Average Error: 53.1 → 0.2
Time: 14.0s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.021894228612228561203778554045129567385:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8867151979513619997064211020187940448523:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{\frac{1}{6}}{1} \cdot \frac{{x}^{3}}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.021894228612228561203778554045129567385:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.8867151979513619997064211020187940448523:\\
\;\;\;\;\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(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right) + x\right)\\

\end{array}
double f(double x) {
        double r111382 = x;
        double r111383 = r111382 * r111382;
        double r111384 = 1.0;
        double r111385 = r111383 + r111384;
        double r111386 = sqrt(r111385);
        double r111387 = r111382 + r111386;
        double r111388 = log(r111387);
        return r111388;
}
↓
double f(double x) {
        double r111389 = x;
        double r111390 = -1.0218942286122286;
        bool r111391 = r111389 <= r111390;
        double r111392 = 0.125;
        double r111393 = 3.0;
        double r111394 = pow(r111389, r111393);
        double r111395 = r111392 / r111394;
        double r111396 = 0.0625;
        double r111397 = 5.0;
        double r111398 = pow(r111389, r111397);
        double r111399 = r111396 / r111398;
        double r111400 = 0.5;
        double r111401 = r111400 / r111389;
        double r111402 = r111399 + r111401;
        double r111403 = r111395 - r111402;
        double r111404 = log(r111403);
        double r111405 = 0.886715197951362;
        bool r111406 = r111389 <= r111405;
        double r111407 = 1.0;
        double r111408 = sqrt(r111407);
        double r111409 = log(r111408);
        double r111410 = r111389 / r111408;
        double r111411 = r111409 + r111410;
        double r111412 = 0.16666666666666666;
        double r111413 = r111412 / r111407;
        double r111414 = r111394 / r111408;
        double r111415 = r111413 * r111414;
        double r111416 = r111411 - r111415;
        double r111417 = r111389 + r111401;
        double r111418 = r111417 - r111395;
        double r111419 = r111418 + r111389;
        double r111420 = log(r111419);
        double r111421 = r111406 ? r111416 : r111420;
        double r111422 = r111391 ? r111404 : r111421;
        return r111422;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.1
Target45.4
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 < -1.0218942286122286

    1. Initial program 63.3

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

    2. Taylor expanded around -inf 0.1

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

    3. Simplified0.1

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

    if -1.0218942286122286 < x < 0.886715197951362

    1. Initial program 58.7

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

    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 0.886715197951362 < x

    1. Initial program 31.9

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

    2. Taylor expanded around inf 0.2

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

    3. Simplified0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.021894228612228561203778554045129567385:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.8867151979513619997064211020187940448523:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{\frac{1}{6}}{1} \cdot \frac{{x}^{3}}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(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)))))