Average Error: 31.7 → 0.1
Time: 10.2s
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 r2914591 = x;
        double r2914592 = r2914591 * r2914591;
        double r2914593 = 1.0;
        double r2914594 = r2914592 - r2914593;
        double r2914595 = sqrt(r2914594);
        double r2914596 = r2914591 + r2914595;
        double r2914597 = log(r2914596);
        return r2914597;
}
↓
double f(double x) {
        double r2914598 = x;
        double r2914599 = 1.0;
        double r2914600 = sqrt(r2914599);
        double r2914601 = r2914598 - r2914600;
        double r2914602 = sqrt(r2914601);
        double r2914603 = r2914598 + r2914600;
        double r2914604 = sqrt(r2914603);
        double r2914605 = r2914602 * r2914604;
        double r2914606 = r2914598 + r2914605;
        double r2914607 = log(r2914606);
        return r2914607;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 31.7

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

  3. Applied add-sqr-sqrt31.7

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

  4. Applied difference-of-squares31.7

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

  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 2019170 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))