Average Error: 31.3 → 0.1
Time: 13.3s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)
double f(double x) {
        double r2050718 = x;
        double r2050719 = r2050718 * r2050718;
        double r2050720 = 1.0;
        double r2050721 = r2050719 - r2050720;
        double r2050722 = sqrt(r2050721);
        double r2050723 = r2050718 + r2050722;
        double r2050724 = log(r2050723);
        return r2050724;
}
↓
double f(double x) {
        double r2050725 = x;
        double r2050726 = 1.0;
        double r2050727 = r2050726 + r2050725;
        double r2050728 = sqrt(r2050727);
        double r2050729 = r2050725 - r2050726;
        double r2050730 = sqrt(r2050729);
        double r2050731 = r2050728 * r2050730;
        double r2050732 = r2050725 + r2050731;
        double r2050733 = log(r2050732);
        return r2050733;
}

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 31.3

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

  3. Applied difference-of-sqr-131.3

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

  4. Applied sqrt-prod0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019139 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))