Average Error: 31.5 → 0.1
Time: 12.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)\]
double f(double x) {
        double r5316906 = x;
        double r5316907 = r5316906 * r5316906;
        double r5316908 = 1.0;
        double r5316909 = r5316907 - r5316908;
        double r5316910 = sqrt(r5316909);
        double r5316911 = r5316906 + r5316910;
        double r5316912 = log(r5316911);
        return r5316912;
}
↓
double f(double x) {
        double r5316913 = x;
        double r5316914 = 1.0;
        double r5316915 = r5316914 + r5316913;
        double r5316916 = sqrt(r5316915);
        double r5316917 = r5316913 - r5316914;
        double r5316918 = sqrt(r5316917);
        double r5316919 = r5316916 * r5316918;
        double r5316920 = r5316913 + r5316919;
        double r5316921 = log(r5316920);
        return r5316921;
}


\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \sqrt{1 + x} \cdot \sqrt{x - 1}\right)

Error

Bits error versus x

Derivation

  1. Initial program 31.5

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

  3. Applied difference-of-sqr-131.5

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

  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 2019102 
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))