Average Error: 32.2 → 0.0
Time: 5.2s
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 r52373 = x;
        double r52374 = r52373 * r52373;
        double r52375 = 1.0;
        double r52376 = r52374 - r52375;
        double r52377 = sqrt(r52376);
        double r52378 = r52373 + r52377;
        double r52379 = log(r52378);
        return r52379;
}
↓
double f(double x) {
        double r52380 = x;
        double r52381 = 1.0;
        double r52382 = sqrt(r52381);
        double r52383 = r52380 + r52382;
        double r52384 = sqrt(r52383);
        double r52385 = r52380 - r52382;
        double r52386 = sqrt(r52385);
        double r52387 = r52384 * r52386;
        double r52388 = r52380 + r52387;
        double r52389 = log(r52388);
        return r52389;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.2

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