Average Error: 32.8 → 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 r73731 = x;
        double r73732 = r73731 * r73731;
        double r73733 = 1.0;
        double r73734 = r73732 - r73733;
        double r73735 = sqrt(r73734);
        double r73736 = r73731 + r73735;
        double r73737 = log(r73736);
        return r73737;
}
↓
double f(double x) {
        double r73738 = x;
        double r73739 = 1.0;
        double r73740 = sqrt(r73739);
        double r73741 = r73738 + r73740;
        double r73742 = sqrt(r73741);
        double r73743 = r73738 - r73740;
        double r73744 = sqrt(r73743);
        double r73745 = r73742 * r73744;
        double r73746 = r73738 + r73745;
        double r73747 = log(r73746);
        return r73747;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.8

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

  3. Applied add-sqr-sqrt32.8

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

  4. Applied difference-of-squares32.8

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

  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 2020020 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))