Actually I just did this above while showing that x+x+x and x*3 are equivalent. But in fact, a compiler optimization does not need to replace code with functionally equivalent code. Rather, the new code only needs to refine the old code. Refinement occurs many times during a typical compilation. For example this C code:

foo (bar(), baz());

can be refined to either this:

tmp1 = bar();
tmp2 = baz();
foo (tmp1, tmp2);

or this:

tmp2 = baz();
tmp1 = bar();
foo (tmp1, tmp2);

These refinements are not (in general) equivalent to each other, nor are they equivalent to the original non-deterministic C code. Analogously, the LLVM instruction mul %x, 3 refines mul nsw %x, 3 but not the other way around.

So how do we teach Z3 to do refinement proofs? I was a bit worried by some wording in the Z3 documentation that says “in Z3 all functions are total” and then I got stuck but Alexey Solovyev, a formal methods postdoc at Utah, helped me out with the following solution. The idea is to use Z3′s ML-style option types to turn functions with undefined behavior into total functions:


Opt = Datatype('Opt')
Opt.declare('some', ('value', BitVecSort(BITWIDTH)))

Opt = Opt.create()
none = Opt.none
some = Opt.some
value = Opt.value

(t, r) = Consts('t r', Opt)

s = Solver()

# print "sat" if f is refined by g, else print "unsat"
# equivalently, you can say "f can be implemented by calling g"
def check_refined(f, g):
    s.add(ForAll([t, r], Implies(f(t, r) != none, f(t, r) == g(t, r))))
    print s.check()

In other words, f() is refined by g() iff g() returns the same value as f() for every member of f()’s domain. For inputs not in the domain of f(), the behavior of g() is a don’t-care.

Here are some LLVM operations that let us reason about the example code above:

INT_MAX = BitVecVal((1 << BITWIDTH - 1) - 1, BITWIDTH)

def signed_add_overflow(y, z):
    ylong = SignExt(BITWIDTH, y)
    zlong = SignExt(BITWIDTH, z)
    rlong = ylong + zlong
    return Or(rlong > INT_MAX_AS_LONG, rlong < INT_MIN_AS_LONG)

def signed_mul_overflow(y, z):
    ylong = SignExt(BITWIDTH, y)
    zlong = SignExt(BITWIDTH, z)
    rlong = ylong * zlong
    return Or(rlong > INT_MAX_AS_LONG, rlong < INT_MIN_AS_LONG)

def add(t, r):
    return If(Or(t == none, r == none), 
              some(value(t) + value(r)))

def add_nsw(t, r):
    return If(Or(t == none, r == none), 
              If(signed_add_overflow(value(t), value(r)), 
                 some(value(t) + value(r))))

def mul(t, r):
    return If(Or(t == none, r == none), 
              some(value(t) * value(r)))

def mul_nsw(t, r):
    return If(Or(t == none, r == none), 
              If(signed_mul_overflow(value(t), value(r)), 
                 some(value(t) * value(r))))

Hopefully this code is pretty clear: add_nsw and its friend mul_nsw act like add and mul as long as there’s no overflow. If an overflow occurs, they map their arguments to none.

At this point I might as well mention a dirty little secret: formal methods code isn’t any easier to get right than regular code, and often it is considerably more difficult. The solution is testing. This is more than a little bit ironic since working around inadequacies of testing is the point of using formal methods in the first place. So here’s some test code. To make it easier to understand I’ve set BITWIDTH to 8 and also put the results of the print statements in comments.

def opt_str(o):
    p = simplify(o)
    if is_const(p):
        return "none"
        return simplify(value(p)).as_signed_long()
ONE = Opt.some(1)
TWO = Opt.some(2)
NEG_ONE = Opt.some(-1)
NEG_TWO = Opt.some(-2)
INT_MIN = Opt.some(INT_MIN)
INT_MAX = Opt.some(INT_MAX)

print opt_str(mul_nsw(ONE, ONE))           #    1
print opt_str(mul_nsw(INT_MAX, ONE))       #  127
print opt_str(mul_nsw(INT_MIN, ONE))       # -128
print opt_str(mul_nsw(INT_MAX, NEG_ONE))   # -127
print opt_str(mul_nsw(INT_MIN, NEG_ONE))   # none
print opt_str(mul_nsw(INT_MAX, TWO))       # none
print opt_str(mul_nsw(INT_MIN, TWO))       # none
print opt_str(mul_nsw(INT_MAX, NEG_TWO))   # none
print opt_str(mul_nsw(INT_MIN, NEG_TWO))   # none

print opt_str(add_nsw(ONE, ONE))           #    2
print opt_str(add_nsw(INT_MAX, ONE))       # none
print opt_str(add_nsw(INT_MIN, ONE))       # -127
print opt_str(add_nsw(INT_MAX, NEG_ONE))   #  126
print opt_str(add_nsw(INT_MIN, NEG_ONE))   # none

check_refined(add, add)                    # sat
check_refined(add, add_nsw)                # unsat

check_refined(add_nsw, add)                # sat
check_refined(add_nsw, add_nsw)            # sat

The formalization passes this quick smoke test. I’ve done some more testing and won’t bore you with it. None of this is conclusive of course.

Finally, we return to the example that motivated this post. Recall that LLVM’s instcombine pass translates two add nsw instructions into a single mul. Let’s make sure that was correct. Again, for convenience, I’ll put outputs in comments.

