Average Error: 32.6 → 0.1
Time: 19.9s
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 r66393 = x;
        double r66394 = r66393 * r66393;
        double r66395 = 1.0;
        double r66396 = r66394 - r66395;
        double r66397 = sqrt(r66396);
        double r66398 = r66393 + r66397;
        double r66399 = log(r66398);
        return r66399;
}
↓
double f(double x) {
        double r66400 = x;
        double r66401 = 1.0;
        double r66402 = sqrt(r66401);
        double r66403 = r66400 + r66402;
        double r66404 = sqrt(r66403);
        double r66405 = r66400 - r66402;
        double r66406 = sqrt(r66405);
        double r66407 = r66404 * r66406;
        double r66408 = r66400 + r66407;
        double r66409 = log(r66408);
        return r66409;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.6

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

  3. Applied add-sqr-sqrt32.6

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

  4. Applied difference-of-squares32.6

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

  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 2019326 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))