Average Error: 31.4 → 0.1
Time: 17.9s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{x - 1} \cdot \left(\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{x - 1} \cdot \left(\sqrt{\sqrt{1 + x}} \cdot \sqrt{\sqrt{1 + x}}\right)\right)
double f(double x) {
        double r2091469 = x;
        double r2091470 = r2091469 * r2091469;
        double r2091471 = 1.0;
        double r2091472 = r2091470 - r2091471;
        double r2091473 = sqrt(r2091472);
        double r2091474 = r2091469 + r2091473;
        double r2091475 = log(r2091474);
        return r2091475;
}
↓
double f(double x) {
        double r2091476 = x;
        double r2091477 = 1.0;
        double r2091478 = r2091476 - r2091477;
        double r2091479 = sqrt(r2091478);
        double r2091480 = r2091477 + r2091476;
        double r2091481 = sqrt(r2091480);
        double r2091482 = sqrt(r2091481);
        double r2091483 = r2091482 * r2091482;
        double r2091484 = r2091479 * r2091483;
        double r2091485 = r2091476 + r2091484;
        double r2091486 = log(r2091485);
        return r2091486;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 31.4

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

  3. Applied *-un-lft-identity31.4

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

  4. Applied difference-of-squares31.4

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

  5. Applied sqrt-prod0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied sqrt-prod0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2019163 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))