Average Error: 32.5 → 0.1
Time: 6.0s
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 r2031504 = x;
        double r2031505 = r2031504 * r2031504;
        double r2031506 = 1.0;
        double r2031507 = r2031505 - r2031506;
        double r2031508 = sqrt(r2031507);
        double r2031509 = r2031504 + r2031508;
        double r2031510 = log(r2031509);
        return r2031510;
}
↓
double f(double x) {
        double r2031511 = x;
        double r2031512 = 1.0;
        double r2031513 = sqrt(r2031512);
        double r2031514 = r2031511 - r2031513;
        double r2031515 = sqrt(r2031514);
        double r2031516 = r2031511 + r2031513;
        double r2031517 = sqrt(r2031516);
        double r2031518 = r2031515 * r2031517;
        double r2031519 = r2031511 + r2031518;
        double r2031520 = log(r2031519);
        return r2031520;
}

Error

Bits error versus x

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Results

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Derivation

  1. Initial program 32.5

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

  3. Applied add-sqr-sqrt32.5

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

  4. Applied difference-of-squares32.5

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

  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 2019171 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))