Average Error: 30.4 → 0.1
Time: 51.1s
Precision: 64
Internal Precision: 2368
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\sqrt{x + \sqrt{1 + x} \cdot \sqrt{x - 1}} \cdot \sqrt{x + \sqrt{1 + x} \cdot \sqrt{x - 1}}\right)\]

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 30.4

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

  2. Initial simplification30.4

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

  4. Applied difference-of-sqr-130.4

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

  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

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

  8. Final simplification0.1

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

Runtime

Time bar (total: 51.1s)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))