Correction (2026-09-13): the Reed-Solomon section costed GF(256) syndrome arithmetic as real EML arithmetic, which this post's own boundary section calls a type error, using a scaling law that is false; its 2037n ceiling is withdrawn. The Nash sentence cited a check that nothing records, and the v5 note at the end called the SuperBEST table complete. The other node counts are April hand counts under SuperBEST v4 and have not been re-costed. Each is now marked as an April count under SuperBEST v4 (add 3n on positive inputs and 11n in general, mul 2n, div 2n, neg 2n, recip 1n, pow 3n, exp and ln 1n); the current table is on /superbest.
From Google’s PageRank to your GPS to the NFL passer rating
Every equation has a cost. Not in dollars or milliseconds — in operator nodes. Under the SuperBEST v4 routing table (div=2n, recip=1n), any arithmetic expression reduces to a tree of EML-family primitives, and that tree has an exact node count. The count is structural: it does not depend on hardware, language, or numerical library. It depends only on the shape of the formula.
We spent nine sessions measuring equations across technology, sports, and nature. Here is what we found.
The cheapest equation in technology
ETA = 2 nodes (April v4 count).
Estimated time of arrival is distance divided by speed: d / v. Under v4 routing, division costs 2 nodes. That is the entire computation. Every time a navigation app shows you “14 minutes,” it performed a 2-node calculation — the same cost as a batting average (H / AB) or a field-goal percentage (FG / FGA).
The 2-node ratio is the simplest non-trivial formula that can exist. Anything cheaper is a constant or a variable, which cost zero nodes by definition. ETA, batting average, and field-goal percentage all sit at the absolute floor: one irreducible arithmetic operation, two nodes.
The remarkable thing is where that floor shows up. ETA is arguably the most-used computation in consumer technology. The Maps app runs it millions of times per minute, worldwide. The cheapest formula in technology is also one of the most frequently evaluated ones.
Search engines
TF-IDF: 7 nodes (April v4 count).
The formula that made Google’s early relevance ranking possible costs 7 nodes. TF-IDF is (f / total_terms) * ln(N / df) — at v4 unit costs, one division (2n), one ln (1n), one division (2n), one multiplication (2n). Total: 7n. That is it. The foundation of a trillion-dollar industry.
PageRank: 5N + 4 nodes for a graph of N pages (April v4 count).
PageRank is more expensive because it sums contributions from N incoming links. Each link contributes a 5-node term (division plus scaling), and the N terms add linearly. The iterative ranking algorithm that shaped the modern web scales as O(N) in the link graph — no transcendentals required.
BM25 (modern search): 34 nodes per query term (April v4 count).
BM25 is the standard ranking function in production search systems today, including Elasticsearch. It costs 34 nodes per term (37 nodes for the query term frequency component, minus 3 shared nodes). The jump from TF-IDF (7n) to BM25 (34n) is the arithmetic cost of saturation — BM25 includes a term-frequency normalization that prevents very common terms from dominating, but that normalization requires extra divisions and additions that multiply the node count by nearly 5x.
Softmax, which converts raw scores to probabilities in neural ranking, was counted at 4N−3 nodes for N classes, from a scaling law that has since been retracted; the sum of exponentials in its denominator, Σ e^{xᵢ}, takes at most 2N − 1 nodes (S₁ = EML(x₁, 1), S_{k+1} = EML(LEAd(x_{k+1}, S_k), 1)). It is cheaper per class than BM25 per term, but it accumulates over every candidate in the retrieval set.
Your GPS
Haversine: 28 nodes (April v4 count).
Every time you open Maps and a route appears, the Haversine formula ran. It computes the great-circle distance between two GPS coordinates on a spherical Earth:
a = sin²(Δlat/2) + cos(lat1)·cos(lat2)·sin²(Δlon/2)
d = 2R·arcsin(√a)The April v4 count is 28: four trigonometric terms (each 1n), four multiplications (each 2n), two divisions for the half-angle reductions (each 2n), three additions (each 3n for positive domain), one square root (3n via pow), and the final arcsin and scaling.
Haversine is almost certainly the most-evaluated “outdoor” equation in human history. Every mapping application — Google Maps, Apple Maps, Waze, ride-share apps, delivery routing — runs this formula for every distance query. The number of daily Haversine evaluations across all devices worldwide is in the billions. It has been running at this scale since smartphones became common. No equation in the physical sciences comes close to this evaluation frequency.
The Vincenty formula (more accurate, accounts for Earth’s elliptical shape) was estimated at about 80 nodes per iteration (April v4). GPS devices and precision surveying use it. Navigation apps use Haversine — 28 nodes is fast enough and accurate enough for routing at human scales.
Video games and 3D graphics
Quaternion rotation: 99 nodes (positive-domain addition), 235 nodes (general-domain addition), both April v4 counts.
This is why 3D rendering is the GPU bottleneck.
