Average Error: 31.8 → 0.0
Time: 15.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\sqrt{x - \sqrt{1}} \cdot \sqrt{\sqrt{1} + x} + x\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\sqrt{x - \sqrt{1}} \cdot \sqrt{\sqrt{1} + x} + x\right)
double f(double x) {
        double r1621636 = x;
        double r1621637 = r1621636 * r1621636;
        double r1621638 = 1.0;
        double r1621639 = r1621637 - r1621638;
        double r1621640 = sqrt(r1621639);
        double r1621641 = r1621636 + r1621640;
        double r1621642 = log(r1621641);
        return r1621642;
}
↓
double f(double x) {
        double r1621643 = x;
        double r1621644 = 1.0;
        double r1621645 = sqrt(r1621644);
        double r1621646 = r1621643 - r1621645;
        double r1621647 = sqrt(r1621646);
        double r1621648 = r1621645 + r1621643;
        double r1621649 = sqrt(r1621648);
        double r1621650 = r1621647 * r1621649;
        double r1621651 = r1621650 + r1621643;
        double r1621652 = log(r1621651);
        return r1621652;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.8

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

  3. Applied add-sqr-sqrt31.8

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

  4. Applied difference-of-squares31.8

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

  5. Applied sqrt-prod0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1.0)))))