Average Error: 32.0 → 0.0
Time: 8.4s
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 r48920 = x;
        double r48921 = r48920 * r48920;
        double r48922 = 1.0;
        double r48923 = r48921 - r48922;
        double r48924 = sqrt(r48923);
        double r48925 = r48920 + r48924;
        double r48926 = log(r48925);
        return r48926;
}
↓
double f(double x) {
        double r48927 = x;
        double r48928 = 1.0;
        double r48929 = sqrt(r48928);
        double r48930 = r48927 + r48929;
        double r48931 = sqrt(r48930);
        double r48932 = r48927 - r48929;
        double r48933 = sqrt(r48932);
        double r48934 = r48931 * r48933;
        double r48935 = r48927 + r48934;
        double r48936 = log(r48935);
        return r48936;
}

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-prod0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))