Average Error: 32.3 → 0.1
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 r69330 = x;
        double r69331 = r69330 * r69330;
        double r69332 = 1.0;
        double r69333 = r69331 - r69332;
        double r69334 = sqrt(r69333);
        double r69335 = r69330 + r69334;
        double r69336 = log(r69335);
        return r69336;
}
↓
double f(double x) {
        double r69337 = x;
        double r69338 = 1.0;
        double r69339 = sqrt(r69338);
        double r69340 = r69337 - r69339;
        double r69341 = sqrt(r69340);
        double r69342 = r69337 + r69339;
        double r69343 = sqrt(r69342);
        double r69344 = r69341 * r69343;
        double r69345 = r69337 + r69344;
        double r69346 = log(r69345);
        return r69346;
}

Error

Bits error versus x

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