Average Error: 32.2 → 0.0
Time: 4.0s
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 r52836 = x;
        double r52837 = r52836 * r52836;
        double r52838 = 1.0;
        double r52839 = r52837 - r52838;
        double r52840 = sqrt(r52839);
        double r52841 = r52836 + r52840;
        double r52842 = log(r52841);
        return r52842;
}
↓
double f(double x) {
        double r52843 = x;
        double r52844 = 1.0;
        double r52845 = sqrt(r52844);
        double r52846 = r52843 + r52845;
        double r52847 = sqrt(r52846);
        double r52848 = r52843 - r52845;
        double r52849 = sqrt(r52848);
        double r52850 = r52847 * r52849;
        double r52851 = r52843 + r52850;
        double r52852 = log(r52851);
        return r52852;
}

Error

Bits error versus x

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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)))))