Average Error: 52.9 → 0.3
Time: 12.0s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.020049338560968577027665560308378189802:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8895740307587358675078803571523167192936:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{\frac{1}{6}}{1} \cdot \frac{{x}^{3}}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x \cdot 2 - \frac{0.125}{{x}^{3}}\right) + \frac{0.5}{x}\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.020049338560968577027665560308378189802:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.8895740307587358675078803571523167192936:\\
\;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{\frac{1}{6}}{1} \cdot \frac{{x}^{3}}{\sqrt{1}}\\

\mathbf{else}:\\
\;\;\;\;\log \left(\left(x \cdot 2 - \frac{0.125}{{x}^{3}}\right) + \frac{0.5}{x}\right)\\

\end{array}
double f(double x) {
        double r148341 = x;
        double r148342 = r148341 * r148341;
        double r148343 = 1.0;
        double r148344 = r148342 + r148343;
        double r148345 = sqrt(r148344);
        double r148346 = r148341 + r148345;
        double r148347 = log(r148346);
        return r148347;
}
↓
double f(double x) {
        double r148348 = x;
        double r148349 = -1.0200493385609686;
        bool r148350 = r148348 <= r148349;
        double r148351 = 0.125;
        double r148352 = 3.0;
        double r148353 = pow(r148348, r148352);
        double r148354 = r148351 / r148353;
        double r148355 = 0.0625;
        double r148356 = 5.0;
        double r148357 = pow(r148348, r148356);
        double r148358 = r148355 / r148357;
        double r148359 = 0.5;
        double r148360 = r148359 / r148348;
        double r148361 = r148358 + r148360;
        double r148362 = r148354 - r148361;
        double r148363 = log(r148362);
        double r148364 = 0.8895740307587359;
        bool r148365 = r148348 <= r148364;
        double r148366 = 1.0;
        double r148367 = sqrt(r148366);
        double r148368 = log(r148367);
        double r148369 = r148348 / r148367;
        double r148370 = r148368 + r148369;
        double r148371 = 0.16666666666666666;
        double r148372 = r148371 / r148366;
        double r148373 = r148353 / r148367;
        double r148374 = r148372 * r148373;
        double r148375 = r148370 - r148374;
        double r148376 = 2.0;
        double r148377 = r148348 * r148376;
        double r148378 = r148377 - r148354;
        double r148379 = r148378 + r148360;
        double r148380 = log(r148379);
        double r148381 = r148365 ? r148375 : r148380;
        double r148382 = r148350 ? r148363 : r148381;
        return r148382;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.9
Target45.4
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 < -1.0200493385609686

    1. Initial program 62.7

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

    2. Taylor expanded around -inf 0.2

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

    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 -1.0200493385609686 < x < 0.8895740307587359

    1. Initial program 58.6

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    if 0.8895740307587359 < x

    1. Initial program 31.1

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

    2. Taylor expanded around inf 0.3

      \[\leadsto \log \color{blue}{\left(\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)}\]

    3. Simplified0.3

      \[\leadsto \log \color{blue}{\left(\frac{0.5}{x} + \left(x \cdot 2 - \frac{0.125}{{x}^{3}}\right)\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.020049338560968577027665560308378189802:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8895740307587358675078803571523167192936:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{\frac{1}{6}}{1} \cdot \frac{{x}^{3}}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x \cdot 2 - \frac{0.125}{{x}^{3}}\right) + \frac{0.5}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))