C++ imposes very few restrictions on the behavior of float and other floating-point types. This can lead to possible inconsistencies in the results, both between compilers, and between runtime/compile-time evaluation by the same compiler. Here is the tl;dr on it:
|
At runtime |
In constant expressions |
| Floating-point errors, like division by zero |
UB, but compilers may support
silent errors through NaN as an extension |
UB in a constant expression
results in a compiler error |
Rounded operations, like 10.0 / 3.0 |
Rounding mode controlled through
floating-point environment; results may vary |
Rounding is implementation-defined,
results can differ from runtime |
Semantics changes through -ffast-math
and other compiler optimizations |
Results can become less precise or more precise
as a result; IEEE-754 conformance is broken |
No effect in practice; at most
implementation-defined effect |
| Calls to math functions |
Same handling of errors and rounding
as arithmetic with + and * |
Some constexpr since C++23,
some constexpr since C++26,
with some errors disallowed at compile-time |
Floating-Point Errors
Some operations can fail, such as division by zero. The C++ standard says:
If the second operand of / or % is zero the behavior is undefined.
- [expr.mul]/4
In constant expressions, this is respected, and so it's not possible to produce NaN through operations or raise FE_DIVBYZERO at compile time.
No exception is made for floating point numbers. However, when std::numeric_limits<float>::is_iec559() is true, most compilers will have IEEE-754 compliance as an extension. For example, division by zero is allowed and produces infinity or NaN depending on the operands.
Rounding Modes
C++ has always allowed differences between compile-time results and runtime results.
For example, you can evaluate:
double x = 10.0f / 3.0;
constexpr double y = 10.0 / 3.0;
assert(x == y); // might fail
The result might not always be the same, because the floating point environment can only be changed at runtime, and thus the rounding mode can be altered.
C++'s approach is to make the effect of the floating point environment implementation-defined. It gives you no portable way to control it (and thus rounding) in constant expressions.
If the [FENVC_ACCESS] pragma is used to enable control over the floating-point environment, this document does not specify the effect on floating-point evaluation in constant expressions.
- [cfenv.syn]/Note 1
Compiler Optimizations
Firstly, compilers can be eager to optimize your code, even if it changes its meaning. For example, GCC will optimize away this call:
// No call to sqrt thanks to constant folding.
// This ignores the fact that this is a runtime evaluation, and would normally be impacted
// by the floating point environment at runtime.
const float x = std::sqrt(2);
The semantics change even more with flags like -ffast-math which allows the compiler to reorder and optimize operations in a way that is not IEEE-754 compliant. For example:
float big() { return 1e20f;}
int main() {
std::cout << big() + 3.14f - big();
}
For IEEE-754 floating point numbers, addition and subtraction are not commutative. We cannot optimize this to: (big() - big()) + 3.14f. The result will be 0, because 3.14f is too small to make any change to big() when added, due to lack of precision. However, with -ffast-math enabled, the result can be 3.14f.
Mathematical Functions
There can be runtime differences to constant expressions for all operations, and that includes calls made to mathematical functions. std::sqrt(2) at compile-time might not be the same as std::sqrt(2) at runtime. However, this issue is not unique to math functions. You can put these functions into the following categories:
No FPENV Dependence / Very Weak Dependence (constexpr since C++23) [P05333r9]
Some functions are completely independent of the floating-point environment, or they simply cannot fail, such as:
std::ceil (round to next greater number)std::fmax (maximum of two numbers)std::signbit (obtains the sign bit of a floating-point number)
Furthermore, there are functions like std::fma which just combine two floating point operations. These are no more problematic than + and * at compile-time. The behavior is is the same as calling these math functions in C (see C23 Standard, Annex F.8.4), however, it is not a constant expression in C++ if exceptions other than FE_INEXACT are raised, errno is set, etc (see [library.c]/3).
Weak FPENV Dependence (constexpr since C++26) [P1383r0]
Other functions are dependent on the floating point environment, such as std::sqrt or std::sin. However, this dependence is called weak, because it's not explicitly stated, and it only exists because floating-point math is inherently imprecise.
It would be arbitrary to allow + and * at compile-time, but not math functions which have the exact same issues.
Mathematical Special Functions (not constexpr yet, possibly in the future)
[P1383r0] deemed it too ambitious to add constexpr to for mathematical special functions, such as:
std::betastd::riemann_zeta- and many more ...
Strong FPENV Dependence (not constexpr yet, possibly never)
Some functions like std::nearbyint are explicitly stated to use the current rounding mode in the standard.
This is problematic, because you cannot control the floating-point environment at compile time using standard means.
Functions like std::nearbyint aren't constexpr, and possibly never will be.
Conclusion
In summary, there are many challenges facing the standard committee and compiler developers when dealing with constexpr math. It has taken decades of discussion to lift some restrictions on constexpr math functions, but we are finally here. The restrictions have ranged from arbitrary in the case of std::fabs, to necessary in the case of std::nearbyint.
We are likely to see further restrictions lifted in the future, at least for mathematical special functions.
