Average Error: 31.9 → 0.1
Time: 9.5s
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 r82836 = x;
        double r82837 = r82836 * r82836;
        double r82838 = 1.0;
        double r82839 = r82837 - r82838;
        double r82840 = sqrt(r82839);
        double r82841 = r82836 + r82840;
        double r82842 = log(r82841);
        return r82842;
}
↓
double f(double x) {
        double r82843 = x;
        double r82844 = 1.0;
        double r82845 = sqrt(r82844);
        double r82846 = r82843 + r82845;
        double r82847 = sqrt(r82846);
        double r82848 = r82843 - r82845;
        double r82849 = sqrt(r82848);
        double r82850 = r82847 * r82849;
        double r82851 = r82843 + r82850;
        double r82852 = log(r82851);
        return r82852;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.9

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

  3. Applied add-sqr-sqrt31.9

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

  4. Applied difference-of-squares31.9

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

  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 2019351 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  :precision binary64
  (log (+ x (sqrt (- (* x x) 1)))))