Average Error: 31.6 → 0.3
Time: 3.7m
Precision: 64
Internal Precision: 128
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[(\frac{1}{4} \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\log 2 - \left(\frac{\frac{3}{32}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \log x\right)\right))_*\]

Error

Bits error versus x

Derivation

  1. Initial program 31.6

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

  2. Simplified31.6

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

  3. Taylor expanded around inf 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019088 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))