Average Error: 30.8 → 0.0
Time: 17.9s
Precision: 64
Internal Precision: 128
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\sqrt{x + \sqrt{1 + x} \cdot \sqrt{x - 1}}\right) + \log \left(\sqrt{x + \sqrt{1 + x} \cdot \sqrt{x - 1}}\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 30.8

    \[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
  2. Using strategy rm

  3. Applied difference-of-sqr-130.8

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

  4. Applied sqrt-prod0.0

    \[\leadsto \log \left(x + \color{blue}{\sqrt{x + 1} \cdot \sqrt{x - 1}}\right)\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt0.0

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

  7. Applied log-prod0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019050 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))