Is there some meaningful statistical data to justify keeping signed integer arithmetic overflow undefined?

Question

I don’t know about studies and statistics, but yes, there are definitely optimizations taking this into account that compilers actually do. And yes, they are very important (tldr loop vectorization for example).

Besides the compiler optimizations, there is another aspect to be taken into account. With UB you get C/C++ signed integers to behave arithmetically as you would expect mathematically. For instance x + 10 > x holds true now (for valid code of course), but would not on a wrap-around behavior.

I’ve found an excellent article How undefined signed overflow enables optimizations in GCC from Krister Walfridsson’s blog listing some optimizations that take signed overflow UB into account. The following examples are from it. I am adding c++ and assembly examples to them.

If the optimizations look too simple, uninteresting or unimpactful, remember that these optimization are just steps in a much much larger chain of optimizations. And the butterfly effect does happen as a seemingly unimportant optimization at an earlier step can trigger a much more impactful optimization at a later step.

If the examples look nonsensical (who would write x * 10 > 0) keep in mind that you can very easily get to this kind of examples in C and C++ with constants, macros, templates. Besides the compiler can get to this kind of examples when applying transformations and optimizations in its IR.

Signed integer expression simplification

Eliminate multiplication in comparison with 0

(x * c) cmp 0   ->   x cmp 0

bool foo(int x) { return x * 10 > 0 }

foo(int):
        test    edi, edi
        setg    al
        ret

Eliminate division after multiplication

(x * c1) / c2 -> x * (c1 / c2) if c1 is divisible by c2

int foo(int x) { return (x * 20) / 10; }

foo(int):
        lea     eax, [rdi+rdi]
        ret

Eliminate negation

(-x) / (-y) -> x / y

int foo(int x, int y) { return (-x) / (-y); }

foo(int, int):
        mov     eax, edi
        cdq
        idiv    esi
        ret

Simplify comparisons that are always true or false

x + c < x       ->   false
x + c <= x      ->   false
x + c > x       ->   true
x + c >= x      ->   true

bool foo(int x) { return x + 10 >= x; }

foo(int):
        mov     eax, 1
        ret

Eliminate negation in comparisons

(-x) cmp (-y)   ->   y cmp x

bool foo(int x, int y) { return -x < -y; }

foo(int, int):
        cmp     edi, esi
        setg    al
        ret

Reduce magnitude of constants

x + c > y       ->   x + (c - 1) >= y
x + c <= y      ->   x + (c - 1) < y

bool foo(int x, int y) { return x + 10 <= y; }

foo(int, int):
        add     edi, 9
        cmp     edi, esi
        setl    al
        ret

Eliminate constants in comparisons

(x + c1) cmp c2         ->   x cmp (c2 - c1)
(x + c1) cmp (y + c2)   ->   x cmp (y + (c2 - c1)) if c1 <= c2

The second transformation is only valid if c1 <= c2, as it would
otherwise introduce an overflow when y has the value INT_MIN.

bool foo(int x) { return x + 42 <= 11; }

foo(int):
        cmp     edi, -30
        setl    al
        ret

Pointer arithmetic and type promotion

If an operation does not overflow, then we will get the same result if
we do the operation in a wider type. This is often useful when doing
things like array indexing on 64-bit architectures — the index
calculations are typically done using 32-bit int, but the pointers are
64-bit, and the compiler may generate more efficient code when signed
overflow is undefined by promoting the 32-bit integers to 64-bit
operations instead of generating type extensions.

One other aspect of this is that undefined overflow ensures that a[i]
and a[i+1] are adjacent. This improves analysis of memory accesses for
vectorization etc.

This is a very important optimization as loop vectorization one of the most efficient and effective optimization algorithms.

This is an example when changing an index from an unsigned index to a signed improves the generated assembly:

Unsigned version

#include <cstddef>

auto foo(int* v, std::size_t start)
{
    int sum = 0;

    for (std::size_t i = start; i < start + 4; ++i)
        sum += v[i];

    return sum;
}

With unsigned the case where start + 4 wraps around must be taken into account and a branch is generated to deal with this case (branches are bad for performance):

; gcc on x64 with -march=skylake

foo1(int*, unsigned long):
        cmp     rsi, -5
        ja      .L3
        vmovdqu xmm0, XMMWORD PTR [rdi+rsi*4]
        vpsrldq xmm1, xmm0, 8
        vpaddd  xmm0, xmm0, xmm1
        vpsrldq xmm1, xmm0, 4
        vpaddd  xmm0, xmm0, xmm1
        vmovd   eax, xmm0
        ret
.L3:
        xor     eax, eax
        ret

; clang on x64 with -march=skylake

foo1(int*, unsigned long):                             # @foo1(int*, unsigned long)
        xor     eax, eax
        cmp     rsi, -4
        jae     .LBB0_2
        vpbroadcastq    xmm0, qword ptr [rdi + 4*rsi + 8]
        vpaddd  xmm0, xmm0, xmmword ptr [rdi + 4*rsi]
        vpshufd xmm1, xmm0, 85                  # xmm1 = xmm0[1,1,1,1]
        vpaddd  xmm0, xmm0, xmm1
        vmovd   eax, xmm0
.LBB0_2:
        ret

As a side note, using a narrower type would result in even worst assembly, inhibiting the use of SSE vectorized instructions:

