Average Error: 31.6 → 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 r46993 = x;
        double r46994 = r46993 * r46993;
        double r46995 = 1.0;
        double r46996 = r46994 - r46995;
        double r46997 = sqrt(r46996);
        double r46998 = r46993 + r46997;
        double r46999 = log(r46998);
        return r46999;
}
↓
double f(double x) {
        double r47000 = x;
        double r47001 = 1.0;
        double r47002 = sqrt(r47001);
        double r47003 = r47000 + r47002;
        double r47004 = sqrt(r47003);
        double r47005 = r47000 - r47002;
        double r47006 = sqrt(r47005);
        double r47007 = r47004 * r47006;
        double r47008 = r47000 + r47007;
        double r47009 = log(r47008);
        return r47009;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.6

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

  3. Applied add-sqr-sqrt31.6

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

  4. Applied difference-of-squares31.6

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

  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 2019298 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))