Average Error: 32.0 → 0.1
Time: 6.1s
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 r41889 = x;
        double r41890 = r41889 * r41889;
        double r41891 = 1.0;
        double r41892 = r41890 - r41891;
        double r41893 = sqrt(r41892);
        double r41894 = r41889 + r41893;
        double r41895 = log(r41894);
        return r41895;
}
↓
double f(double x) {
        double r41896 = x;
        double r41897 = 1.0;
        double r41898 = sqrt(r41897);
        double r41899 = r41896 + r41898;
        double r41900 = sqrt(r41899);
        double r41901 = r41896 - r41898;
        double r41902 = sqrt(r41901);
        double r41903 = r41900 * r41902;
        double r41904 = r41896 + r41903;
        double r41905 = log(r41904);
        return r41905;
}

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 2019209 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))