Average Error: 52.8 → 0.2
Time: 20.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0869167841195677:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.9419391586635008:\\ \;\;\;\;\mathsf{fma}\left(\frac{3}{40}, {x}^{5}, x\right) - \frac{1}{6} \cdot \left(\left(x \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{1}{x}, \frac{1}{2} - \frac{\frac{1}{8}}{x \cdot x}, x\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0869167841195677:\\
\;\;\;\;\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 0.9419391586635008:\\
\;\;\;\;\mathsf{fma}\left(\frac{3}{40}, {x}^{5}, x\right) - \frac{1}{6} \cdot \left(\left(x \cdot x\right) \cdot x\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\mathsf{fma}\left(\frac{1}{x}, \frac{1}{2} - \frac{\frac{1}{8}}{x \cdot x}, x\right) + x\right)\\

\end{array}
double f(double x) {
        double r5036364 = x;
        double r5036365 = r5036364 * r5036364;
        double r5036366 = 1.0;
        double r5036367 = r5036365 + r5036366;
        double r5036368 = sqrt(r5036367);
        double r5036369 = r5036364 + r5036368;
        double r5036370 = log(r5036369);
        return r5036370;
}
↓
double f(double x) {
        double r5036371 = x;
        double r5036372 = -1.0869167841195677;
        bool r5036373 = r5036371 <= r5036372;
        double r5036374 = 0.125;
        double r5036375 = r5036371 * r5036371;
        double r5036376 = r5036374 / r5036375;
        double r5036377 = r5036376 / r5036371;
        double r5036378 = 0.5;
        double r5036379 = r5036378 / r5036371;
        double r5036380 = r5036377 - r5036379;
        double r5036381 = 0.0625;
        double r5036382 = 5.0;
        double r5036383 = pow(r5036371, r5036382);
        double r5036384 = r5036381 / r5036383;
        double r5036385 = r5036380 - r5036384;
        double r5036386 = log(r5036385);
        double r5036387 = 0.9419391586635008;
        bool r5036388 = r5036371 <= r5036387;
        double r5036389 = 0.075;
        double r5036390 = fma(r5036389, r5036383, r5036371);
        double r5036391 = 0.16666666666666666;
        double r5036392 = r5036375 * r5036371;
        double r5036393 = r5036391 * r5036392;
        double r5036394 = r5036390 - r5036393;
        double r5036395 = 1.0;
        double r5036396 = r5036395 / r5036371;
        double r5036397 = r5036378 - r5036376;
        double r5036398 = fma(r5036396, r5036397, r5036371);
        double r5036399 = r5036398 + r5036371;
        double r5036400 = log(r5036399);
        double r5036401 = r5036388 ? r5036394 : r5036400;
        double r5036402 = r5036373 ? r5036386 : r5036401;
        return r5036402;
}

Error

Bits error versus x

Target

Original52.8
Target45.2
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 < -1.0869167841195677

    1. Initial program 61.7

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

    2. Simplified60.9

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

    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 -1.0869167841195677 < x < 0.9419391586635008

    1. Initial program 58.6

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

    2. Simplified58.6

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

    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

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

    if 0.9419391586635008 < x

    1. Initial program 31.8

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

    2. Simplified0.0

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

    3. Taylor expanded around inf 0.2

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

    4. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0869167841195677:\\ \;\;\;\;\log \left(\left(\frac{\frac{\frac{1}{8}}{x \cdot x}}{x} - \frac{\frac{1}{2}}{x}\right) - \frac{\frac{1}{16}}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.9419391586635008:\\ \;\;\;\;\mathsf{fma}\left(\frac{3}{40}, {x}^{5}, x\right) - \frac{1}{6} \cdot \left(\left(x \cdot x\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(\frac{1}{x}, \frac{1}{2} - \frac{\frac{1}{8}}{x \cdot x}, x\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +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)))))