Average Error: 31.6 → 0.1
Time: 15.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\sqrt{x + \sqrt{1}} \cdot \left(\sqrt[3]{\sqrt{x - \sqrt{1}}} \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)\right) \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\sqrt{x + \sqrt{1}} \cdot \left(\sqrt[3]{\sqrt{x - \sqrt{1}}} \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)\right) \cdot \sqrt[3]{\sqrt{x - \sqrt{1}}}\right)
double f(double x) {
        double r3609013 = x;
        double r3609014 = r3609013 * r3609013;
        double r3609015 = 1.0;
        double r3609016 = r3609014 - r3609015;
        double r3609017 = sqrt(r3609016);
        double r3609018 = r3609013 + r3609017;
        double r3609019 = log(r3609018);
        return r3609019;
}
↓
double f(double x) {
        double r3609020 = x;
        double r3609021 = 1.0;
        double r3609022 = sqrt(r3609021);
        double r3609023 = r3609020 + r3609022;
        double r3609024 = sqrt(r3609023);
        double r3609025 = r3609020 - r3609022;
        double r3609026 = sqrt(r3609025);
        double r3609027 = cbrt(r3609026);
        double r3609028 = r3609027 * r3609027;
        double r3609029 = r3609024 * r3609028;
        double r3609030 = r3609029 * r3609027;
        double r3609031 = r3609020 + r3609030;
        double r3609032 = log(r3609031);
        return r3609032;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.6

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

  3. Applied add-sqr-sqrt31.6

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

  4. Applied difference-of-squares31.6

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

  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-cube-cbrt0.1

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

  8. Applied associate-*r*0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))