Average Error: 31.3 → 0.1
Time: 10.0s
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 r3663328 = x;
        double r3663329 = r3663328 * r3663328;
        double r3663330 = 1.0;
        double r3663331 = r3663329 - r3663330;
        double r3663332 = sqrt(r3663331);
        double r3663333 = r3663328 + r3663332;
        double r3663334 = log(r3663333);
        return r3663334;
}
↓
double f(double x) {
        double r3663335 = x;
        double r3663336 = 1.0;
        double r3663337 = sqrt(r3663336);
        double r3663338 = r3663335 + r3663337;
        double r3663339 = sqrt(r3663338);
        double r3663340 = r3663335 - r3663337;
        double r3663341 = sqrt(r3663340);
        double r3663342 = r3663339 * r3663341;
        double r3663343 = r3663335 + r3663342;
        double r3663344 = log(r3663343);
        return r3663344;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.3

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

  3. Applied add-sqr-sqrt31.3

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

  4. Applied difference-of-squares31.3

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

  5. Applied sqrt-prod0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2019173 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))