Average Error: 31.3 → 0.0
Time: 21.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt[3]{\sqrt{1 + x} \cdot \sqrt{x - 1}} \cdot \left(\sqrt[3]{\sqrt{1 + x} \cdot \sqrt{x - 1}} \cdot \sqrt[3]{\sqrt{1 + x} \cdot \sqrt{x - 1}}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt[3]{\sqrt{1 + x} \cdot \sqrt{x - 1}} \cdot \left(\sqrt[3]{\sqrt{1 + x} \cdot \sqrt{x - 1}} \cdot \sqrt[3]{\sqrt{1 + x} \cdot \sqrt{x - 1}}\right)\right)
double f(double x) {
        double r1247023 = x;
        double r1247024 = r1247023 * r1247023;
        double r1247025 = 1.0;
        double r1247026 = r1247024 - r1247025;
        double r1247027 = sqrt(r1247026);
        double r1247028 = r1247023 + r1247027;
        double r1247029 = log(r1247028);
        return r1247029;
}
↓
double f(double x) {
        double r1247030 = x;
        double r1247031 = 1.0;
        double r1247032 = r1247031 + r1247030;
        double r1247033 = sqrt(r1247032);
        double r1247034 = r1247030 - r1247031;
        double r1247035 = sqrt(r1247034);
        double r1247036 = r1247033 * r1247035;
        double r1247037 = cbrt(r1247036);
        double r1247038 = r1247037 * r1247037;
        double r1247039 = r1247037 * r1247038;
        double r1247040 = r1247030 + r1247039;
        double r1247041 = log(r1247040);
        return r1247041;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.3

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

  3. Applied *-un-lft-identity31.3

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

  4. Applied difference-of-squares31.3

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

  5. Applied sqrt-prod0.0

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

  7. Applied add-cube-cbrt0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019151 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))