Average Error: 32.4 → 0.1
Time: 4.9s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\sqrt{x + \sqrt{1}} \cdot \sqrt{\sqrt{x - \sqrt{1}}}\right) \cdot \sqrt{\sqrt{x - \sqrt{1}}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\sqrt{x + \sqrt{1}} \cdot \sqrt{\sqrt{x - \sqrt{1}}}\right) \cdot \sqrt{\sqrt{x - \sqrt{1}}}\right)
double f(double x) {
        double r77886 = x;
        double r77887 = r77886 * r77886;
        double r77888 = 1.0;
        double r77889 = r77887 - r77888;
        double r77890 = sqrt(r77889);
        double r77891 = r77886 + r77890;
        double r77892 = log(r77891);
        return r77892;
}
↓
double f(double x) {
        double r77893 = x;
        double r77894 = 1.0;
        double r77895 = sqrt(r77894);
        double r77896 = r77893 + r77895;
        double r77897 = sqrt(r77896);
        double r77898 = r77893 - r77895;
        double r77899 = sqrt(r77898);
        double r77900 = sqrt(r77899);
        double r77901 = r77897 * r77900;
        double r77902 = r77901 * r77900;
        double r77903 = r77893 + r77902;
        double r77904 = log(r77903);
        return r77904;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.4

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

  3. Applied add-sqr-sqrt32.4

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

  4. Applied difference-of-squares32.4

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

  5. Applied sqrt-prod0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied sqrt-prod0.1

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

  9. Applied associate-*r*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020060 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))