Average Error: 30.6 → 0.3
Time: 8.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x - 1}\right)\]
↓
\[\log \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \mathsf{fma}\left(2, x, \frac{\frac{-1}{2}}{x}\right)\right)\]
\log \left(x + \sqrt{x \cdot x - 1}\right)
↓
\log \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \mathsf{fma}\left(2, x, \frac{\frac{-1}{2}}{x}\right)\right)
double f(double x) {
        double r1222205 = x;
        double r1222206 = r1222205 * r1222205;
        double r1222207 = 1.0;
        double r1222208 = r1222206 - r1222207;
        double r1222209 = sqrt(r1222208);
        double r1222210 = r1222205 + r1222209;
        double r1222211 = log(r1222210);
        return r1222211;
}
↓
double f(double x) {
        double r1222212 = -0.125;
        double r1222213 = x;
        double r1222214 = r1222213 * r1222213;
        double r1222215 = r1222214 * r1222213;
        double r1222216 = r1222212 / r1222215;
        double r1222217 = 2.0;
        double r1222218 = -0.5;
        double r1222219 = r1222218 / r1222213;
        double r1222220 = fma(r1222217, r1222213, r1222219);
        double r1222221 = r1222216 + r1222220;
        double r1222222 = log(r1222221);
        return r1222222;
}

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

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

  5. Final simplification0.3

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

Reproduce

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