Average Error: 30.9 → 0.1
Time: 14.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)
double f(double x) {
        double r1969452 = x;
        double r1969453 = r1969452 * r1969452;
        double r1969454 = 1.0;
        double r1969455 = r1969453 - r1969454;
        double r1969456 = sqrt(r1969455);
        double r1969457 = r1969452 + r1969456;
        double r1969458 = log(r1969457);
        return r1969458;
}
↓
double f(double x) {
        double r1969459 = x;
        double r1969460 = 1.0;
        double r1969461 = r1969460 + r1969459;
        double r1969462 = sqrt(r1969461);
        double r1969463 = r1969459 - r1969460;
        double r1969464 = sqrt(r1969463);
        double r1969465 = r1969462 * r1969464;
        double r1969466 = r1969459 + r1969465;
        double r1969467 = log(r1969466);
        return r1969467;
}

Error

Bits error versus x

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Results

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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-prod0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019144 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))