Same bits on every machine

A weighting filter designed by phonometry is the same filter, byte for byte, on every machine that shares a C library: any CPU, with or without AVX-512; any BLAS kernel set; any thread count; any hash seed. Across C libraries it is not, yet: Linux, macOS and Windows each design their own bytes for the same filter, because a handful of transcendentals still come from the platform’s math once per design, and this page names them and measures what they do. On one platform the coefficients a laptop designs are the coefficients a cluster designs, so a conformance verdict, a published figure or a filter shipped inside a measurement chain is a reproducible artifact rather than one machine’s account of itself.

That sentence is not how scientific Python normally behaves, and this page records what it cost, with the numbers.

IEEE 754 pins every single operation: given two doubles, their sum, difference, product, quotient and square root are the same bit pattern on every conforming machine. Everything beyond a single operation is where reproducibility leaks, through three separate doors.

Reductions are free to reassociate. A sum of many terms has no defined order, and a BLAS picks the order that suits the vector registers it finds at run time. phonometry measured what that costs on its own weighting fit: two OpenBLAS kernel sets agreed on every accept decision of the Levenberg–Marquardt search for thirty-nine consecutive steps while the costs they were comparing drifted from 3 × 10⁻¹³ to 1 × 10⁻⁴ apart, and the two runs then landed on different filters: 5 × 10⁻⁵ apart in the leading coefficient, 0.002 dB apart in response, a visible 0.0225 pt apart in a plotted curve. An optimiser in a shallow valley amplifies whatever the last bit does.

Vectorised transcendentals are dispatched. numpy chooses log, exp, tan, power, arctan and sin kernels by what the CPU offers, and the AVX-512 kernels do not agree to the last bit with the ones a machine without AVX-512 runs. Measured under Intel SDE emulating Skylake-X against the same host natively: every one of those six returns a different digest over the same inputs. The leak reaches further than the obvious call sites: numpy.geomspace is 10 ** linspace(...), so the frequency grid every residual of the fit is evaluated on came out different under AVX-512, and every step downstream inherited it.

The platform’s own libm is a platform choice. math.log is whatever log the C library ships. glibc’s and Apple’s are both accurate to about half an ulp, which means that on a handful of inputs they legitimately round to opposite neighbours, so a design loop built on math.log returned different filters on macOS than on Linux with nothing anywhere to say so. The same holds for every other transcendental the C library ships, which is why the evidence below still finds three sets of bytes.

The deterministic design path answers each door in kind, in filters._weighting_design and filters._pinned_log:

• Its own reductions. Every sum the fit compares or solves with is folded pairwise in a shape fixed by the code, and the normal equations are solved by Gaussian elimination written out, the same algorithm LAPACK’s dgesv runs with the order no longer a library’s choice. Pairwise folding is also the more accurate order: O(ε log n) rounding against the O(ε n) of a running total, so nothing was traded away.
• Real arithmetic where complex arithmetic dispatches. Complex multiply and complex magnitude fuse operations differently per CPU, so the two places the path handed numpy a complex expression are written out in real arithmetic.
• Pinned transcendentals. Outside the iteration, each transcendental is taken from the C library once per design. Those few calls, an exp, a tan, an atan, a pow, a log10 or two and a sin or two, are the one place a platform’s libm still enters a design, and they are enough to separate the platforms: the evidence below has the three digests. Spelling them in pinned numpy the way the logarithm is spelled is the remaining step to one digest everywhere. Inside the iteration, the logarithm runs three quarters of a million times per design, and the loop that once pinned it cost a factor of four; it is now glibc’s own table-driven log algorithm spelled in numpy operations IEEE 754 fixes exactly, plain arithmetic, integer bit work and table lookups, with the fused multiply-adds of the original reproduced through exact error-free transformations. Its bits are its own on every machine, which is stronger than calling anyone’s libm.
• The grid spelled out. The geometric frequency grid is generated term for term with the endpoints pinned, so the fit evaluates the same frequencies everywhere.

Every claim above is measured, and most of them are re-measured by tests that run in CI:

• 91,658,333 inputs, zero exceptions. Every distinct value the whole design corpus feeds its logarithm was captured and compared bit for bit against glibc’s log: identical on all of them, plus roughly a hundred million adversarial draws over the full exponent range, the near-one band, the subnormals and the branch edges. An adversarial search also found the honest limit: about one input in thirty million, in one narrow band, whose true logarithm sits within half an ulp of both neighbours and where the two implementations part by the last bit. None of those can reach a design, because on the design path the pinned routine is the definition.
• The corpus is byte-identical. The designs of seven curves at nineteen sampling rates, 133 in all, hash to the same single SHA-256 before and after the logarithm was vectorised: no shipped coefficient, figure or conformance value moved.
• Digest for digest across environments. The designs are bit-identical between this host and the same host under AVX-512 emulation, under every selectable OpenBLAS kernel set, every thread count and every hash seed; the environment-versus-environment comparisons are pinned as tests.
• Each platform its own bytes, measured. With the logarithm pinned, a design no longer depends on which log the C library ships, but the once-per-design calls still do, and they are enough: the A weighting at 48 kHz hashes to 991833ff389afe91 on Linux (glibc, x86-64), 4efa1818bd80e23a on macOS (arm64) and 79852a57e60961c8 on Windows (x86-64), the same value under Python 3.13 and 3.14 on each. The three digests are pinned as a test in the CI matrix, so a platform that drifts from its own value turns CI red. The response tests that run on the same three platforms hold the A design within 0.013 dB of its analog prototype everywhere in its fit band, which bounds what the three byte strings can differ by in response.

Two designs are the same design, and the bytes have a name:

import hashlib
from phonometry import filters

once = filters.WeightingFilter(48000, "A").sos
again = filters.WeightingFilter(48000, "A").sos
print(once.tobytes() == again.tobytes())                 # True
digest = hashlib.sha256(once.tobytes()).hexdigest()
print(digest[:16])                                       # 991833ff389afe91 on Linux

The second line is the point: those sixteen hex digits, the head of the SHA-256, are not “what my machine got”, they are the digest of this release’s A weighting at 48 kHz on any glibc Linux machine on x86-64, the leg the CI matrix pins, and the evidence above gives the macOS and Windows values to expect. If a future release moves a coefficient deliberately, the digest moves with it and the change is a documented event, never a property of your hardware.

The guarantee covers the deterministic weighting design path end to end on one platform, and block processing composes with it: a stateful cascade fed block by block is bit-identical to one pass over the whole record, so streaming does not spend the guarantee. General analysis functions built on numpy and SciPy in their ordinary form remain reproducible on one machine but may differ in the last bits across CPUs, which is the normal condition of scientific Python; the deterministic treatment is applied where a filter’s identity matters.

The full engineering account lives with the code, in the module docstrings of filters/_weighting_design.py and filters/_pinned_log.py, and the tests that hold it are in tests/filters/test_weighting_design.py and tests/filters/test_pinned_log.py.

• Conformance report — what the numbers are checked against; this page is why the checks read the same everywhere.
• Errata in published sources — the other half of the evidence story: where the printed expected value is the thing that is wrong.
• Frequency weightings — the guide to the curves the deterministic design realises.
• Block processing — the streaming identity the scope section leans on.