Average Error: 31.3 → 0.1
Time: 15.5s
Precision: 64
Internal Precision: 2432
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{x + 1} \cdot \sqrt{x - 1}\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 31.3

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

  3. Applied difference-of-sqr-131.3

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

  4. Applied sqrt-prod0.1

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

Runtime

Time bar (total: 15.5s)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))