Average Error: 53.1 → 0.2
Time: 12.4s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.002814715336328044159586170280817896128:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \le 0.9017023301953626113203199565759859979153:\\ \;\;\;\;\frac{x + \frac{{x}^{3}}{\frac{1}{\frac{-1}{6}}}}{\sqrt{1}} + \log \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{0.5}{x} + \left(x - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.002814715336328044159586170280817896128:\\
\;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\

\mathbf{elif}\;x \le 0.9017023301953626113203199565759859979153:\\
\;\;\;\;\frac{x + \frac{{x}^{3}}{\frac{1}{\frac{-1}{6}}}}{\sqrt{1}} + \log \left(\sqrt{1}\right)\\

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

\end{array}
double f(double x) {
        double r93361 = x;
        double r93362 = r93361 * r93361;
        double r93363 = 1.0;
        double r93364 = r93362 + r93363;
        double r93365 = sqrt(r93364);
        double r93366 = r93361 + r93365;
        double r93367 = log(r93366);
        return r93367;
}
↓
double f(double x) {
        double r93368 = x;
        double r93369 = -1.002814715336328;
        bool r93370 = r93368 <= r93369;
        double r93371 = 0.125;
        double r93372 = 3.0;
        double r93373 = pow(r93368, r93372);
        double r93374 = r93371 / r93373;
        double r93375 = 0.0625;
        double r93376 = 5.0;
        double r93377 = pow(r93368, r93376);
        double r93378 = r93375 / r93377;
        double r93379 = r93374 - r93378;
        double r93380 = 0.5;
        double r93381 = r93380 / r93368;
        double r93382 = r93379 - r93381;
        double r93383 = log(r93382);
        double r93384 = 0.9017023301953626;
        bool r93385 = r93368 <= r93384;
        double r93386 = 1.0;
        double r93387 = -0.16666666666666666;
        double r93388 = r93386 / r93387;
        double r93389 = r93373 / r93388;
        double r93390 = r93368 + r93389;
        double r93391 = sqrt(r93386);
        double r93392 = r93390 / r93391;
        double r93393 = log(r93391);
        double r93394 = r93392 + r93393;
        double r93395 = r93368 - r93374;
        double r93396 = r93381 + r93395;
        double r93397 = r93368 + r93396;
        double r93398 = log(r93397);
        double r93399 = r93385 ? r93394 : r93398;
        double r93400 = r93370 ? r93383 : r93399;
        return r93400;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -1.002814715336328

    1. Initial program 63.1

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

    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

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

    if -1.002814715336328 < x < 0.9017023301953626

    1. Initial program 58.7

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

      \[\leadsto \color{blue}{\log \left(\sqrt{1}\right) + \left(\frac{-1}{6} \cdot \frac{x \cdot x}{1} + 1\right) \cdot \frac{x}{\sqrt{1}}}\]
    4. Using strategy rm

    5. Applied *-un-lft-identity0.2

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

    6. Applied associate-*l*0.2

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

    7. Simplified0.2

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

    if 0.9017023301953626 < x

    1. Initial program 32.2

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

    2. Taylor expanded around inf 0.1

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

    3. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.002814715336328044159586170280817896128:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \le 0.9017023301953626113203199565759859979153:\\ \;\;\;\;\frac{x + \frac{{x}^{3}}{\frac{1}{\frac{-1}{6}}}}{\sqrt{1}} + \log \left(\sqrt{1}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{0.5}{x} + \left(x - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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)))))