Average Error: 32.0 → 0.1
Time: 2.4s
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 r63866 = x;
        double r63867 = r63866 * r63866;
        double r63868 = 1.0;
        double r63869 = r63867 - r63868;
        double r63870 = sqrt(r63869);
        double r63871 = r63866 + r63870;
        double r63872 = log(r63871);
        return r63872;
}
↓
double f(double x) {
        double r63873 = x;
        double r63874 = 1.0;
        double r63875 = sqrt(r63874);
        double r63876 = r63873 + r63875;
        double r63877 = sqrt(r63876);
        double r63878 = r63873 - r63875;
        double r63879 = sqrt(r63878);
        double r63880 = r63877 * r63879;
        double r63881 = r63873 + r63880;
        double r63882 = log(r63881);
        return r63882;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.0

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

  3. Applied add-sqr-sqrt32.0

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

  4. Applied difference-of-squares32.0

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

  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 2020062 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))