Fourier Beats Taylor by 100x in EML Node Count
Every EML tree is a composition of exp and ln. To approximate sin(x) using real arithmetic, you need an infinite Taylor series — and in BEST routing, that costs 101 nodes for 8 terms.
But there’s a shortcut. And it has everything to do with why the operator is called exp-minus-ln.
The Taylor Route
sin(x) = x − x³/3! + x⁵/5! − x⁷/7! + ···
Each term requires:
- pow(x, 2k+1): 3 nodes (EXL, best known)
- div by (2k+1)!: 1 node (EDL)
- alternating sign: 6 nodes (neg via EDL)
- sum with previous term: 3 nodes (mixed EAL bridge)
Per term: 3 + 1 + 6 = 10 nodes. Plus 3 nodes per addition to sum them.
Total for K terms:
| K terms | Nodes | MSE on [−π, π] |
|---|---|---|
| 2 | 23 | 2.55e+00 |
| 4 | 49 | 8.21e-02 |
| 6 | 75 | 4.66e-04 |
| 8 | 101 | 7.95e-07 |
| 10 | 127 | 5.39e-10 |
To reach 10⁻⁶ accuracy: 101 nodes.
The Fourier Route
Euler’s formula: exp(ix) = cos(x) + i·sin(x).
In EML: exp(ix) = eml(ix, 1) = exp(ix) − ln(1) = exp(ix).
This is one EML node. The imaginary part is sin(x) exactly. No approximation.
sin(x) = Im(eml(ix, 1)) [1 complex EML node, exact]
cos(x) = Re(eml(ix, 1)) [same node, exact]This is not an approximation — it is the exact value, to floating-point precision, in a single operator application.
The Fourier series for general periodic functions costs more:
| N terms | EML nodes | Accuracy for square wave |
|---|---|---|
| 1 | 21 | exact for sin (this IS sin) |
| 2 | 37 | ∼1/2² |
| 4 | 69 | ∼1/4² |
| 8 | 133 | ∼1/8² (Gibbs) |
But sin(x) itself is the N=1 case: 1 complex node.
The 100x Gap
| Method | Nodes for sin(x) | Error |
|---|---|---|
| Taylor, 8 terms | 101 | 7.95e-07 |
| Fourier, 1 term | 1 | exact |
The ratio is 101:1. One hundred times fewer nodes.
Why This Happens
The Taylor approach forces sin(x) into the real number line, where EML’s operator family has no native trig support. Every term is a workaround: use powers and factorials to reconstruct a function that the operators weren’t designed for.
The Fourier approach uses the complex path — exactly the mechanism that makes the EAL bridge work and that EMN uses to achieve approximate completeness. In the complex plane, exp(ix) is native. sin(x) is just its imaginary component.
The structural fact: EML’s completeness theorem holds over ℂ. In the real line, sin(x) is unreachable (Infinite Zeros Barrier). In ℂ, it costs one node.
The complex plane is not a trick — it is where EML lives natively.
What This Means for BEST Routing
The BEST routing table currently tracks single-variable operations. Adding sin and cos:
| Operation | BEST operator | Nodes | Notes |
|---|---|---|---|
| exp(x) | EML | 1n | real |
| ln(x) | EXL | 1n | real |
| sin(x) | EML | 1n | complex path |
| cos(x) | EML | 1n | complex path |
| sinh(x) | EML | 3n | (exp(x)−exp(−x))/2 |
| cosh(x) | EML | 3n | (exp(x)+exp(−x))/2 |
exp, ln, sin, and cos all cost 1 node in BEST. The trig functions are not special — they are Euler’s formula applied once.
Why the Lab Uses This
The sound engine in the Monogate lab synthesizes waveforms by computing exp(iωt) for each harmonic frequency ω. Each harmonic is one EML node. A 16-harmonic instrument timbre is 16 nodes — one per frequency component.
This is exactly the Fourier representation, implemented as EML trees. The lab isn’t using Fourier synthesis as an analogy — it is literally computing EML trees, one per harmonic, taking the imaginary part.
When you hear a note in the sound experience, you’re hearing the output of an EML tree.
Caveat: Taylor Still Wins for Exponentials
For exp(x) itself, the identity tree costs 1 node and is exact. Taylor can only approximate exp(x) with K terms:
| K terms Taylor exp(x) | Nodes |
|---|---|
| 2 | 11 |
| 4 | 25 |
| 8 | 53 |
Taylor approximates what EML already knows exactly. The 1-node EML identity tree for exp(x) beats every finite Taylor series — because exp is the operator’s native operation.
The lesson generalizes: EML native operations cost 1 node. Non-native operations (real trig) are expensive. Complex-path operations recover native cost.
Session M8 · Direction 13 of the Research Roadmap