Average Error: 52.9 → 0.2
Time: 7.4s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0196250072526523:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.890120300530239716:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0196250072526523:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.890120300530239716:\\
\;\;\;\;\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 \left(x + \left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\\

\end{array}
double f(double x) {
        double r179298 = x;
        double r179299 = r179298 * r179298;
        double r179300 = 1.0;
        double r179301 = r179299 + r179300;
        double r179302 = sqrt(r179301);
        double r179303 = r179298 + r179302;
        double r179304 = log(r179303);
        return r179304;
}
↓
double f(double x) {
        double r179305 = x;
        double r179306 = -1.0196250072526523;
        bool r179307 = r179305 <= r179306;
        double r179308 = 0.125;
        double r179309 = 3.0;
        double r179310 = pow(r179305, r179309);
        double r179311 = r179308 / r179310;
        double r179312 = 0.5;
        double r179313 = r179312 / r179305;
        double r179314 = 0.0625;
        double r179315 = -r179314;
        double r179316 = 5.0;
        double r179317 = pow(r179305, r179316);
        double r179318 = r179315 / r179317;
        double r179319 = r179313 - r179318;
        double r179320 = r179311 - r179319;
        double r179321 = log(r179320);
        double r179322 = 0.8901203005302397;
        bool r179323 = r179305 <= r179322;
        double r179324 = 1.0;
        double r179325 = sqrt(r179324);
        double r179326 = log(r179325);
        double r179327 = r179305 / r179325;
        double r179328 = r179326 + r179327;
        double r179329 = 0.16666666666666666;
        double r179330 = pow(r179325, r179309);
        double r179331 = r179310 / r179330;
        double r179332 = r179329 * r179331;
        double r179333 = r179328 - r179332;
        double r179334 = r179305 + r179313;
        double r179335 = r179334 - r179311;
        double r179336 = r179305 + r179335;
        double r179337 = log(r179336);
        double r179338 = r179323 ? r179333 : r179337;
        double r179339 = r179307 ? r179321 : r179338;
        return r179339;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.9
Target45.3
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 < -1.0196250072526523

    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 -1.0196250072526523 < x < 0.8901203005302397

    1. Initial program 58.9

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

    2. Taylor expanded around 0 0.2

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

    if 0.8901203005302397 < x

    1. Initial program 31.4

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

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0196250072526523:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.890120300530239716:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\left(x + \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 
(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)))))