Average Error: 53.0 → 0.3
Time: 13.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0208992259036227:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.94854207071551944:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log 2 - \left(\left(\frac{0.09375}{{x}^{4}} - \log x\right) - \frac{\frac{0.25}{x}}{x}\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0208992259036227:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.94854207071551944:\\
\;\;\;\;\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 2 - \left(\left(\frac{0.09375}{{x}^{4}} - \log x\right) - \frac{\frac{0.25}{x}}{x}\right)\\

\end{array}
double f(double x) {
        double r170863 = x;
        double r170864 = r170863 * r170863;
        double r170865 = 1.0;
        double r170866 = r170864 + r170865;
        double r170867 = sqrt(r170866);
        double r170868 = r170863 + r170867;
        double r170869 = log(r170868);
        return r170869;
}
↓
double f(double x) {
        double r170870 = x;
        double r170871 = -1.0208992259036227;
        bool r170872 = r170870 <= r170871;
        double r170873 = 0.125;
        double r170874 = 3.0;
        double r170875 = pow(r170870, r170874);
        double r170876 = r170873 / r170875;
        double r170877 = 0.5;
        double r170878 = r170877 / r170870;
        double r170879 = 0.0625;
        double r170880 = -r170879;
        double r170881 = 5.0;
        double r170882 = pow(r170870, r170881);
        double r170883 = r170880 / r170882;
        double r170884 = r170878 - r170883;
        double r170885 = r170876 - r170884;
        double r170886 = log(r170885);
        double r170887 = 0.9485420707155194;
        bool r170888 = r170870 <= r170887;
        double r170889 = 1.0;
        double r170890 = sqrt(r170889);
        double r170891 = log(r170890);
        double r170892 = r170870 / r170890;
        double r170893 = r170891 + r170892;
        double r170894 = 0.16666666666666666;
        double r170895 = pow(r170890, r170874);
        double r170896 = r170875 / r170895;
        double r170897 = r170894 * r170896;
        double r170898 = r170893 - r170897;
        double r170899 = 2.0;
        double r170900 = log(r170899);
        double r170901 = 0.09375;
        double r170902 = 4.0;
        double r170903 = pow(r170870, r170902);
        double r170904 = r170901 / r170903;
        double r170905 = log(r170870);
        double r170906 = r170904 - r170905;
        double r170907 = 0.25;
        double r170908 = r170907 / r170870;
        double r170909 = r170908 / r170870;
        double r170910 = r170906 - r170909;
        double r170911 = r170900 - r170910;
        double r170912 = r170888 ? r170898 : r170911;
        double r170913 = r170872 ? r170886 : r170912;
        return r170913;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.0
Target45.5
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 < -1.0208992259036227

    1. Initial program 63.2

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

    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 -1.0208992259036227 < x < 0.9485420707155194

    1. Initial program 58.5

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

    2. Taylor expanded around 0 0.4

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

    if 0.9485420707155194 < x

    1. Initial program 32.2

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

    2. Taylor expanded around inf 0.3

      \[\leadsto \color{blue}{\left(0.25 \cdot \frac{1}{{x}^{2}} + \log 2\right) - \left(\log \left(\frac{1}{x}\right) + 0.09375 \cdot \frac{1}{{x}^{4}}\right)}\]

    3. Simplified0.3

      \[\leadsto \color{blue}{\log 2 - \left(\left(\frac{0.09375}{{x}^{4}} - \log x\right) - \frac{\frac{0.25}{x}}{x}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0208992259036227:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.94854207071551944:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log 2 - \left(\left(\frac{0.09375}{{x}^{4}} - \log x\right) - \frac{\frac{0.25}{x}}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +o rules:numerics
(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)))))