Average Error: 32.1 → 0.1
Time: 4.3s
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 r91514 = x;
        double r91515 = r91514 * r91514;
        double r91516 = 1.0;
        double r91517 = r91515 - r91516;
        double r91518 = sqrt(r91517);
        double r91519 = r91514 + r91518;
        double r91520 = log(r91519);
        return r91520;
}
↓
double f(double x) {
        double r91521 = x;
        double r91522 = 1.0;
        double r91523 = sqrt(r91522);
        double r91524 = r91521 + r91523;
        double r91525 = sqrt(r91524);
        double r91526 = r91521 - r91523;
        double r91527 = sqrt(r91526);
        double r91528 = r91525 * r91527;
        double r91529 = r91521 + r91528;
        double r91530 = log(r91529);
        return r91530;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.1

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

  3. Applied add-sqr-sqrt32.1

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

  4. Applied difference-of-squares32.1

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

  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 2020035 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))