Average Error: 31.0 → 0.1
Time: 13.1s
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 r3004630 = x;
        double r3004631 = r3004630 * r3004630;
        double r3004632 = 1.0;
        double r3004633 = r3004631 - r3004632;
        double r3004634 = sqrt(r3004633);
        double r3004635 = r3004630 + r3004634;
        double r3004636 = log(r3004635);
        return r3004636;
}
↓
double f(double x) {
        double r3004637 = x;
        double r3004638 = 1.0;
        double r3004639 = r3004638 + r3004637;
        double r3004640 = sqrt(r3004639);
        double r3004641 = r3004637 - r3004638;
        double r3004642 = sqrt(r3004641);
        double r3004643 = r3004640 * r3004642;
        double r3004644 = r3004637 + r3004643;
        double r3004645 = log(r3004644);
        return r3004645;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 31.0

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

  3. Applied *-un-lft-identity31.0

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

  4. Applied difference-of-squares31.0

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

  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 2019165 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))