Average Error: 31.4 → 0.1
Time: 9.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} + (2 \cdot x + \left(\frac{\frac{-1}{2}}{x}\right))_*\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\frac{\frac{\frac{-1}{8}}{x}}{x \cdot x} + (2 \cdot x + \left(\frac{\frac{-1}{2}}{x}\right))_*\right)
double f(double x) {
        double r11646815 = x;
        double r11646816 = r11646815 * r11646815;
        double r11646817 = 1.0;
        double r11646818 = r11646816 - r11646817;
        double r11646819 = sqrt(r11646818);
        double r11646820 = r11646815 + r11646819;
        double r11646821 = log(r11646820);
        return r11646821;
}
↓
double f(double x) {
        double r11646822 = -0.125;
        double r11646823 = x;
        double r11646824 = r11646822 / r11646823;
        double r11646825 = r11646823 * r11646823;
        double r11646826 = r11646824 / r11646825;
        double r11646827 = 2.0;
        double r11646828 = -0.5;
        double r11646829 = r11646828 / r11646823;
        double r11646830 = fma(r11646827, r11646823, r11646829);
        double r11646831 = r11646826 + r11646830;
        double r11646832 = log(r11646831);
        return r11646832;
}

Error

Bits error versus x

Derivation

  1. Initial program 31.4

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

  2. Simplified31.4

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

  3. Taylor expanded around inf 0.1

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

  4. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))