Average Error: 53.1 → 0.3
Time: 15.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.004281043358186797931352884916123002768:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \le 0.8904752561288424850260980747407302260399:\\ \;\;\;\;\log \left(\sqrt{1}\right) + \frac{x + \frac{{x}^{3}}{\frac{1}{\frac{-1}{6}}}}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{0.5}{x} + \left(x - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.004281043358186797931352884916123002768:\\
\;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\

\mathbf{elif}\;x \le 0.8904752561288424850260980747407302260399:\\
\;\;\;\;\log \left(\sqrt{1}\right) + \frac{x + \frac{{x}^{3}}{\frac{1}{\frac{-1}{6}}}}{\sqrt{1}}\\

\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 r110305 = x;
        double r110306 = r110305 * r110305;
        double r110307 = 1.0;
        double r110308 = r110306 + r110307;
        double r110309 = sqrt(r110308);
        double r110310 = r110305 + r110309;
        double r110311 = log(r110310);
        return r110311;
}
↓
double f(double x) {
        double r110312 = x;
        double r110313 = -1.0042810433581868;
        bool r110314 = r110312 <= r110313;
        double r110315 = 0.125;
        double r110316 = 3.0;
        double r110317 = pow(r110312, r110316);
        double r110318 = r110315 / r110317;
        double r110319 = 0.0625;
        double r110320 = 5.0;
        double r110321 = pow(r110312, r110320);
        double r110322 = r110319 / r110321;
        double r110323 = r110318 - r110322;
        double r110324 = 0.5;
        double r110325 = r110324 / r110312;
        double r110326 = r110323 - r110325;
        double r110327 = log(r110326);
        double r110328 = 0.8904752561288425;
        bool r110329 = r110312 <= r110328;
        double r110330 = 1.0;
        double r110331 = sqrt(r110330);
        double r110332 = log(r110331);
        double r110333 = -0.16666666666666666;
        double r110334 = r110330 / r110333;
        double r110335 = r110317 / r110334;
        double r110336 = r110312 + r110335;
        double r110337 = r110336 / r110331;
        double r110338 = r110332 + r110337;
        double r110339 = r110312 - r110318;
        double r110340 = r110325 + r110339;
        double r110341 = r110312 + r110340;
        double r110342 = log(r110341);
        double r110343 = r110329 ? r110338 : r110342;
        double r110344 = r110314 ? r110327 : r110343;
        return r110344;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.1
Target45.4
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0042810433581868

    1. Initial program 62.8

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

    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 -1.0042810433581868 < x < 0.8904752561288425

    1. Initial program 58.6

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    5. Applied associate-*r/0.3

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

    6. Simplified0.3

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

    if 0.8904752561288425 < x

    1. Initial program 32.2

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

    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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.004281043358186797931352884916123002768:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.0625}{{x}^{5}}\right) - \frac{0.5}{x}\right)\\ \mathbf{elif}\;x \le 0.8904752561288424850260980747407302260399:\\ \;\;\;\;\log \left(\sqrt{1}\right) + \frac{x + \frac{{x}^{3}}{\frac{1}{\frac{-1}{6}}}}{\sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \left(\frac{0.5}{x} + \left(x - \frac{0.125}{{x}^{3}}\right)\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(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)))))