Average Error: 32.2 → 0.0
Time: 3.8s
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 r80445 = x;
        double r80446 = r80445 * r80445;
        double r80447 = 1.0;
        double r80448 = r80446 - r80447;
        double r80449 = sqrt(r80448);
        double r80450 = r80445 + r80449;
        double r80451 = log(r80450);
        return r80451;
}
↓
double f(double x) {
        double r80452 = x;
        double r80453 = 1.0;
        double r80454 = sqrt(r80453);
        double r80455 = r80452 + r80454;
        double r80456 = sqrt(r80455);
        double r80457 = r80452 - r80454;
        double r80458 = sqrt(r80457);
        double r80459 = r80456 * r80458;
        double r80460 = r80452 + r80459;
        double r80461 = log(r80460);
        return r80461;
}

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)))))