How we check our maths

How the checks work

We made one sample file of invented signals: 4,000 readings taken 1,000 times a second, with two tones, random noise, a sudden step, a column with empty cells and a few columns for fatigue counts and prices. Then we ran every calculation in the Math dialog on it twice: once in MegaRows, exactly as the dialog does it, and once with NumPy and SciPy, the free scientific libraries for Python that most engineers and researchers use, with the same settings. Every output value was compared.

The sample file is public: download it and you can repeat any check yourself. It is also the sample signal behind FFT from CSV.

How we check

These are our own checks, not an independent audit: MegaRows writes and runs them, and the recipe below lets you repeat three of them yourself. The full comparison runs every calculation of the Math dialog, with every option it offers, through MegaRows and through NumPy and SciPy, and compares every output value; it was last run on 8 October 2026. Between full runs, our automated tests check the same operations and options against SciPy’s stored answers on every change to the site, so a change that moved a result would be caught.

What the differences mean

Computers store numbers with about 16 significant digits, and two programs that add up the same numbers in a different order can disagree in the last one or two. So a check does not ask for identical answers. It measures the largest difference between the two results, divided by the size of the result, and passes when that is tiny:

• none: identical to the last digit (341 of the 526 checks);
• 10⁻¹⁵ means one part in a million billion: the last digit a computer keeps. Single functions such as a sine or a logarithm are held to this;
• 10⁻¹² is the limit for sums, statistics, fits and spectra: still a thousand times finer than anything a plot or a table can show;
• 10⁻¹⁰ is the limit for the filters, which work out their own coefficients and carry the last digits through every sample.

The largest difference of all is 1.2 × 10⁻¹¹ of the result, in Savitzky-Golay smoothing, and there SciPy is the less precise of the two: against the exact answer, MegaRows agrees to the last digit.

Where we follow a different convention on purpose

In a few places MegaRows deliberately does something other than the library’s default, usually to match what spreadsheet users expect. There the reference was set to the same convention, so the numbers could still be compared, and the difference is explained here.

