Average Error: 32.3 → 0.0
Time: 6.2s
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 r56904 = x;
        double r56905 = r56904 * r56904;
        double r56906 = 1.0;
        double r56907 = r56905 - r56906;
        double r56908 = sqrt(r56907);
        double r56909 = r56904 + r56908;
        double r56910 = log(r56909);
        return r56910;
}
↓
double f(double x) {
        double r56911 = x;
        double r56912 = 1.0;
        double r56913 = sqrt(r56912);
        double r56914 = r56911 + r56913;
        double r56915 = sqrt(r56914);
        double r56916 = r56911 - r56913;
        double r56917 = sqrt(r56916);
        double r56918 = r56915 * r56917;
        double r56919 = r56911 + r56918;
        double r56920 = log(r56919);
        return r56920;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.3

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

  3. Applied add-sqr-sqrt32.3

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

  4. Applied difference-of-squares32.3

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

  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 2019212 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))