Average Error: 52.9 → 0.3
Time: 5.9s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.01388879764755901:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.888575674591003:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.01388879764755901:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.888575674591003:\\
\;\;\;\;\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(\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)\\

\end{array}
double f(double x) {
        double r138975 = x;
        double r138976 = r138975 * r138975;
        double r138977 = 1.0;
        double r138978 = r138976 + r138977;
        double r138979 = sqrt(r138978);
        double r138980 = r138975 + r138979;
        double r138981 = log(r138980);
        return r138981;
}
↓
double f(double x) {
        double r138982 = x;
        double r138983 = -1.013888797647559;
        bool r138984 = r138982 <= r138983;
        double r138985 = 0.125;
        double r138986 = 3.0;
        double r138987 = pow(r138982, r138986);
        double r138988 = r138985 / r138987;
        double r138989 = 0.5;
        double r138990 = r138989 / r138982;
        double r138991 = 0.0625;
        double r138992 = -r138991;
        double r138993 = 5.0;
        double r138994 = pow(r138982, r138993);
        double r138995 = r138992 / r138994;
        double r138996 = r138990 - r138995;
        double r138997 = r138988 - r138996;
        double r138998 = log(r138997);
        double r138999 = 0.888575674591003;
        bool r139000 = r138982 <= r138999;
        double r139001 = 1.0;
        double r139002 = sqrt(r139001);
        double r139003 = log(r139002);
        double r139004 = r138982 / r139002;
        double r139005 = r139003 + r139004;
        double r139006 = 0.16666666666666666;
        double r139007 = pow(r139002, r138986);
        double r139008 = r138987 / r139007;
        double r139009 = r139006 * r139008;
        double r139010 = r139005 - r139009;
        double r139011 = 2.0;
        double r139012 = r139011 * r138982;
        double r139013 = 1.0;
        double r139014 = r139013 / r138982;
        double r139015 = r138989 * r139014;
        double r139016 = r139012 + r139015;
        double r139017 = r139013 / r138987;
        double r139018 = r138985 * r139017;
        double r139019 = r139016 - r139018;
        double r139020 = log(r139019);
        double r139021 = r139000 ? r139010 : r139020;
        double r139022 = r138984 ? r138998 : r139021;
        return r139022;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.9
Target44.8
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 < -1.013888797647559

    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

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

    if -1.013888797647559 < x < 0.888575674591003

    1. Initial program 58.5

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

    2. Taylor expanded around 0 0.3

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

    if 0.888575674591003 < x

    1. Initial program 31.3

      \[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
    2. Using strategy rm

    3. Applied flip-+62.9

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

    4. Simplified62.9

      \[\leadsto \log \left(\frac{\color{blue}{0 - 1}}{x - \sqrt{x \cdot x + 1}}\right)\]

    5. Taylor expanded around inf 0.2

      \[\leadsto \log \color{blue}{\left(\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.01388879764755901:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.888575674591003:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(2 \cdot x + 0.5 \cdot \frac{1}{x}\right) - 0.125 \cdot \frac{1}{{x}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 
(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)))))