Average Error: 31.2 → 0.1
Time: 11.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)
double f(double x) {
        double r3461342 = x;
        double r3461343 = r3461342 * r3461342;
        double r3461344 = 1.0;
        double r3461345 = r3461343 - r3461344;
        double r3461346 = sqrt(r3461345);
        double r3461347 = r3461342 + r3461346;
        double r3461348 = log(r3461347);
        return r3461348;
}
↓
double f(double x) {
        double r3461349 = x;
        double r3461350 = 1.0;
        double r3461351 = r3461350 + r3461349;
        double r3461352 = sqrt(r3461351);
        double r3461353 = r3461349 - r3461350;
        double r3461354 = sqrt(r3461353);
        double r3461355 = r3461352 * r3461354;
        double r3461356 = r3461349 + r3461355;
        double r3461357 = log(r3461356);
        return r3461357;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.2

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

  3. Applied difference-of-sqr-131.2

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

  4. Applied sqrt-prod0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019120 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))