The SuperBEST Cost of Calculus
Tier: OBSERVATION (computed results, reproduce commands below)
How expensive is calculus in EML nodes? This post measures Taylor series, integrals, ODEs, automatic differentiation, and transforms — all with SuperBEST FINAL routing (T08).
CAL-1: Taylor Series
A Taylor series for sin(x) with N terms costs 9N − 3 nodes under SuperBEST routing.
Per term: one pow (3n) + one mul (3n) = 6n. Per join: one sub or add = 3n. For N terms: N terms × 6n + (N−1) joins × 3n = 9N − 3.
| N terms | SuperBEST nodes | Old BEST nodes |
|---|---|---|
| 4 | 33n | 27n |
| 6 | 51n | 39n |
| 8 | 69n | 51n |
| 10 | 87n | 63n |
| 12 | 105n | 75n |
SuperBEST is slightly more expensive for Taylor series than old routing. Why? Old routing used mul=7n and sub=5n; new routing has mul=3n but the formula structure penalizes the joins. The savings from mul (7→3) are offset by the per-join cost holding at 3n.
Key contrast: Fourier series via Euler gateway (T16) needs 1 complex EML node for the kernel exp(iωt), regardless of N.
CAL-2: Integration
Elementary integration (closed-form antiderivatives) is free — an antiderivative is itself an elementary function, so it has the same EML node count as the integrand.
Non-elementary integrals (erf, Li₂, Fresnel, elliptic) have no finite EML tree (T01 generalization). They require numerical integration, which costs O(N·f_nodes) for N quadrature points.
| Function | Integrable? | EML cost |
|---|---|---|
| exp(x) | Yes | 1n |
| x^n | Yes | 3n (pow) |
| 1/(1+x²) | Yes (arctan) | 4n |
| sin(x) | Yes (-cos) | 1n complex |
| erf(x) | No | ∞ exact |
| x^x | No | ∞ exact |
CAL-4: Ordinary Differential Equations
Heat equation mode: 2 nodes per mode (OBSERVATION).
The solution to u_t = k·u_xx with mode n is: exp(−n²π²kt) · sin(nπx)
This decomposes as: 1 DEML node for the exponential decay factor × 1 complex EML node for the spatial oscillation. Total: 2 nodes, exact, per mode.
Harmonic oscillator y” + ω²y = 0: solution cos(ωx) or sin(ωx) = 1 complex EML node via T03 (Euler gateway).
CAL-5: Automatic Differentiation
The k-th derivative of an N-node EML tree costs approximately 3^k · N nodes.
Each differentiation order multiplies the tree size by roughly 3 (chain rule on EML introduces exp and 1/x terms). For N=1, exp(x):
| k | d^k/dx^k exp(x) | Cost |
|---|---|---|
| 1 | exp(x) | 1n |
| 2 | exp(x) | 1n |
| k | exp(x) | 1n |
exp is its own derivative — 0 cost growth. For general EML trees with N>1, cost grows as 3^k·N.
Crossover: for N≤2, EML autodiff stays competitive with dual-number forward mode.
CAL-9: Fourier and Laplace Transforms
Fourier kernel: 1 complex EML node (OBSERVATION, optimal).
exp(−iωt) = ceml(−iωt, 1) — this is T03 (Euler Gateway) applied to the transform kernel.
| Transform | Kernel | EML cost |
|---|---|---|
| Fourier: e^{−iωt} | ceml(−iωt,1) | 1 node |
| Laplace: e^{−st} | ceml(−st,1) | 1 node (complex s) |
| Z-transform: z^{−n} | epL | 1 node |
| Wavelet: ψ(t/a) | depends on ψ | varies |
The Fourier kernel is a single complex EML node — this is why Fourier series beats Taylor by 100× (T16).
Reproduce
python python/scripts/research_cal_quantum.pyResults in python/results/cal_quantum_results.json.
Cite: Monogate Research (2026). “The SuperBEST Cost of Calculus.” monogate research blog. https://monogate.org/blog/calculus-costs