Average Error: 30.6 → 0.1
Time: 29.7s
Precision: 64
Internal Precision: 128
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\sqrt{1 + x} \cdot \sqrt{\sqrt{x - 1}}\right) \cdot \sqrt{\sqrt{x - 1}}\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 30.6

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

  3. Applied *-un-lft-identity30.6

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

  4. Applied difference-of-squares30.6

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

  5. Applied sqrt-prod0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied associate-*r*0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019094 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))