A Fixed-Point Construction of the Elementary Transcendental Functions
arXiv:2607.23196
2026
Other
1 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper replaces direct iteration of duplication identities with a residual coordinate in which the refinement map is contractive and numerically well-conditioned. The most transferable construction is the logarithm residual \(\rho(u)\), whose dyadic square-root refinement avoids the exponential error amplification caused by reconstructing the macroscopic logarithm at every step. This can become a table-free, vectorizable \(\log1p\) primitive for neural-network kernels, especially softplus, log-sum-exp, negative-log-likelihood, and attention normalization paths. The opportunity is primarily inference and training throughput plus numerical stability, rather than a new model architecture.
Ideas from this paper
Unverified
2026
Implement \(\log(1+u)\) through the bounded residual \(\rho(u)\), refining the argument by repeated square roots and propagating the residual backward with a rational recurrence. Use this table-free approximation in softplus and log-sum-exp kernels, where millions of logarithms are evaluated and predictable arithmetic can outperform general-purpose library calls on vectorized hardware.
Useful5/10
Difficulty5/10
Novelty6/10