Average Error: 32.4 → 0.0
Time: 5.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\sqrt{x + \sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}} \cdot \sqrt{x + \sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\sqrt{x + \sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}} \cdot \sqrt{x + \sqrt{x + \sqrt{1}} \cdot \sqrt{x - \sqrt{1}}}\right)
double f(double x) {
        double r57890 = x;
        double r57891 = r57890 * r57890;
        double r57892 = 1.0;
        double r57893 = r57891 - r57892;
        double r57894 = sqrt(r57893);
        double r57895 = r57890 + r57894;
        double r57896 = log(r57895);
        return r57896;
}
↓
double f(double x) {
        double r57897 = x;
        double r57898 = 1.0;
        double r57899 = sqrt(r57898);
        double r57900 = r57897 + r57899;
        double r57901 = sqrt(r57900);
        double r57902 = r57897 - r57899;
        double r57903 = sqrt(r57902);
        double r57904 = r57901 * r57903;
        double r57905 = r57897 + r57904;
        double r57906 = sqrt(r57905);
        double r57907 = r57906 * r57906;
        double r57908 = log(r57907);
        return r57908;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.4

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

  3. Applied add-sqr-sqrt32.4

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

  4. Applied difference-of-squares32.4

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

  5. Applied sqrt-prod0.0

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

  7. Applied add-sqr-sqrt0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))