Average Error: 31.1 → 0.4
Time: 1.5m
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[(\frac{1}{4} \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\log 2 - \left(\frac{\frac{3}{32}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \log x\right)\right))_*\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
(\frac{1}{4} \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\log 2 - \left(\frac{\frac{3}{32}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \log x\right)\right))_*
double f(double x) {
        double r8071890 = x;
        double r8071891 = r8071890 * r8071890;
        double r8071892 = 1.0;
        double r8071893 = r8071891 - r8071892;
        double r8071894 = sqrt(r8071893);
        double r8071895 = r8071890 + r8071894;
        double r8071896 = log(r8071895);
        return r8071896;
}
↓
double f(double x) {
        double r8071897 = 0.25;
        double r8071898 = -1.0;
        double r8071899 = x;
        double r8071900 = r8071899 * r8071899;
        double r8071901 = r8071898 / r8071900;
        double r8071902 = 2.0;
        double r8071903 = log(r8071902);
        double r8071904 = 0.09375;
        double r8071905 = r8071900 * r8071900;
        double r8071906 = r8071904 / r8071905;
        double r8071907 = log(r8071899);
        double r8071908 = r8071906 - r8071907;
        double r8071909 = r8071903 - r8071908;
        double r8071910 = fma(r8071897, r8071901, r8071909);
        return r8071910;
}

Error

Bits error versus x

Derivation

  1. Initial program 31.1

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

  2. Simplified31.1

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

  3. Taylor expanded around inf 0.4

    \[\leadsto \color{blue}{\log 2 - \left(\log \left(\frac{1}{x}\right) + \left(\frac{3}{32} \cdot \frac{1}{{x}^{4}} + \frac{1}{4} \cdot \frac{1}{{x}^{2}}\right)\right)}\]

  4. Simplified0.4

    \[\leadsto \color{blue}{(\frac{1}{4} \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\log 2 - \left(\frac{\frac{3}{32}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \log x\right)\right))_*}\]

  5. Final simplification0.4

    \[\leadsto (\frac{1}{4} \cdot \left(\frac{-1}{x \cdot x}\right) + \left(\log 2 - \left(\frac{\frac{3}{32}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} - \log x\right)\right))_*\]

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))