The F16 Framework
Every two-input function formed by combining exp and ln with arithmetic is one of exactly 16 operators. Every elementary function is a finite binary tree over F16.
The generating operator
A single binary function that generates every elementary function as a finite tree. The 16 operators below are its complete family — all sign combinations of the exponent argument and all branch choices of the logarithm.
The 16 F16 operators
exp(x) − ln(y)all x, y > 0Native gate. eml(x,1) = eˣ.exp(x) − ln(−y)all x, y < 0Negative-branch ln.exp(−x) − ln(y)all x, y > 0Dual gate. deml(x,1) = e^(−x).exp(−x) − ln(−y)all x, y < 0Dual negative-branch.exp(y) − ln(x)x > 0, all yArgs swapped.exp(−y) − ln(x)x > 0, all yDual-swap.exp(y) − ln(−x)x < 0, all ySwap + negative branch.exp(−y) − ln(−x)x < 0, all yDual swap + negative branch.x − ln(y)all x, y > 0lediv(x,eml(y,1)) = x−y.x − ln(−y)all x, y < 0Negative-branch LEdiv.ln(eˣ + y)all x, y > −eˣLog-sum. LEAd(x,1) = softplus(x).ln(eˣ − y)all x, y < eˣLog-difference.exp(x · ln(y))all x, y > 0y^x — pow primitive. EPL(n,x)=xⁿ (1 node).exp(x + ln(y))all x, y > 0y·eˣ — mul primitive.exp(x + ln(−y))all x, y < 0(−y)·eˣ.exp(ln(x) + ln(y))x, y > 0x·y — product via log-sum.ELC(ℝ) — Exp-Log Closure
The Exp-Log Closure of ℝ (ELC(ℝ)) is the smallest set containing ℝ that is closed under the EML gate and the constant 1.
Theorem (Odrzywołek 2026): ELC(ℝ) equals the set of all functions expressible as a finite composition of real exp and ln. Every classical elementary function belongs to ELC(ℝ).
What is in ELC(ℝ)
- ✓ All polynomials and rational functions
- ✓ exp(x), ln(x), x^r for any rational r
- ✓ sin(x), cos(x) over ℂ (via Euler: e^(ix) = cos x + i sin x)
- ✓ Sigmoid, softplus, and all activation functions built from exp/ln
- ✓ Every function in the classical elementary function hierarchy
What is NOT in ELC(ℝ) over ℝ
- ✗ sin(x) and cos(x) over ℝ — T01: every real F16 tree has finitely many zeros; sin/cos have infinitely many
- ✗ erf(x), Γ(x) — same zero-count obstruction
- ✗ |x| — no real F16 tree computes abs(x) (sign barrier)
SuperBEST: minimum depth within ELC(ℝ)
For each operation in ELC(ℝ), the SuperBEST table records the minimum number of F16 nodes needed. Every arithmetic primitive costs ≤ 2 nodes on the positive domain.