Average Error: 30.9 → 0.1
Time: 34.3s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\sqrt{x + \sqrt{1 + x} \cdot \sqrt{x - 1}}\right) + \log \left(\sqrt{x + \sqrt{1 + x} \cdot \sqrt{x - 1}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\sqrt{x + \sqrt{1 + x} \cdot \sqrt{x - 1}}\right) + \log \left(\sqrt{x + \sqrt{1 + x} \cdot \sqrt{x - 1}}\right)
double f(double x) {
        double r5064560 = x;
        double r5064561 = r5064560 * r5064560;
        double r5064562 = 1.0;
        double r5064563 = r5064561 - r5064562;
        double r5064564 = sqrt(r5064563);
        double r5064565 = r5064560 + r5064564;
        double r5064566 = log(r5064565);
        return r5064566;
}
↓
double f(double x) {
        double r5064567 = x;
        double r5064568 = 1.0;
        double r5064569 = r5064568 + r5064567;
        double r5064570 = sqrt(r5064569);
        double r5064571 = r5064567 - r5064568;
        double r5064572 = sqrt(r5064571);
        double r5064573 = r5064570 * r5064572;
        double r5064574 = r5064567 + r5064573;
        double r5064575 = sqrt(r5064574);
        double r5064576 = log(r5064575);
        double r5064577 = r5064576 + r5064576;
        return r5064577;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.9

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

  3. Applied *-un-lft-identity30.9

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

  4. Applied difference-of-squares30.9

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

  5. Applied sqrt-prod0.0

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied log-prod0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019128 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))