Average Error: 30.6 → 0.3
Time: 32.3s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\mathsf{fma}\left(2, x, \frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x}\right) - \frac{\frac{1}{2}}{x}\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\mathsf{fma}\left(2, x, \frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x}\right) - \frac{\frac{1}{2}}{x}\right)
double f(double x) {
        double r976022 = x;
        double r976023 = r976022 * r976022;
        double r976024 = 1.0;
        double r976025 = r976023 - r976024;
        double r976026 = sqrt(r976025);
        double r976027 = r976022 + r976026;
        double r976028 = log(r976027);
        return r976028;
}
↓
double f(double x) {
        double r976029 = 2.0;
        double r976030 = x;
        double r976031 = -0.125;
        double r976032 = r976030 * r976030;
        double r976033 = r976032 * r976030;
        double r976034 = r976031 / r976033;
        double r976035 = fma(r976029, r976030, r976034);
        double r976036 = 0.5;
        double r976037 = r976036 / r976030;
        double r976038 = r976035 - r976037;
        double r976039 = log(r976038);
        return r976039;
}

Error

Bits error versus x

Derivation

  1. Initial program 30.6

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

  2. Simplified30.6

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

  3. Taylor expanded around inf 0.3

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

  4. Simplified0.3

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

  5. Final simplification0.3

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

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arc-cosine"
  (log (+ x (sqrt (- (* x x) 1)))))