Average Error: 32.5 → 0.1
Time: 2.5s
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 r70768 = x;
        double r70769 = r70768 * r70768;
        double r70770 = 1.0;
        double r70771 = r70769 - r70770;
        double r70772 = sqrt(r70771);
        double r70773 = r70768 + r70772;
        double r70774 = log(r70773);
        return r70774;
}
↓
double f(double x) {
        double r70775 = x;
        double r70776 = 1.0;
        double r70777 = sqrt(r70776);
        double r70778 = r70775 + r70777;
        double r70779 = sqrt(r70778);
        double r70780 = r70775 - r70777;
        double r70781 = sqrt(r70780);
        double r70782 = r70779 * r70781;
        double r70783 = r70775 + r70782;
        double r70784 = log(r70783);
        return r70784;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.5

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

  3. Applied add-sqr-sqrt32.5

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

  4. Applied difference-of-squares32.5

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

  5. Applied sqrt-prod0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))