Average Error: 31.4 → 0.1
Time: 2.3s
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 r65587 = x;
        double r65588 = r65587 * r65587;
        double r65589 = 1.0;
        double r65590 = r65588 - r65589;
        double r65591 = sqrt(r65590);
        double r65592 = r65587 + r65591;
        double r65593 = log(r65592);
        return r65593;
}
↓
double f(double x) {
        double r65594 = x;
        double r65595 = 1.0;
        double r65596 = sqrt(r65595);
        double r65597 = r65594 + r65596;
        double r65598 = sqrt(r65597);
        double r65599 = r65594 - r65596;
        double r65600 = sqrt(r65599);
        double r65601 = r65598 * r65600;
        double r65602 = r65594 + r65601;
        double r65603 = log(r65602);
        return r65603;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.4

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

  3. Applied add-sqr-sqrt31.4

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

  4. Applied difference-of-squares31.4

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

  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 2019356 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))