• FRF and coherence at 0 Hz with a rectangular window. The 0 Hz row means nothing here, in MegaRows and in SciPy alike, so it is left out of the comparison. Each segment's mean is removed before the FFT (scipy's default, and the only choice in the dialog). With a rectangular window that makes the 0 Hz input and output spectra zero up to rounding, so H1, H2 and the coherence at 0 Hz are 0/0 in both tools: MegaRows shows |H1| 0.099 there and scipy 0.269, both rounding noise. Every other row agrees to 1e-12 or better, and with any other window the 0 Hz row is a real value and is compared.
• Powers with a missing or infinite exponent. pow(1, missing) stays missing in MegaRows, where NumPy says 1. pow and ^ follow JavaScript: a NaN exponent gives NaN even for a base of 1, and (±1) ^ (±∞) is NaN (C and numpy give 1); x ^ 0.5 is computed as sqrt(x), so (−∞) ^ 0.5 is NaN where numpy.power gives +∞. The sample has no such values; formula.json covers them with a reference that follows this rule.
• Exponential moving average with gaps. A missing row stays missing in MegaRows; pandas repeats the previous value there. The smoothing state is held across the gap in both, so every other row is the same; the reference sets the missing rows back to NaN. With a time constant and an X column the smoothing factor follows each X step, which pandas cannot do with adjust=False, so that case uses the recurrence written out in numpy.
• Rainflow range histogram. Cycle ranges are binned like numpy.histogram, not like the rainflow package's own binning. The bins are numpy.histogram(ranges, bins, range=(0, largest range), weights=counts); the package's count_cycles(nbins=...) uses right-closed bins sized from the signal's span, so its table would differ while the cycles themselves are identical.
• Rounding. round(2.5) is 3, like Excel; NumPy rounds halves to even (2). round() rounds halves away from zero (−2.5 → −3), like Excel and MATLAB. The reference is numpy written out for half-away rounding, sign(x)·floor(|x| + ½), computed exactly; with numpy.round the prices ending in a half cent would differ.
• Standard deviation. The standard deviation is the sample one, divided by n − 1, like MATLAB's std and Excel's STDEV. NumPy's nanstd divides by n unless asked otherwise, so the reference is numpy.nanstd(y, ddof=1). The same holds for the statistics panel and the Compare view. The z-score keeps SciPy's default (n) and the moving and expanding standard deviations pandas' (n − 1); both have a choice in the dialog.
• Missing values in comparisons and logic. A comparison with a missing value is missing, not false, so a gap stays a gap. numpy would say NaN > 0 is false. and / or follow three-valued logic: false and missing is false, true or missing is true. pandas' nullable arrays behave exactly like this, so they are the reference.
• Gaps in filters and spectra. Gaps are bridged by a straight line before filtering, then shown as gaps again. Operations that skip missing samples leave them out (pandas rolling(min_periods=1), nanpercentile). Filters, spectra, Hilbert, Savitzky-Golay, decimation and SRS first fill gaps by linear interpolation and, for results on the same rows, set the filled rows back to missing; the references do the same.
• Detrend against X. With an X column the trend line is fitted against X, not against the row number. The reference is numpy.polyfit(time_s, y, 1); scipy.signal.detrend fits against the row number and is the reference when X is the row number.
• Rainflow hysteresis. No standard library has a hysteresis gate, so the reference gate is a short independent Python function. A turning point is kept once the signal has come back from it by the gate; the kept points then go through rainflow.extract_cycles.
• Savitzky-Golay with long windows. Here SciPy is the less precise one; MegaRows matches exact arithmetic. SciPy solves each window's least-squares fit on raw powers of the sample offsets, which leaves about 1e-11 of its own rounding for a 51-sample window (polynomial order 4). MegaRows uses an orthonormal basis; against the same filter computed in exact rational arithmetic it agrees to 1e-15, so that case has a second row with the exact reference and a 1e-13 tolerance.

Check it yourself in Python

Three checks you can run in a minute: download the sample file, then run these lines with NumPy, SciPy and pandas installed. They read the signal_a column and work out what the statistics panel, the FFT spectrum and the PSD show for it in MegaRows.

They print:

Open the same file in the CSV plotter and you get the same numbers: the statistics panel for the whole file (Mean and Std dev, the sample standard deviation), Math › FFT spectrum with a rectangular window (the amplitude at 50 Hz and 120 Hz), and Math › PSD (Welch) with a Hann window, segments of 1,024 samples and 50 % overlap (the Grms readout). We ran these lines on 8 October 2026 with NumPy 2.1.3, SciPy 1.14.1, pandas 2.2.3 and Python 3.12; a test checks on every change that MegaRows still gives these numbers. The worked example on FFT from CSV goes through the PSD step by step.

The worked examples of the signal pages

Each signal page has a worked example on a made-up sample file of its own. A test checks on every change that MegaRows gives the numbers the page prints and that they match SciPy’s to about 12 significant digits:

• FFT from CSV: the PSD of the sample signal against scipy.signal.welch, and the FFT spectrum against NumPy’s FFT.
• PSD calculator: the Grms, the band’s level, the tone and the dB view of the vibration sample against scipy.signal.welch.
• Spectrogram from CSV: every slice’s loudest frequency and the values of the middle one against scipy.signal.spectrogram, with the Tukey and Hann windows and a frequency range.
• FRF and coherence: H1, H2, the phase and the coherence of the mast sample against scipy.signal.csd, welch and coherence.

Repeat the checks

The full comparison was last run on 8 October 2026, with numpy 2.1.3, scipy 1.14.1, pandas 2.2.3, rainflow 3.2.0, Python 3.12.15, Node.js 24.20.0. Between full runs, our automated tests check the same operations against SciPy’s stored answers on every change to the site.

Download the sample file (CSV, 185 KB), open it in FFT from CSV or the plotter, and compare any result with your own tools.