Average Error: 31.2 → 0.3
Time: 9.1m
Precision: 64
Internal Precision: 128
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\frac{\frac{\frac{-1}{8}}{x \cdot x}}{x} + (2 \cdot x + \left(\frac{\frac{-1}{2}}{x}\right))_*\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 31.2

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

  2. Simplified31.2

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

  3. Taylor expanded around inf 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))