Average Error: 52.9 → 0.1
Time: 15.7s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.03845383551809634:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.00121273027943644379:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.03845383551809634:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.00121273027943644379:\\
\;\;\;\;\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(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\\

\end{array}
double f(double x) {
        double r223005 = x;
        double r223006 = r223005 * r223005;
        double r223007 = 1.0;
        double r223008 = r223006 + r223007;
        double r223009 = sqrt(r223008);
        double r223010 = r223005 + r223009;
        double r223011 = log(r223010);
        return r223011;
}
↓
double f(double x) {
        double r223012 = x;
        double r223013 = -1.0384538355180963;
        bool r223014 = r223012 <= r223013;
        double r223015 = 0.125;
        double r223016 = 3.0;
        double r223017 = pow(r223012, r223016);
        double r223018 = r223015 / r223017;
        double r223019 = 0.5;
        double r223020 = r223019 / r223012;
        double r223021 = 0.0625;
        double r223022 = -r223021;
        double r223023 = 5.0;
        double r223024 = pow(r223012, r223023);
        double r223025 = r223022 / r223024;
        double r223026 = r223020 - r223025;
        double r223027 = r223018 - r223026;
        double r223028 = log(r223027);
        double r223029 = 0.0012127302794364438;
        bool r223030 = r223012 <= r223029;
        double r223031 = 1.0;
        double r223032 = sqrt(r223031);
        double r223033 = log(r223032);
        double r223034 = r223012 / r223032;
        double r223035 = r223033 + r223034;
        double r223036 = 0.16666666666666666;
        double r223037 = pow(r223032, r223016);
        double r223038 = r223017 / r223037;
        double r223039 = r223036 * r223038;
        double r223040 = r223035 - r223039;
        double r223041 = hypot(r223012, r223032);
        double r223042 = r223041 + r223012;
        double r223043 = log(r223042);
        double r223044 = r223030 ? r223040 : r223043;
        double r223045 = r223014 ? r223028 : r223044;
        return r223045;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original52.9
Target44.8
Herbie0.1
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.0384538355180963

    1. Initial program 63.0

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

    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 -1.0384538355180963 < x < 0.0012127302794364438

    1. Initial program 58.8

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

    2. Taylor expanded around 0 0.1

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

    if 0.0012127302794364438 < x

    1. Initial program 32.3

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

    3. Applied add-log-exp32.3

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

    4. Simplified0.1

      \[\leadsto \log \color{blue}{\left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.03845383551809634:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.00121273027943644379:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))))