Average Error: 32.0 → 0.1
Time: 9.6s
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 r3368373 = x;
        double r3368374 = r3368373 * r3368373;
        double r3368375 = 1.0;
        double r3368376 = r3368374 - r3368375;
        double r3368377 = sqrt(r3368376);
        double r3368378 = r3368373 + r3368377;
        double r3368379 = log(r3368378);
        return r3368379;
}
↓
double f(double x) {
        double r3368380 = x;
        double r3368381 = 1.0;
        double r3368382 = sqrt(r3368381);
        double r3368383 = r3368380 + r3368382;
        double r3368384 = sqrt(r3368383);
        double r3368385 = r3368380 - r3368382;
        double r3368386 = sqrt(r3368385);
        double r3368387 = r3368384 * r3368386;
        double r3368388 = r3368380 + r3368387;
        double r3368389 = log(r3368388);
        return r3368389;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.0

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

  3. Applied add-sqr-sqrt32.0

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

  4. Applied difference-of-squares32.0

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

  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 2019174 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))