A quaternion rotation requires multiplying two quaternions — four components each — then applying the result to a 3D vector. The algebra expands into many cross-multiplications and additions. Under v4 routing with add_gen=11n (required because quaternion components can be negative), the full general-domain count reaches 235 nodes per rotation.
GPUs exist because of this number. A single rendered frame at 60 fps requires quaternion rotations for every moving object, every bone in an animation rig, every camera transformation. At 235 nodes each, the only way to hit frame-rate targets is massive parallelism — thousands of shader cores doing quaternion arithmetic simultaneously.
For comparison:
| Operation | Nodes (April v4) |
|---|---|
| Perspective projection | 8n |
| Verlet integration (physics) | 12n |
| Phong lighting (per pixel, per light) | 43n |
| Quaternion rotation (general) | 235n |
Verlet integration — the algorithm used for cloth simulation, soft bodies, and rigid-body physics in games — costs only 12 nodes. Physics engines are arithmetically cheap. It is the rotation and lighting that consume GPU budget.
Phong lighting at 43 nodes is evaluated per pixel, per light source. A scene with four lights and a 1080p framebuffer requires 4 × 1920 × 1080 × 43 = 357 million node evaluations per frame. At 60 fps, that is 21 billion node evaluations per second, for lighting alone. This is why rasterization is a parallel problem.
Sports
NFL passer rating: 33 nodes. Hodgkin-Huxley neuron: 30 nodes. (April v4 counts)
The NFL passer rating formula is more arithmetically complex than the equation that describes how neurons fire.
The passer rating formula, introduced in 1973, involves four components (completion percentage, yards per attempt, touchdown rate, interception rate), each clamped to a range and scaled by specific constants, then averaged and converted to a 0–158.3 scale. The clamping and averaging require multiple additions and divisions that accumulate to 33 nodes.
The Hodgkin-Huxley model — the 1963 Nobel Prize equation describing how action potentials propagate along nerve axons — costs 30 nodes. It contains three conductance terms with voltage-gated variables, but the terms are structurally simpler than the NFL’s clamped polynomial.
This is not a joke about sports versus science. It is a factual observation about arithmetic structure: the NFL’s particular choice of formula, with its four normalized components and range restrictions, happens to require more irreducible operations than the biophysical model of a neuron. Both equations are what they are. Cost theory just counts them.
Other results:
| Formula | Nodes (April v4) |
|---|---|
| Batting average | 2n |
| ELO rating update | 26n |
| Kelly criterion | 8n |
| Nash equilibrium (2-player) | 19n |
| Pythagorean expectation | 11n |
Under v4, ELO (26n) costs more than batting average (2n) because it includes a logistic sigmoid transformation on the rating difference — that sigmoid costs 7n on its own under v4 (neg 2n, exp 1n, add 3n, recip 1n; /superbest’s current F16 route for sigmoid is 4n) — plus the update arithmetic. Nash equilibrium (19n) was counted with April’s general-domain addition, add_gen = 11n, in the denominator, because game payoffs can be negative; addition now takes 2n for all real inputs (ADD-T1), so 19n is out of date. This sentence also said “the counterexample exists and was verified”, without saying what the counterexample was or what checked it, and nothing on record does.
The EML boundary
CRC-32, Hamming distance, parity check: outside EML.
This is an important boundary to state honestly.
EML arithmetic handles real and complex numbers via exp, ln, and their combinations. It does not handle GF(2) — the finite field with two elements, where addition is XOR and multiplication is AND. CRC-32, Hamming error-correcting codes, and parity checks all operate in GF(2). They are not more or less expensive than EML formulas; they are in a categorically different domain.
Asking for the SuperBEST node count of a CRC checksum is a type error, the way asking for the temperature of a poem is a type error. The arithmetic is not comparable. Any cost measurement in this series applies only within the real/complex exp-ln domain. Bitwise polynomial arithmetic is elsewhere.
Cross-domain isomorphisms
The deepest result of the nine sessions is not any individual equation cost. It is that the universe reuses the same arithmetic templates across completely unrelated physics.
The 5-node exponential decay template: A · exp(−B·x) (April v4 count)
Five nodes. This structure appears in:
- Beer-Lambert law (optical absorption through a medium)
- CT scan attenuation (X-ray through tissue)
- Radioactive decay (nuclear physics)
- RC circuit discharge (electrical engineering)
- Atmospheric absorption (meteorology)
These equations describe different physical phenomena with different variables and different units. They are structurally identical. The operator tree is mul(A, exp(mul(neg(B), x))) — 5 nodes every time. The universe has one algorithm for exponential attenuation and applies it everywhere.
