Average Error: 52.6 → 0.2
Time: 27.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.061520745717705:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9645338519110289:\\ \;\;\;\;\left(\frac{3}{40} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{1}{2}}{x} - \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(x + x\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.061520745717705:\\
\;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.9645338519110289:\\
\;\;\;\;\left(\frac{3}{40} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right) + x\\

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

\end{array}
double f(double x) {
        double r6114487 = x;
        double r6114488 = r6114487 * r6114487;
        double r6114489 = 1.0;
        double r6114490 = r6114488 + r6114489;
        double r6114491 = sqrt(r6114490);
        double r6114492 = r6114487 + r6114491;
        double r6114493 = log(r6114492);
        return r6114493;
}
↓
double f(double x) {
        double r6114494 = x;
        double r6114495 = -1.061520745717705;
        bool r6114496 = r6114494 <= r6114495;
        double r6114497 = -0.0625;
        double r6114498 = r6114494 * r6114494;
        double r6114499 = r6114498 * r6114494;
        double r6114500 = r6114498 * r6114499;
        double r6114501 = r6114497 / r6114500;
        double r6114502 = 0.125;
        double r6114503 = r6114502 / r6114494;
        double r6114504 = r6114503 / r6114498;
        double r6114505 = 0.5;
        double r6114506 = r6114505 / r6114494;
        double r6114507 = r6114504 - r6114506;
        double r6114508 = r6114501 + r6114507;
        double r6114509 = log(r6114508);
        double r6114510 = 0.9645338519110289;
        bool r6114511 = r6114494 <= r6114510;
        double r6114512 = 0.075;
        double r6114513 = r6114512 * r6114500;
        double r6114514 = -0.16666666666666666;
        double r6114515 = r6114499 * r6114514;
        double r6114516 = r6114513 + r6114515;
        double r6114517 = r6114516 + r6114494;
        double r6114518 = r6114494 + r6114494;
        double r6114519 = r6114504 - r6114518;
        double r6114520 = r6114506 - r6114519;
        double r6114521 = log(r6114520);
        double r6114522 = r6114511 ? r6114517 : r6114521;
        double r6114523 = r6114496 ? r6114509 : r6114522;
        return r6114523;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.6
Target45.0
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 < -1.061520745717705

    1. Initial program 61.6

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

    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

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

    if -1.061520745717705 < x < 0.9645338519110289

    1. Initial program 58.6

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

    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

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

    if 0.9645338519110289 < x

    1. Initial program 31.9

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

    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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.061520745717705:\\ \;\;\;\;\log \left(\frac{\frac{-1}{16}}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \frac{\frac{1}{2}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9645338519110289:\\ \;\;\;\;\left(\frac{3}{40} \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) + \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{-1}{6}\right) + x\\ \mathbf{else}:\\ \;\;\;\;\log \left(\frac{\frac{1}{2}}{x} - \left(\frac{\frac{\frac{1}{8}}{x}}{x \cdot x} - \left(x + x\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(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)))))