Average Error: 32.3 → 0.1
Time: 6.3s
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 r51989 = x;
        double r51990 = r51989 * r51989;
        double r51991 = 1.0;
        double r51992 = r51990 - r51991;
        double r51993 = sqrt(r51992);
        double r51994 = r51989 + r51993;
        double r51995 = log(r51994);
        return r51995;
}
↓
double f(double x) {
        double r51996 = x;
        double r51997 = 1.0;
        double r51998 = sqrt(r51997);
        double r51999 = r51996 - r51998;
        double r52000 = sqrt(r51999);
        double r52001 = r51996 + r51998;
        double r52002 = sqrt(r52001);
        double r52003 = r52000 * r52002;
        double r52004 = r51996 + r52003;
        double r52005 = log(r52004);
        return r52005;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Derivation

  1. Initial program 32.3

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

  3. Applied add-sqr-sqrt32.3

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

  4. Applied difference-of-squares32.3

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

  5. Applied sqrt-prod0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))