Average Error: 32.2 → 0.0
Time: 5.8s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}\right)
double f(double x) {
        double r39890 = x;
        double r39891 = r39890 * r39890;
        double r39892 = 1.0;
        double r39893 = r39891 - r39892;
        double r39894 = sqrt(r39893);
        double r39895 = r39890 + r39894;
        double r39896 = log(r39895);
        return r39896;
}
↓
double f(double x) {
        double r39897 = x;
        double r39898 = 1.0;
        double r39899 = sqrt(r39898);
        double r39900 = r39897 + r39899;
        double r39901 = sqrt(r39900);
        double r39902 = r39897 - r39899;
        double r39903 = sqrt(r39902);
        double r39904 = r39901 * r39903;
        double r39905 = r39897 + r39904;
        double r39906 = log(r39905);
        return r39906;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.2

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

  3. Applied add-sqr-sqrt32.2

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

  4. Applied difference-of-squares32.2

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

  5. Applied sqrt-prod0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))