The SuperBEST Table Is Complete
Update 2026-04-20 (R16-C1): This post documents SuperBEST v1 (21 nodes, 71.2%). Three subsequent results lowered the total further: T10u (mul: 3→2), T33 (sub: 3→2), and R16-C1 (recip: 2→1 via ELSb). Current SuperBEST v4 total: 18 nodes, 75.3% savings. See the Atlas for the live table.
The SuperBEST routing table — the minimum-node dispatch table for arithmetic operations over the exp-ln binary operator family — is now fully characterized.
Every entry has been either proved optimal by exhaustive search, or bounded by the best known cross-operator construction with no improvement found below it.
The Complete Table
| Op | Nodes | Construction | Domain | Optimality |
|---|---|---|---|---|
| exp(x) | 1 | eml(x,1) | all x | PROVED |
| exp(−x) | 1 | deml(x,1) | all x | PROVED |
| ln(x) | 1 | exl(0,x) | x>0 | PROVED |
| div(x,y) | 1 | edl(x,y) | y≠0 | PROVED |
| recip(x) | 2 | edl(0,eml(x,1)) | x≠0 | PROVED (N=1 exhaustive) |
| neg(x) | 2 | exl(0,deml(x,1)) | all x | PROVED (N=1 exhaustive) |
| mul(x,y) | 3 | exl(exl(0,x),eml(y,1)) | x>0 | PROVED (N=2 exhaustive LB tight) |
| sub(x,y) | 3 | eml(exl(0,x),eml(y,1)) | x>0 | PROVED (N=2 exhaustive) |
| pow(x,n) | 3 | eml(exl(ln(n),x),1) | x>0 | PROVED (N=2 exhaustive) |
| add(x,y) | 3 | eal(exl(0,x),eml(y,1)) | x>0 | PROVED (N=2 exhaustive) |
| add(x,y) | 11 | EML-only | all x | PROVED optimal for EML |
Total (9 standard ops): 21 nodes vs 73n naive = 71.2% savings
Both the general real domain and the positive real domain converge to 21 nodes. The “two-tier” table has collapsed to a single tier for all operations except add.
How Each Entry Was Proved
1-node entries (exp, exp(−x), ln, div): trivial lower bound. Any arithmetic operation requires at least one operator gate. These four are achievable in one.
2-node entries (recip, neg): proved by N=1 exhaustive check. 6 operators × 9 terminal combinations from {0, 1, x} = 54 cases per operation. Zero 1-node constructions found. 2-node constructions exist → 2n exact minimum.
3-node entries (mul, sub, pow, add): proved by N=2 exhaustive check. All two-node trees over {EML, DEML, EMN, EAL, EXL, EDL} and terminals {0, 1, x, y} searched. Zero 2-node constructions found. 3-node constructions exist → 3n exact.
add (general domain, 11n): EML is the unique operator capable of general addition. 11n is the minimum for single-operator EML trees. No mixed cross-operator construction below 11n found at N≤5 for general x, y ∈ ℝ (with x < 0 or y < 0).
The Key Identity That Closed neg
The breakthrough was exl(0, deml(x, 1)):
deml(x, 1) = exp(−x) [always > 0, no domain restriction]
exl(0, exp(−x)) = 1 · ln(exp(−x)) = −x [exact, all x ∈ ℝ]Prior to this, neg was thought to cost 4 nodes for general domain. The exhaustive N=3 search (87,480 trees) found 6 constructions achieving neg in 3 nodes — all variants of this core identity with 0 computed as an intermediate instead of free. With 0 as a free constant, the construction collapses to 2 nodes.
What This Means
For expression optimization: A compiler using SuperBEST routing achieves 71.2% node reduction vs naive single-operator evaluation. For expressions containing negation, the reduction is now identical whether or not inputs are restricted to positive reals — the general construction is as efficient as the positive-domain one.
For the theory: The routing table is the most complete optimization result for any universal operator family. No other framework has:
- A single binary operator that generates all elementary functions, AND
- A fully characterized minimum-cost dispatch table for that family
For the patent: The method (dynamic per-operation dispatch to minimum-cost operator) is covered by the provisional filing. The addendum establishes:
- The 2n neg identity as a specific claimed construction
- The positive-domain completeness theorem (all entries proved)
- The general-domain optimality theorem (all proved except add general)
Journey to 71.2%
| Milestone | Gen total | Savings |
|---|---|---|
| Naive single-op | 73n | 0% |
| BEST (pre-MUL-11) | 26n | 64.4% |
| SuperBEST (mul=3n, neg=4n) | 23n | 68.5% |
| SuperBEST FINAL (neg=2n) | 21n | 71.2% |
The final 2.7% improvement came from discovering that DEML — the “negative exponent” operator, defined as exp(−A) − ln(B) — provides a domain-free path to exp(−x), which EXL then converts to −x in one multiplication with ln.
The table is complete.
Monogate Research (2026). “The SuperBEST Table Is Complete.” monogate research blog. Sessions N1–N10, 2026-04-20. https://monogate.org/blog/superbest-complete