(a, b) = Consts('a b', Opt)

def slow(a, b):
    return add_nsw(a, add_nsw(a, a))

def fast1(a, b):
    return mul(a, Opt.some(3))

check_refined(slow, fast1)           # sat
check_refined(fast1, slow)           # unsat

As expected, LLVM’s transformation is correct: x+x+x where both + operators have undefined overflow can be refined by x*3 where the * operator has two’s complement wraparound. The converse is not true.

Next, let’s explore the idea that instcombine could emit a mul nsw instruction.

def fast2(a, b):
    return mul_nsw(a, Opt.some(3))

check_refined(slow, fast2)           # sat
check_refined(fast2, slow)           # sat

Not only does this optimization work, but the refinement property also holds for the converse, indicating that the two formulations are equivalent.

We can also ask Z3 to prove equivalence of these different formulations:

prove (slow(a, b) == fast1(a, b))    # counterexample [a = some(137)]
prove (slow(a, b) == fast2(a, b))    # proved

As expected, fast1 is not equivalent to the original code and additionally Z3 has given us a value for a that demonstrates this (for the 8-bit case). The second equivalence check passes.

What have we learned here? First, automated proofs about compiler optimizations in the presence of integer undefined behaviors are not too difficult. This is nice because manual reasoning about undefined behavior is error-prone. Second, there’s room for at least a little bit of improvement in LLVM’s instcombine pass (I’m sure the LLVM people know this). One idea that I quite like is using a solver to improve LLVM’s peephole optimizations by automatically figuring out when qualifiers like “nsw”, “nuw”, and “exact” can be used correctly.

Here’s the Python code all in one piece. You can run it locally after installing Z3 or else you can paste it into the Z3 website’s Python interface.

That Was Easy

I wanted to play around with the Python interface to Z3 and the classic SEND + MORE = MONEY puzzle seemed like a good way to get started. This turned out to be so easy that my code worked almost on the first try:

from z3 import *

# find the (distinct) integer value in 0..9 for each letter that makes
# this equation work:
#   SEND
# + MORE
# ------

S = Int('S')
E = Int('E')
N = Int('N')
D = Int('D')
M = Int('M')
O = Int('O')
R = Int('R')
Y = Int('Y')

s = Solver()

s.add (S >= 0, S < 10, 
       E >= 0, E < 10, 
       N >= 0, N < 10, 
       D >= 0, D < 10, 
       M >= 0, M < 10, 
       O >= 0, O < 10, 
       R >= 0, R < 10, 
       Y >= 0, Y < 10)

s.add ((D + 10*N + 100*E + 1000*S + 
        E + 10*R + 100*O + 1000*M) == 
       (Y + 10*E + 100*N + 1000*O + 10000*M))

s.add (S != E, S != N, S != D, S != M, S != O, S != R, S != Y, 
       E != N, E != D, E != M, E != O, E != R, E != Y, 
       N != D, N != M, N != O, N != R, N != Y, 
       D != M, D != O, D != R, D != Y, 
       M != O, M != R, M != Y, 
       O != R, O != Y, 
       R != Y)

print s.check()
m = s.model()
for d in m.decls():
    print "%s = %s" % (, m[d])


$ python ./
D = 7
S = 9
N = 6
O = 0
R = 8
E = 5
M = 1
Y = 2

Is there a non-quadratic way to express the uniqueness constraint? That would be nice. Anyhow, it’s really a pleasure when research software is this easy to use.

UPDATE: There is a better way! See comment #1 below. Thanks Taylor.

Reproducibility in Computer Systems Research

These results about reproducibility in CS have been the subject of lively discussion at Facebook and G+ lately. The question is, for 613 papers, can the associated software be located, compiled, and run? In contrast with something I often worry about — producing good software — the bar here is low, since even a system that runs might be without value and may not even support the results in the paper.

It is great that these researchers are doing this work. Probably everyone who has worked in applied CS research for any length of time has wanted to compare their results against previous research and been thwarted by crappy or unavailable software; certainly this has happened to me many times. The most entertaining part of the paper is Section 4.3, called “So, What Were Their Excuses? (Or, The Dog Ate My Program)”. The recommendations in Section 5 are also good, I think all CS grad students and faculty should read over this material. Anecdote 2 (at the very end of the paper) is also great.

On the other hand, it’s not too hard to quibble with the methods here. For example, one of my papers was placed in the irreproducible category with the reason of:

could not install: style, flex, GNU indent, LLVM/clang 3.2

It’s not clear why installing these dependencies was difficult, but whatever. In a Facebook thread a number of other people reported learning that they have been doing irreproducible research for similar reasons. The G+ thread also contains some strong opinions about, for example, this build failure.

Another thing we might take issue with is the fact that “reproducible research” usually means something entirely different than successfully compiling some code. Usually, we would say that research is reproducible if an independent team of researchers can repeat the experiment using information from the paper and end up with similar (or at least non-conflicting) results. For a CS paper, you would use the description in the paper to implement the system yourself, run the experiments, and then see if the results agree with those reported.

Although it’s not too hard to put some code on the web, producing code that will compile in 10 or 20 years is not an easy problem. A stable platform is needed. The most obvious one is x86, meaning that we should distribute VM images with our papers, which seems clunky and heavyweight but perhaps it is the right solution. Another reasonable choice is the Linux system call API, meaning that we should distribute statically linked executables or equivalent. Yet another choice (for research areas where this works) might be a particular version of Matlab or some similar language.

UPDATE: You can help watch the watchmen here.