Average Error: 31.1 → 0.1
Time: 20.4s
Precision: 64
Internal Precision: 2368
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\sqrt{x - 1} \cdot \sqrt[3]{\sqrt{1 + x}}\right) \cdot \left(\sqrt[3]{\sqrt{1 + x}} \cdot \sqrt[3]{\sqrt{1 + x}}\right)\right)\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.1

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

  3. Applied difference-of-sqr-131.1

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

  4. Applied sqrt-prod0.1

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

  6. Applied add-cube-cbrt0.1

    \[\leadsto \log \left(x + \color{blue}{\left(\left(\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt[3]{\sqrt{x + 1}}\right) \cdot \sqrt[3]{\sqrt{x + 1}}\right)} \cdot \sqrt{x - 1}\right)\]

  7. Applied associate-*l*0.1

    \[\leadsto \log \left(x + \color{blue}{\left(\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt[3]{\sqrt{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt{x + 1}} \cdot \sqrt{x - 1}\right)}\right)\]

  8. Final simplification0.1

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

Runtime

Time bar (total: 20.4s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes0.10.10.10.0100%
herbie shell --seed 2018285 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))