Average Error: 32.6 → 0.0
Time: 2.6s
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 r63786 = x;
        double r63787 = r63786 * r63786;
        double r63788 = 1.0;
        double r63789 = r63787 - r63788;
        double r63790 = sqrt(r63789);
        double r63791 = r63786 + r63790;
        double r63792 = log(r63791);
        return r63792;
}
↓
double f(double x) {
        double r63793 = x;
        double r63794 = 1.0;
        double r63795 = sqrt(r63794);
        double r63796 = r63793 + r63795;
        double r63797 = sqrt(r63796);
        double r63798 = r63793 - r63795;
        double r63799 = sqrt(r63798);
        double r63800 = r63797 * r63799;
        double r63801 = r63793 + r63800;
        double r63802 = log(r63801);
        return r63802;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 32.6

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

  3. Applied add-sqr-sqrt32.6

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

  4. Applied difference-of-squares32.6

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

  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

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

Reproduce

herbie shell --seed 2019354 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))