Average Error: 32.2 → 0.0
Time: 4.3s
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 r68298 = x;
        double r68299 = r68298 * r68298;
        double r68300 = 1.0;
        double r68301 = r68299 - r68300;
        double r68302 = sqrt(r68301);
        double r68303 = r68298 + r68302;
        double r68304 = log(r68303);
        return r68304;
}
↓
double f(double x) {
        double r68305 = x;
        double r68306 = 1.0;
        double r68307 = sqrt(r68306);
        double r68308 = r68305 + r68307;
        double r68309 = sqrt(r68308);
        double r68310 = r68305 - r68307;
        double r68311 = sqrt(r68310);
        double r68312 = r68309 * r68311;
        double r68313 = r68305 + r68312;
        double r68314 = log(r68313);
        return r68314;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.2

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

  3. Applied add-sqr-sqrt32.2

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

  4. Applied difference-of-squares32.2

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

  5. Applied sqrt-prod0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2020047 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))