Average Error: 30.7 → 0.2
Time: 9.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} - \left(\frac{\frac{1}{2}}{x} - x\right)\right)\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 30.7

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

  2. Simplified30.7

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

  3. Taylor expanded around inf 0.2

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

  4. Simplified0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019100 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))