Average Error: 31.9 → 0.1
Time: 8.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}\right)
double f(double x) {
        double r86338 = x;
        double r86339 = r86338 * r86338;
        double r86340 = 1.0;
        double r86341 = r86339 - r86340;
        double r86342 = sqrt(r86341);
        double r86343 = r86338 + r86342;
        double r86344 = log(r86343);
        return r86344;
}
↓
double f(double x) {
        double r86345 = x;
        double r86346 = 1.0;
        double r86347 = sqrt(r86346);
        double r86348 = r86345 + r86347;
        double r86349 = sqrt(r86348);
        double r86350 = r86345 - r86347;
        double r86351 = sqrt(r86350);
        double r86352 = r86349 * r86351;
        double r86353 = r86345 + r86352;
        double r86354 = log(r86353);
        return r86354;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.9

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

  3. Applied add-sqr-sqrt31.9

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

  4. Applied difference-of-squares31.9

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

  5. Applied sqrt-prod0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019351 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))