Average Error: 52.5 → 0.2
Time: 10.6s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0757153356474918:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.9619268999163055:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{6}, \left(x \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{3}{40}, {x}^{5}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \mathsf{fma}\left(2, x, \frac{\frac{1}{2}}{x}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0757153356474918:\\
\;\;\;\;\log \left(\left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 0.9619268999163055:\\
\;\;\;\;\mathsf{fma}\left(\frac{-1}{6}, \left(x \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{3}{40}, {x}^{5}, x\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\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)\\

\end{array}
double f(double x) {
        double r2570323 = x;
        double r2570324 = r2570323 * r2570323;
        double r2570325 = 1.0;
        double r2570326 = r2570324 + r2570325;
        double r2570327 = sqrt(r2570326);
        double r2570328 = r2570323 + r2570327;
        double r2570329 = log(r2570328);
        return r2570329;
}
↓
double f(double x) {
        double r2570330 = x;
        double r2570331 = -1.0757153356474918;
        bool r2570332 = r2570330 <= r2570331;
        double r2570333 = 0.125;
        double r2570334 = r2570330 * r2570330;
        double r2570335 = r2570333 / r2570334;
        double r2570336 = r2570335 / r2570330;
        double r2570337 = 0.5;
        double r2570338 = r2570337 / r2570330;
        double r2570339 = r2570336 - r2570338;
        double r2570340 = 0.0625;
        double r2570341 = 5.0;
        double r2570342 = pow(r2570330, r2570341);
        double r2570343 = r2570340 / r2570342;
        double r2570344 = r2570339 - r2570343;
        double r2570345 = log(r2570344);
        double r2570346 = 0.9619268999163055;
        bool r2570347 = r2570330 <= r2570346;
        double r2570348 = -0.16666666666666666;
        double r2570349 = r2570334 * r2570330;
        double r2570350 = 0.075;
        double r2570351 = fma(r2570350, r2570342, r2570330);
        double r2570352 = fma(r2570348, r2570349, r2570351);
        double r2570353 = -0.125;
        double r2570354 = r2570353 / r2570349;
        double r2570355 = 2.0;
        double r2570356 = fma(r2570355, r2570330, r2570338);
        double r2570357 = r2570354 + r2570356;
        double r2570358 = log(r2570357);
        double r2570359 = r2570347 ? r2570352 : r2570358;
        double r2570360 = r2570332 ? r2570345 : r2570359;
        return r2570360;
}

Error

Bits error versus x

Target

Original52.5
Target45.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0757153356474918

    1. Initial program 61.6

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

    2. Simplified60.8

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

    3. Taylor expanded around -inf 0.2

      \[\leadsto \log \color{blue}{\left(\frac{1}{8} \cdot \frac{1}{{x}^{3}} - \left(\frac{1}{16} \cdot \frac{1}{{x}^{5}} + \frac{1}{2} \cdot \frac{1}{x}\right)\right)}\]

    4. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)}\]

    if -1.0757153356474918 < x < 0.9619268999163055

    1. Initial program 58.7

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

    2. Simplified58.7

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

    3. Taylor expanded around 0 0.2

      \[\leadsto \color{blue}{\left(x + \frac{3}{40} \cdot {x}^{5}\right) - \frac{1}{6} \cdot {x}^{3}}\]

    4. Simplified0.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, \left(x \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{3}{40}, {x}^{5}, x\right)\right)}\]

    if 0.9619268999163055 < x

    1. Initial program 31.5

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

    2. Simplified0.1

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

    3. Taylor expanded around inf 0.2

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

    4. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \mathsf{fma}\left(2, x, \frac{\frac{1}{2}}{x}\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0757153356474918:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.9619268999163055:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{6}, \left(x \cdot x\right) \cdot x, \mathsf{fma}\left(\frac{3}{40}, {x}^{5}, x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{-1}{8}}{\left(x \cdot x\right) \cdot x} + \mathsf{fma}\left(2, x, \frac{\frac{1}{2}}{x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

  (log (+ x (sqrt (+ (* x x) 1)))))