Average Error: 32.0 → 0.1
Time: 2.4s
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 r61401 = x;
        double r61402 = r61401 * r61401;
        double r61403 = 1.0;
        double r61404 = r61402 - r61403;
        double r61405 = sqrt(r61404);
        double r61406 = r61401 + r61405;
        double r61407 = log(r61406);
        return r61407;
}
↓
double f(double x) {
        double r61408 = x;
        double r61409 = 1.0;
        double r61410 = sqrt(r61409);
        double r61411 = r61408 + r61410;
        double r61412 = sqrt(r61411);
        double r61413 = r61408 - r61410;
        double r61414 = sqrt(r61413);
        double r61415 = r61412 * r61414;
        double r61416 = r61408 + r61415;
        double r61417 = log(r61416);
        return r61417;
}

Error

Bits error versus x

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Your Program's Arguments

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 2020002 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))