Average Error: 32.1 → 0.1
Time: 7.7s
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 r67688 = x;
        double r67689 = r67688 * r67688;
        double r67690 = 1.0;
        double r67691 = r67689 - r67690;
        double r67692 = sqrt(r67691);
        double r67693 = r67688 + r67692;
        double r67694 = log(r67693);
        return r67694;
}
↓
double f(double x) {
        double r67695 = x;
        double r67696 = 1.0;
        double r67697 = sqrt(r67696);
        double r67698 = r67695 + r67697;
        double r67699 = sqrt(r67698);
        double r67700 = r67695 - r67697;
        double r67701 = sqrt(r67700);
        double r67702 = r67699 * r67701;
        double r67703 = r67695 + r67702;
        double r67704 = log(r67703);
        return r67704;
}

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 2019208 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))