Average Error: 32.5 → 0.1
Time: 2.5s
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 r46494 = x;
        double r46495 = r46494 * r46494;
        double r46496 = 1.0;
        double r46497 = r46495 - r46496;
        double r46498 = sqrt(r46497);
        double r46499 = r46494 + r46498;
        double r46500 = log(r46499);
        return r46500;
}
↓
double f(double x) {
        double r46501 = x;
        double r46502 = 1.0;
        double r46503 = sqrt(r46502);
        double r46504 = r46501 + r46503;
        double r46505 = sqrt(r46504);
        double r46506 = r46501 - r46503;
        double r46507 = sqrt(r46506);
        double r46508 = r46505 * r46507;
        double r46509 = r46501 + r46508;
        double r46510 = log(r46509);
        return r46510;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.5

    \[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
  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 2019353 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))