Average Error: 52.3 → 0.3
Time: 50.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0506541446768847:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9667367906799228:\\ \;\;\;\;\frac{3}{40} \cdot {x}^{5} + \left(x + \left(x \cdot \frac{-1}{6}\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{\frac{1}{2}}{x} - \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} - x\right)\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0506541446768847:\\
\;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.9667367906799228:\\
\;\;\;\;\frac{3}{40} \cdot {x}^{5} + \left(x + \left(x \cdot \frac{-1}{6}\right) \cdot \left(x \cdot x\right)\right)\\

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

\end{array}
double f(double x) {
        double r27330332 = x;
        double r27330333 = r27330332 * r27330332;
        double r27330334 = 1.0;
        double r27330335 = r27330333 + r27330334;
        double r27330336 = sqrt(r27330335);
        double r27330337 = r27330332 + r27330336;
        double r27330338 = log(r27330337);
        return r27330338;
}
↓
double f(double x) {
        double r27330339 = x;
        double r27330340 = -1.0506541446768847;
        bool r27330341 = r27330339 <= r27330340;
        double r27330342 = -0.5;
        double r27330343 = r27330342 / r27330339;
        double r27330344 = 0.0625;
        double r27330345 = 5.0;
        double r27330346 = pow(r27330339, r27330345);
        double r27330347 = r27330344 / r27330346;
        double r27330348 = 0.125;
        double r27330349 = r27330348 / r27330339;
        double r27330350 = r27330349 / r27330339;
        double r27330351 = r27330350 / r27330339;
        double r27330352 = r27330347 - r27330351;
        double r27330353 = r27330343 - r27330352;
        double r27330354 = log(r27330353);
        double r27330355 = 0.9667367906799228;
        bool r27330356 = r27330339 <= r27330355;
        double r27330357 = 0.075;
        double r27330358 = r27330357 * r27330346;
        double r27330359 = -0.16666666666666666;
        double r27330360 = r27330339 * r27330359;
        double r27330361 = r27330339 * r27330339;
        double r27330362 = r27330360 * r27330361;
        double r27330363 = r27330339 + r27330362;
        double r27330364 = r27330358 + r27330363;
        double r27330365 = 0.5;
        double r27330366 = r27330365 / r27330339;
        double r27330367 = r27330339 * r27330361;
        double r27330368 = r27330348 / r27330367;
        double r27330369 = r27330368 - r27330339;
        double r27330370 = r27330366 - r27330369;
        double r27330371 = r27330370 + r27330339;
        double r27330372 = log(r27330371);
        double r27330373 = r27330356 ? r27330364 : r27330372;
        double r27330374 = r27330341 ? r27330354 : r27330373;
        return r27330374;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.3
Target44.5
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0506541446768847

    1. Initial program 61.6

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

    2. Taylor expanded around -inf 0.3

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

    3. Simplified0.3

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

    if -1.0506541446768847 < x < 0.9667367906799228

    1. Initial program 58.6

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 0.9667367906799228 < x

    1. Initial program 30.2

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0506541446768847:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{\frac{\frac{1}{8}}{x}}{x}}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9667367906799228:\\ \;\;\;\;\frac{3}{40} \cdot {x}^{5} + \left(x + \left(x \cdot \frac{-1}{6}\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\frac{\frac{1}{2}}{x} - \left(\frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)} - x\right)\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019124 
(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)))))