Average Error: 31.3 → 0.4
Time: 29.5s
Precision: 64
Internal Precision: 128
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\frac{\frac{\frac{-1}{4}}{x}}{x} + \left(\left(\log x + \log 2\right) - \frac{\frac{3}{32}}{{x}^{4}}\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 31.3

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

  2. Simplified31.3

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

  3. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{\log 2 - \left(\log \left(\frac{1}{x}\right) + \left(\frac{3}{32} \cdot \frac{1}{{x}^{4}} + \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right)\right)}\]

  4. Simplified0.4

    \[\leadsto \color{blue}{\left(\left(\log x + \log 2\right) - \frac{\frac{3}{32}}{{x}^{4}}\right) + \frac{\frac{\frac{-1}{4}}{x}}{x}}\]

  5. Final simplification0.4

    \[\leadsto \frac{\frac{\frac{-1}{4}}{x}}{x} + \left(\left(\log x + \log 2\right) - \frac{\frac{3}{32}}{{x}^{4}}\right)\]

Reproduce

herbie shell --seed 2019068 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))