Average Error: 32.0 → 0.1
Time: 6.7s
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 r2956406 = x;
        double r2956407 = r2956406 * r2956406;
        double r2956408 = 1.0;
        double r2956409 = r2956407 - r2956408;
        double r2956410 = sqrt(r2956409);
        double r2956411 = r2956406 + r2956410;
        double r2956412 = log(r2956411);
        return r2956412;
}
↓
double f(double x) {
        double r2956413 = x;
        double r2956414 = 1.0;
        double r2956415 = sqrt(r2956414);
        double r2956416 = r2956413 + r2956415;
        double r2956417 = sqrt(r2956416);
        double r2956418 = r2956413 - r2956415;
        double r2956419 = sqrt(r2956418);
        double r2956420 = r2956417 * r2956419;
        double r2956421 = r2956413 + r2956420;
        double r2956422 = log(r2956421);
        return r2956422;
}

Error

Bits error versus x

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.0

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

  3. Applied add-sqr-sqrt32.0

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

  4. Applied difference-of-squares32.0

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

  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 2019174 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))