Average Error: 31.1 → 0.1
Time: 32.3s
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 r9857778 = x;
        double r9857779 = r9857778 * r9857778;
        double r9857780 = 1.0;
        double r9857781 = r9857779 - r9857780;
        double r9857782 = sqrt(r9857781);
        double r9857783 = r9857778 + r9857782;
        double r9857784 = log(r9857783);
        return r9857784;
}
↓
double f(double x) {
        double r9857785 = x;
        double r9857786 = 1.0;
        double r9857787 = r9857786 + r9857785;
        double r9857788 = sqrt(r9857787);
        double r9857789 = r9857785 - r9857786;
        double r9857790 = sqrt(r9857789);
        double r9857791 = r9857788 * r9857790;
        double r9857792 = r9857785 + r9857791;
        double r9857793 = log(r9857792);
        return r9857793;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 31.1

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

  3. Applied *-un-lft-identity31.1

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

  4. Applied difference-of-squares31.1

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

  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 2019121 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))