#include <cstddef>

auto foo(int* v, unsigned start)
{
    int sum = 0;

    for (unsigned i = start; i < start + 4; ++i)
        sum += v[i];

    return sum;
}

; gcc on x64 with -march=skylake

foo(int*, unsigned int):
        cmp     esi, -5
        ja      .L3
        mov     eax, esi
        mov     eax, DWORD PTR [rdi+rax*4]
        lea     edx, [rsi+1]
        add     eax, DWORD PTR [rdi+rdx*4]
        lea     edx, [rsi+2]
        add     eax, DWORD PTR [rdi+rdx*4]
        lea     edx, [rsi+3]
        add     eax, DWORD PTR [rdi+rdx*4]
        ret
.L3:
        xor     eax, eax
        ret

; clang on x64 with -march=skylake

foo(int*, unsigned int):                              # @foo(int*, unsigned int)
        xor     eax, eax
        cmp     esi, -5
        ja      .LBB0_3
        mov     ecx, esi
        add     esi, 4
        mov     eax, dword ptr [rdi + 4*rcx]
        lea     rdx, [rcx + 1]
        cmp     rdx, rsi
        jae     .LBB0_3
        add     eax, dword ptr [rdi + 4*rcx + 4]
        add     eax, dword ptr [rdi + 4*rcx + 8]
        add     eax, dword ptr [rdi + 4*rcx + 12]
.LBB0_3:
        ret

Signed version

Using a signed index however results in nice vectorized branchless code:

#include <cstddef>

auto foo(int* v, std::ptrdiff_t start)
{
    int sum = 0;

    for (std::ptrdiff_t i = start; i < start + 4; ++i)
        sum += v[i];

    return sum;
}

; gcc on x64 with -march=skylake

foo(int*, long):
        vmovdqu xmm0, XMMWORD PTR [rdi+rsi*4]
        vpsrldq xmm1, xmm0, 8
        vpaddd  xmm0, xmm0, xmm1
        vpsrldq xmm1, xmm0, 4
        vpaddd  xmm0, xmm0, xmm1
        vmovd   eax, xmm0
        ret

; clang on x64 with -march=skylake

foo(int*, long):                              # @foo(int*, long)
        vpbroadcastq    xmm0, qword ptr [rdi + 4*rsi + 8]
        vpaddd  xmm0, xmm0, xmmword ptr [rdi + 4*rsi]
        vpshufd xmm1, xmm0, 85                  # xmm1 = xmm0[1,1,1,1]
        vpaddd  xmm0, xmm0, xmm1
        vmovd   eax, xmm0
        ret

Vectorized instruction are still used when using a narrower signed type:

#include <cstddef>

auto foo(int* v, int start)
{
    int sum = 0;

    for (int i = start; i < start + 4; ++i)
        sum += v[i];

    return sum;
}

; gcc on x64 with -march=skylake

foo(int*, int):
        movsx   rsi, esi
        vmovdqu xmm0, XMMWORD PTR [rdi+rsi*4]
        vpsrldq xmm1, xmm0, 8
        vpaddd  xmm0, xmm0, xmm1
        vpsrldq xmm1, xmm0, 4
        vpaddd  xmm0, xmm0, xmm1
        vmovd   eax, xmm0
        ret

; clang on x64 with -march=skylake

foo(int*, int):                              # @foo(int*, int)
        movsxd  rax, esi
        vpbroadcastq    xmm0, qword ptr [rdi + 4*rax + 8]
        vpaddd  xmm0, xmm0, xmmword ptr [rdi + 4*rax]
        vpshufd xmm1, xmm0, 85                  # xmm1 = xmm0[1,1,1,1]
        vpaddd  xmm0, xmm0, xmm1
        vmovd   eax, xmm0
        ret

Value range calculations

The compiler keeps track of the variables’ range of possible values at
each point in the program, i.e. for code such as
int x = foo();
if (x > 0) {
  int y = x + 5;
  int z = y / 4;
it determines that x has the range [1, INT_MAX] after the
if-statement, and can thus determine that y has the range [6, INT_MAX] as overflow is not allowed. And the next line can be
optimized to int z = y >> 2; as the compiler knows that y is
non-negative.

auto foo(int x)
{
    if (x <= 0)
        __builtin_unreachable();
    
    return (x + 5) / 4;
}

foo(int):
        lea     eax, [rdi+5]
        sar     eax, 2
        ret

The undefined overflow helps optimizations that need to compare two
values (as the wrapping case would give possible values of the form
[INT_MIN, (INT_MIN+4)] or [6, INT_MAX] that prevents all useful
comparisons with < or >), such as

Changing comparisons x<y to true or false if the ranges for x and y does not overlap

Changing min(x,y) or max(x,y) to x or y if the ranges do not overlap

Changing abs(x) to x or -x if the range does not cross 0

Changing x/c to x>>log2(c) if x>0 and the constant c is a power of 2

Changing x%c to x&(c-1) if x>0 and the constant c is a power of 2

Loop analysis and optimization

The canonical example of why undefined signed overflow helps loop
optimizations is that loops like
for (int i = 0; i <= m; i++)
are guaranteed to terminate for undefined overflow. This helps
architectures that have specific loop instructions, as they do in
general not handle infinite loops.

But undefined signed overflow helps many more loop optimizations. All
analysis such as determining number of iteration, transforming
induction variables, and keeping track of memory accesses are using
everything in the previous sections in order to do its work. In
particular, the set of loops that can be vectorized are severely
reduced when signed overflow is allowed.