Average Error: 52.7 → 0.3
Time: 16.5s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0840635159626253:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{elif}\;x \le 0.9540314055762552:\\ \;\;\;\;\left(x - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{6}\right) + \frac{3}{40} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0840635159626253:\\
\;\;\;\;\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 0.9540314055762552:\\
\;\;\;\;\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(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\

\end{array}
double f(double x) {
        double r7616348 = x;
        double r7616349 = r7616348 * r7616348;
        double r7616350 = 1.0;
        double r7616351 = r7616349 + r7616350;
        double r7616352 = sqrt(r7616351);
        double r7616353 = r7616348 + r7616352;
        double r7616354 = log(r7616353);
        return r7616354;
}
↓
double f(double x) {
        double r7616355 = x;
        double r7616356 = -1.0840635159626253;
        bool r7616357 = r7616355 <= r7616356;
        double r7616358 = -0.5;
        double r7616359 = r7616358 / r7616355;
        double r7616360 = 0.0625;
        double r7616361 = 5.0;
        double r7616362 = pow(r7616355, r7616361);
        double r7616363 = r7616360 / r7616362;
        double r7616364 = 0.125;
        double r7616365 = r7616355 * r7616355;
        double r7616366 = r7616355 * r7616365;
        double r7616367 = r7616364 / r7616366;
        double r7616368 = r7616363 - r7616367;
        double r7616369 = r7616359 - r7616368;
        double r7616370 = log(r7616369);
        double r7616371 = 0.9540314055762552;
        bool r7616372 = r7616355 <= r7616371;
        double r7616373 = 0.16666666666666666;
        double r7616374 = r7616366 * r7616373;
        double r7616375 = r7616355 - r7616374;
        double r7616376 = 0.075;
        double r7616377 = r7616376 * r7616362;
        double r7616378 = r7616375 + r7616377;
        double r7616379 = 0.5;
        double r7616380 = r7616379 / r7616355;
        double r7616381 = r7616380 - r7616367;
        double r7616382 = r7616355 + r7616381;
        double r7616383 = r7616382 + r7616355;
        double r7616384 = log(r7616383);
        double r7616385 = r7616372 ? r7616378 : r7616384;
        double r7616386 = r7616357 ? r7616370 : r7616385;
        return r7616386;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.7
Target44.8
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 < -1.0840635159626253

    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 -1.0840635159626253 < x < 0.9540314055762552

    1. Initial program 58.4

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

    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

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

    if 0.9540314055762552 < x

    1. Initial program 32.3

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

    2. Taylor expanded around inf 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0840635159626253:\\ \;\;\;\;\log \left(\frac{\frac{-1}{2}}{x} - \left(\frac{\frac{1}{16}}{{x}^{5}} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right)\\ \mathbf{elif}\;x \le 0.9540314055762552:\\ \;\;\;\;\left(x - \left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{6}\right) + \frac{3}{40} \cdot {x}^{5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(x + \left(\frac{\frac{1}{2}}{x} - \frac{\frac{1}{8}}{x \cdot \left(x \cdot x\right)}\right)\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(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)))))