Average Error: 31.2 → 0.0
Time: 14.3s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)
double f(double x) {
        double r1779901 = x;
        double r1779902 = r1779901 * r1779901;
        double r1779903 = 1.0;
        double r1779904 = r1779902 - r1779903;
        double r1779905 = sqrt(r1779904);
        double r1779906 = r1779901 + r1779905;
        double r1779907 = log(r1779906);
        return r1779907;
}
↓
double f(double x) {
        double r1779908 = x;
        double r1779909 = 1.0;
        double r1779910 = r1779909 + r1779908;
        double r1779911 = sqrt(r1779910);
        double r1779912 = r1779908 - r1779909;
        double r1779913 = sqrt(r1779912);
        double r1779914 = r1779911 * r1779913;
        double r1779915 = r1779908 + r1779914;
        double r1779916 = log(r1779915);
        return r1779916;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.2

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

  3. Applied difference-of-sqr-131.2

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

  4. Applied sqrt-prod0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019149 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))