The 5-node exponential growth template: A · exp(B·t) (April v4 count)
Same cost, different sign:
- Population growth (biology)
- Compound interest (finance)
- Bacterial colony growth (microbiology)
- Logarithmic spiral geometry (mathematics)
The 7-node logarithmic ratio template: 10 · log10(ratio) (April v4 count)
Seven nodes. This template underlies:
- Sound pressure level in decibels (acoustics)
- Signal-to-noise ratio in communications
- Richter magnitude scale (seismology)
- dBm power measurement (radio engineering)
- Perceptual audio loudness
All five measure ratios on a logarithmic scale. All five cost 7 nodes. The ear, the seismograph, and the radio antenna solve the same 7-node problem.
The 11-node Hill/Pythagorean template: x^k / (x^k + y^k) (April v4 count)
Eleven nodes. Three disciplines:
- Pythagorean expectation (sports analytics): predicts win probability from runs scored and allowed
- Hill equation (pharmacology): models receptor-ligand binding cooperativity
- Michaelis-Menten kinetics (biochemistry): enzyme saturation model
A baseball statistician in 1983, a pharmacologist in 1910, and a biochemist in 1913 each discovered a different interpretation of the same 11-node rational function.
The most expensive equation computed (withdrawn)
This section gave Reed-Solomon syndrome computation for RS(255,223) a cost of 2037 nodes, the most expensive formula in the catalog by a wide margin.
Every QR code, every DVD, every deep-space transmission from the Voyager probes uses Reed-Solomon error correction. The syndrome computation for the standard RS(255,223) code — 255 total symbols, 223 data symbols, capable of correcting up to 16 symbol errors — evaluates a degree-254 polynomial at 32 points over GF(256), the finite field with 256 elements.
That count is withdrawn. Finite-field arithmetic is outside EML: by this post’s own boundary section above, asking for its node count is a type error. The 2037 came from translating the XOR-based GF arithmetic into real arithmetic and evaluating a real polynomial under 8N − 3 with N = 255, and 8N − 3 comes from a scaling law, (α₀+3)N − 3, that was stated as exact and is false. A real polynomial evaluation is not what a Reed-Solomon decoder computes, so 2037 is not a cost of Reed-Solomon at all, and neither this section’s comparison with Haversine nor its explanation of dedicated RS hardware stands.
Among the counts that remain in this post, the largest is quaternion rotation with general-domain addition, 235n, itself an April count made before addition dropped to 2n (see the update at the end).
Conclusion
A formula is a formula.
The same node-counting that works for chemistry and neuroscience works for sports analytics, GPS navigation, and error correction. The SuperBEST cost is structural — it does not care whether the equation came from a physics textbook, an NFL rulebook, or an IETF standard.
What the nine sessions established:
- The floor is 2n (ETA, batting average, FG%) — any ratio (April v4 counts).
- The 2037n Reed-Solomon ceiling is withdrawn: syndrome computation is finite-field arithmetic, outside EML.
- The most frequently evaluated formula in outdoor navigation is 28n (Haversine, April v4).
- Sports statistics can exceed neuroscience in arithmetic complexity (April v4: NFL 33n > HH 30n).
- The universe uses five or six canonical templates — exponential decay, exponential growth, log ratio, Hill function — across every domain of science and engineering.
- GF(2) bitwise arithmetic is outside EML. The boundary is real and important.
Domain-2 expansion (FIN, INFO, QM, THERMO, CHEM, BIO, ECON sessions) extended the catalog beyond these nine sessions to 295+ equations across 12+ domains. The floor is unchanged, and the ceiling was the withdrawn Reed-Solomon count; the cross-domain isomorphism count grew from 8 to 12+ families. See 295+ Equations →.
SuperBEST cost theory does not tell you what an equation means. It tells you how many irreducible operations it requires. Sometimes that number surprises you. The NFL passer rating surprised us. The Reed-Solomon count should have stopped us: it measured arithmetic this post says EML does not do.
The cost of everything is countable. We are counting.
Monogate Research (2026). “The SuperBEST Cost of Everything.” monogate research blog. https://monogate.org/blog/cost-of-everything
Sessions: TECH-1 through TECH-5, SPORT-1 through SPORT-2, NAT-1 through NAT-2. SuperBEST v4 routing table (div=2n, recip=1n). The counts are April hand counts; they have not been re-costed, and no public script reproduces them.
Update — 2026-04-20 (SuperBEST v5): General-domain addition drops from 11n to 2n via
lediv(x, deml(y,1)). The two-tier table (add_pos=3n / add_gen=11n) collapses to a single entry: add=2n for all reals. New total: 18n, savings 75.3%. This note called the table complete. It was not: later constructions cut mul to 1 node for x, y > 0 and pow and sqrt to 1 node for x > 0, and the current positive-domain total is 15n in F16 (79.5%; 14n if ln counts as one EXL node, see /superbest). Equations that previously used add_gen=11n (ELO, Nash equilibrium, quaternion rotation, Henderson-Hasselbalch) benefit proportionally — e.g. quaternion rotation (general) drops from 235n substantially. See General Addition in 2 Nodes →.