Average Error: 31.9 → 0.1
Time: 15.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{x + \sqrt{1}} \cdot \left(\left|\sqrt[3]{x - \sqrt{1}}\right| \cdot \sqrt{\sqrt[3]{x - \sqrt{1}}}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{x + \sqrt{1}} \cdot \left(\left|\sqrt[3]{x - \sqrt{1}}\right| \cdot \sqrt{\sqrt[3]{x - \sqrt{1}}}\right)\right)
double f(double x) {
        double r71941 = x;
        double r71942 = r71941 * r71941;
        double r71943 = 1.0;
        double r71944 = r71942 - r71943;
        double r71945 = sqrt(r71944);
        double r71946 = r71941 + r71945;
        double r71947 = log(r71946);
        return r71947;
}
↓
double f(double x) {
        double r71948 = x;
        double r71949 = 1.0;
        double r71950 = sqrt(r71949);
        double r71951 = r71948 + r71950;
        double r71952 = sqrt(r71951);
        double r71953 = r71948 - r71950;
        double r71954 = cbrt(r71953);
        double r71955 = fabs(r71954);
        double r71956 = sqrt(r71954);
        double r71957 = r71955 * r71956;
        double r71958 = r71952 * r71957;
        double r71959 = r71948 + r71958;
        double r71960 = log(r71959);
        return r71960;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.9

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

  3. Applied add-sqr-sqrt31.9

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

  4. Applied difference-of-squares31.9

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

  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-cube-cbrt0.1

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

  8. Applied sqrt-prod0.1

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

  9. Simplified0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2019294 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))