Average Error: 32.1 → 0.1
Time: 11.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{\sqrt{x + \sqrt{1}}} \cdot \left(\sqrt{\sqrt{x + \sqrt{1}}} \cdot \sqrt{x - \sqrt{1}}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{\sqrt{x + \sqrt{1}}} \cdot \left(\sqrt{\sqrt{x + \sqrt{1}}} \cdot \sqrt{x - \sqrt{1}}\right)\right)
double f(double x) {
        double r43876 = x;
        double r43877 = r43876 * r43876;
        double r43878 = 1.0;
        double r43879 = r43877 - r43878;
        double r43880 = sqrt(r43879);
        double r43881 = r43876 + r43880;
        double r43882 = log(r43881);
        return r43882;
}
↓
double f(double x) {
        double r43883 = x;
        double r43884 = 1.0;
        double r43885 = sqrt(r43884);
        double r43886 = r43883 + r43885;
        double r43887 = sqrt(r43886);
        double r43888 = sqrt(r43887);
        double r43889 = r43883 - r43885;
        double r43890 = sqrt(r43889);
        double r43891 = r43888 * r43890;
        double r43892 = r43888 * r43891;
        double r43893 = r43883 + r43892;
        double r43894 = log(r43893);
        return r43894;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 32.1

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

  3. Applied add-sqr-sqrt32.1

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

  4. Applied difference-of-squares32.1

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

  5. Applied sqrt-prod0.0

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

  7. Applied add-sqr-sqrt0.0

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

  8. Applied sqrt-prod0.1

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

  9. Applied associate-*l*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))