Average Error: 30.9 → 0.2
Time: 11.4s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(x + \left(\left(x - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(x + \left(\left(x - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x}\right)\right)
double f(double x) {
        double r1113936 = x;
        double r1113937 = r1113936 * r1113936;
        double r1113938 = 1.0;
        double r1113939 = r1113937 - r1113938;
        double r1113940 = sqrt(r1113939);
        double r1113941 = r1113936 + r1113940;
        double r1113942 = log(r1113941);
        return r1113942;
}
↓
double f(double x) {
        double r1113943 = x;
        double r1113944 = 0.5;
        double r1113945 = r1113944 / r1113943;
        double r1113946 = r1113943 - r1113945;
        double r1113947 = 0.125;
        double r1113948 = r1113943 * r1113943;
        double r1113949 = r1113948 * r1113943;
        double r1113950 = r1113947 / r1113949;
        double r1113951 = r1113946 - r1113950;
        double r1113952 = r1113943 + r1113951;
        double r1113953 = log(r1113952);
        return r1113953;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 30.9

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

  2. Simplified30.9

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

  3. Taylor expanded around inf 0.2

    \[\leadsto \log \left(x + \color{blue}{\left(x - \left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\right)\]

  4. Simplified0.2

    \[\leadsto \log \left(x + \color{blue}{\left(\left(x - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)}\right)\]

  5. Final simplification0.2

    \[\leadsto \log \left(x + \left(\left(x - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{8}}{\left(x \cdot x\right) \cdot x}\right)\right)\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))