Average Error: 30.9 → 0.1
Time: 28.5s
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 r3952128 = x;
        double r3952129 = r3952128 * r3952128;
        double r3952130 = 1.0;
        double r3952131 = r3952129 - r3952130;
        double r3952132 = sqrt(r3952131);
        double r3952133 = r3952128 + r3952132;
        double r3952134 = log(r3952133);
        return r3952134;
}
↓
double f(double x) {
        double r3952135 = x;
        double r3952136 = 1.0;
        double r3952137 = r3952136 + r3952135;
        double r3952138 = sqrt(r3952137);
        double r3952139 = r3952135 - r3952136;
        double r3952140 = sqrt(r3952139);
        double r3952141 = r3952138 * r3952140;
        double r3952142 = cbrt(r3952141);
        double r3952143 = r3952142 * r3952142;
        double r3952144 = r3952142 * r3952143;
        double r3952145 = r3952135 + r3952144;
        double r3952146 = log(r3952145);
        return r3952146;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.9

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

  3. Applied *-un-lft-identity30.9

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

  4. Applied difference-of-squares30.9

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

  5. Applied sqrt-prod0.1

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

  7. Applied add-cube-cbrt0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019112 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))