Average Error: 31.1 → 0.3
Time: 56.2s
Precision: 64
Internal Precision: 128
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\frac{\frac{-1}{4}}{x \cdot x} - \left(\frac{\frac{3}{32}}{{x}^{4}} - \left(\log 2 + \log x\right)\right)\]

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.1

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

  2. Initial simplification31.1

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

  3. Taylor expanded around inf 0.3

    \[\leadsto \color{blue}{\log 2 - \left(\log \left(\frac{1}{x}\right) + \left(\frac{3}{32} \cdot \frac{1}{{x}^{4}} + \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right)\right)}\]

  4. Simplified0.3

    \[\leadsto \color{blue}{\frac{\frac{-1}{4}}{x \cdot x} - \left(\frac{\frac{3}{32}}{{x}^{4}} - \left(\log x + \log 2\right)\right)}\]

  5. Final simplification0.3

    \[\leadsto \frac{\frac{-1}{4}}{x \cdot x} - \left(\frac{\frac{3}{32}}{{x}^{4}} - \left(\log 2 + \log x\right)\right)\]

Runtime

Time bar (total: 56.2s)Debug logProfile

herbie shell --seed 2018346 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))