Average Error: 52.2 → 0.2
Time: 14.0s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0682549790295666:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{elif}\;x \le 0.9560728927963928:\\ \;\;\;\;\left(x - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{6}\right) + \frac{3}{40} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{1}{2}}{x} + \left(\left(x + x\right) - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0682549790295666:\\
\;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\

\mathbf{elif}\;x \le 0.9560728927963928:\\
\;\;\;\;\left(x - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{6}\right) + \frac{3}{40} \cdot {x}^{5}\\

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

\end{array}
double f(double x) {
        double r5698483 = x;
        double r5698484 = r5698483 * r5698483;
        double r5698485 = 1.0;
        double r5698486 = r5698484 + r5698485;
        double r5698487 = sqrt(r5698486);
        double r5698488 = r5698483 + r5698487;
        double r5698489 = log(r5698488);
        return r5698489;
}
↓
double f(double x) {
        double r5698490 = x;
        double r5698491 = -1.0682549790295666;
        bool r5698492 = r5698490 <= r5698491;
        double r5698493 = -0.5;
        double r5698494 = r5698493 / r5698490;
        double r5698495 = 0.0625;
        double r5698496 = 5.0;
        double r5698497 = pow(r5698490, r5698496);
        double r5698498 = r5698495 / r5698497;
        double r5698499 = 0.125;
        double r5698500 = r5698490 * r5698490;
        double r5698501 = r5698490 * r5698500;
        double r5698502 = r5698499 / r5698501;
        double r5698503 = r5698498 - r5698502;
        double r5698504 = r5698494 - r5698503;
        double r5698505 = log(r5698504);
        double r5698506 = 0.9560728927963928;
        bool r5698507 = r5698490 <= r5698506;
        double r5698508 = 0.16666666666666666;
        double r5698509 = r5698501 * r5698508;
        double r5698510 = r5698490 - r5698509;
        double r5698511 = 0.075;
        double r5698512 = r5698511 * r5698497;
        double r5698513 = r5698510 + r5698512;
        double r5698514 = 0.5;
        double r5698515 = r5698514 / r5698490;
        double r5698516 = r5698490 + r5698490;
        double r5698517 = r5698516 - r5698502;
        double r5698518 = r5698515 + r5698517;
        double r5698519 = log(r5698518);
        double r5698520 = r5698507 ? r5698513 : r5698519;
        double r5698521 = r5698492 ? r5698505 : r5698520;
        return r5698521;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.2
Target44.7
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 < -1.0682549790295666

    1. Initial program 61.8

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

    2. Taylor expanded around -inf 0.2

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

    3. Simplified0.2

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

    if -1.0682549790295666 < x < 0.9560728927963928

    1. Initial program 58.4

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

    if 0.9560728927963928 < x

    1. Initial program 29.9

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0682549790295666:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{elif}\;x \le 0.9560728927963928:\\ \;\;\;\;\left(x - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{6}\right) + \frac{3}{40} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{1}{2}}{x} + \left(\left(x + x\right) - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(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)))))