Average Error: 31.2 → 0.2
Time: 1.7m
Precision: 64
Internal Precision: 2432
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\left(x \cdot 2 - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 31.2

    \[\log \left(x + \sqrt{x \cdot x - 1}\right)\]

  2. Taylor expanded around inf 0.2

    \[\leadsto \log \color{blue}{\left(2 \cdot x - \left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]

  3. Applied simplify0.2

    \[\leadsto \color{blue}{\log \left(\left(x \cdot 2 - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{\frac{1}{8}}{x}}{x \cdot x}\right)}\]

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))