Average Error: 53.2 → 0.3
Time: 5.4s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0090202414523164:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.963811647786171011:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log 2 - \left(\left(\frac{0.09375}{{x}^{4}} - \log x\right) - \frac{\frac{0.25}{x}}{x}\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0090202414523164:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.963811647786171011:\\
\;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\log 2 - \left(\left(\frac{0.09375}{{x}^{4}} - \log x\right) - \frac{\frac{0.25}{x}}{x}\right)\\

\end{array}
double f(double x) {
        double r244366 = x;
        double r244367 = r244366 * r244366;
        double r244368 = 1.0;
        double r244369 = r244367 + r244368;
        double r244370 = sqrt(r244369);
        double r244371 = r244366 + r244370;
        double r244372 = log(r244371);
        return r244372;
}
↓
double f(double x) {
        double r244373 = x;
        double r244374 = -1.0090202414523164;
        bool r244375 = r244373 <= r244374;
        double r244376 = 0.125;
        double r244377 = 3.0;
        double r244378 = pow(r244373, r244377);
        double r244379 = r244376 / r244378;
        double r244380 = 0.5;
        double r244381 = r244380 / r244373;
        double r244382 = 0.0625;
        double r244383 = -r244382;
        double r244384 = 5.0;
        double r244385 = pow(r244373, r244384);
        double r244386 = r244383 / r244385;
        double r244387 = r244381 - r244386;
        double r244388 = r244379 - r244387;
        double r244389 = log(r244388);
        double r244390 = 0.963811647786171;
        bool r244391 = r244373 <= r244390;
        double r244392 = 1.0;
        double r244393 = sqrt(r244392);
        double r244394 = log(r244393);
        double r244395 = r244373 / r244393;
        double r244396 = r244394 + r244395;
        double r244397 = 0.16666666666666666;
        double r244398 = pow(r244393, r244377);
        double r244399 = r244378 / r244398;
        double r244400 = r244397 * r244399;
        double r244401 = r244396 - r244400;
        double r244402 = 2.0;
        double r244403 = log(r244402);
        double r244404 = 0.09375;
        double r244405 = 4.0;
        double r244406 = pow(r244373, r244405);
        double r244407 = r244404 / r244406;
        double r244408 = log(r244373);
        double r244409 = r244407 - r244408;
        double r244410 = 0.25;
        double r244411 = r244410 / r244373;
        double r244412 = r244411 / r244373;
        double r244413 = r244409 - r244412;
        double r244414 = r244403 - r244413;
        double r244415 = r244391 ? r244401 : r244414;
        double r244416 = r244375 ? r244389 : r244415;
        return r244416;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.2
Target45.5
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0090202414523164

    1. Initial program 62.9

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

    2. Taylor expanded around -inf 0.2

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)}\]

    3. Simplified0.2

      \[\leadsto \log \color{blue}{\left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)}\]

    if -1.0090202414523164 < x < 0.963811647786171

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.4

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

    if 0.963811647786171 < x

    1. Initial program 32.3

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

    2. Taylor expanded around inf 0.3

      \[\leadsto \color{blue}{\left(0.25 \cdot \frac{1}{{x}^{2}} + \log 2\right) - \left(\log \left(\frac{1}{x}\right) + 0.09375 \cdot \frac{1}{{x}^{4}}\right)}\]

    3. Simplified0.3

      \[\leadsto \color{blue}{\log 2 - \left(\left(\frac{0.09375}{{x}^{4}} - \log x\right) - \frac{\frac{0.25}{x}}{x}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0090202414523164:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.963811647786171011:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log 2 - \left(\left(\frac{0.09375}{{x}^{4}} - \log x\right) - \frac{\frac{0.25}{x}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(FPCore (x)
  :name "Hyperbolic arcsine"
  :precision binary64

  :herbie-target
  (if (< x 0.0) (log (/ -1 (- x (sqrt (+ (* x x) 1))))) (log (+ x (sqrt (+ (* x x) 1)))))

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