Average Error: 31.8 → 0.0
Time: 3.0s
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 r55804 = x;
        double r55805 = r55804 * r55804;
        double r55806 = 1.0;
        double r55807 = r55805 - r55806;
        double r55808 = sqrt(r55807);
        double r55809 = r55804 + r55808;
        double r55810 = log(r55809);
        return r55810;
}
↓
double f(double x) {
        double r55811 = x;
        double r55812 = 1.0;
        double r55813 = sqrt(r55812);
        double r55814 = r55811 + r55813;
        double r55815 = sqrt(r55814);
        double r55816 = r55811 - r55813;
        double r55817 = sqrt(r55816);
        double r55818 = r55815 * r55817;
        double r55819 = r55811 + r55818;
        double r55820 = log(r55819);
        return r55820;
}

Error

Bits error versus x

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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

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

Reproduce

herbie shell --seed 2020100 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))