Average Error: 31.4 → 0.1
Time: 2.3s
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 r35285 = x;
        double r35286 = r35285 * r35285;
        double r35287 = 1.0;
        double r35288 = r35286 - r35287;
        double r35289 = sqrt(r35288);
        double r35290 = r35285 + r35289;
        double r35291 = log(r35290);
        return r35291;
}
↓
double f(double x) {
        double r35292 = x;
        double r35293 = 1.0;
        double r35294 = sqrt(r35293);
        double r35295 = r35292 + r35294;
        double r35296 = sqrt(r35295);
        double r35297 = r35292 - r35294;
        double r35298 = sqrt(r35297);
        double r35299 = r35296 * r35298;
        double r35300 = r35292 + r35299;
        double r35301 = log(r35300);
        return r35301;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.4

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

  3. Applied add-sqr-sqrt31.4

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

  4. Applied difference-of-squares31.4

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

  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 2019356 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))