Average Error: 30.9 → 0.2
Time: 32.1s
Precision: 64
Internal Precision: 2368
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left((\left(-\frac{1}{x}\right) \cdot \left((\left(\frac{1}{x}\right) \cdot \left(\frac{\frac{1}{8}}{x}\right) + \frac{1}{2})_*\right) + \left(x \cdot 2\right))_*\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 30.9

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

  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

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

Runtime

Time bar (total: 32.1s)Debug logProfile

herbie shell --seed 2018198 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))