Average Error: 31.4 → 0.1
Time: 2.2s
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 r105492 = x;
        double r105493 = r105492 * r105492;
        double r105494 = 1.0;
        double r105495 = r105493 - r105494;
        double r105496 = sqrt(r105495);
        double r105497 = r105492 + r105496;
        double r105498 = log(r105497);
        return r105498;
}
↓
double f(double x) {
        double r105499 = x;
        double r105500 = 1.0;
        double r105501 = sqrt(r105500);
        double r105502 = r105499 + r105501;
        double r105503 = sqrt(r105502);
        double r105504 = r105499 - r105501;
        double r105505 = sqrt(r105504);
        double r105506 = r105503 * r105505;
        double r105507 = r105499 + r105506;
        double r105508 = log(r105507);
        return r105508;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.4

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

  3. Applied add-sqr-sqrt31.4

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

  4. Applied difference-of-squares31.4

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

  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 2020024 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))