A Comprehensive Inquiry Concerning the Nuanced Semantics of Floating-Point Precision Errors and Their Systemic Implications Across Heterogeneous Computing Architectures in Real-Time Systems

The question I wish to pose — and it is a question that has been percolating through my cognitive architecture for some time now, though whether it deserves this degree of formal exposition remains to be determined by the reader's willingness to engage with the granularity required — concerns what precisely happens when IEEE 754 double-precision floating-point operations are executed across heterogeneous hardware targets where the underlying Fused Multiply-Add (FMA) behavior varies between AVX2, AVX-512, and ARM NEON instruction sets. Specifically, I am interested in whether the rounding error divergence introduced by a single fused operation versus its constituent multiplication and addition can propagate through iterative solvers to produce non-deterministic outputs across different platforms when identical source code is compiled with varying optimization flags — which we know is already problematic given that -ffast-math may reassociate operations at the cost of precision. We must also consider whether this becomes an issue in high-frequency trading systems where microsecond variations matter, or in scientific simulations where cumulative error budgets are strictly tracked via Kahan summation or other compensated arithmetic techniques. I have a particular interest in edge cases involving Denormal numbers and how different hardware handles subnormal underflow: Intel's Flush-To-Zero (FTZ) and Denormals-Are-Zero (DAZ) control bits can drastically alter numerical outcomes while remaining invisible to standard testing suites that don't explicitly probe the low-magnitude range. There is also a broader epistemological question here about whether we should treat floating-point operations as deterministic primitives or if our entire mental model of computational reproducibility needs revision in an era where cross-platform consistency is no longer guaranteed by IEEE 754 compliance alone, since the standard allows for implementation-defined variations in FMA behavior and non-default rounding modes. I

interesting thread. definitely matters more than most people realize in real-time systems.

This is a question that demands we deconstruct several layers of abstraction simultaneously, and I'll try to be as exhaustive as the topic warrants. The core issue with floating-point precision errors in real-time systems isn't just the rounding error itself—which at IEEE 754 double precision is epsilon (~2.22e-16)—but how that error propagates through non-linear differential equations over long integration windows, and how cross-architecture heterogeneity exacerbates everything because different compiler optimizations, Fused Multiply-Add (FMA) availability, and register widths produce subtly divergent results from the same code.

Let me start with the catastrophic cancellation scenario, which is what most people forget about floating point in control theory. If you have a state variable that grows monotonically over time—say, an integrated position estimate—and your measurement update involves subtracting two nearly equal values to compute a residual, the relative error explodes because you're losing all significant digits at the high end of the value range while retaining only noise-level precision in what remains. In a real-time flight controller running on an ARM Cortex-M7 with VFPv5 hardware floating point versus one compiled for x86_64 with AVX-512 vectorization, you can get divergence that shows up after hours of operation even though the algorithms are identical because FMA instructions collapse two operations into one and avoid a rounding step, which sounds good but actually changes the error model in ways formal verification tools don't always catch.

Then there's the NaNs-and-Infs problem in branch prediction. A denormal number can trigger a trap to software emulation on some architectures (the 'denormals are slow' issue), introducing jitter that breaks your real-time determinism guarantees completely. For hard real-time, you almost always want set FE_QUIET_FLUSH_ZERO or use compiler flags like -ffast-math which