Average Error: 53.0 → 0.1
Time: 12.9s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.02086207945877905:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 9.5203500651170323 \cdot 10^{-4}:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.02086207945877905:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 9.5203500651170323 \cdot 10^{-4}:\\
\;\;\;\;\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 r199061 = x;
        double r199062 = r199061 * r199061;
        double r199063 = 1.0;
        double r199064 = r199062 + r199063;
        double r199065 = sqrt(r199064);
        double r199066 = r199061 + r199065;
        double r199067 = log(r199066);
        return r199067;
}
↓
double f(double x) {
        double r199068 = x;
        double r199069 = -1.020862079458779;
        bool r199070 = r199068 <= r199069;
        double r199071 = 0.125;
        double r199072 = 3.0;
        double r199073 = pow(r199068, r199072);
        double r199074 = r199071 / r199073;
        double r199075 = 0.5;
        double r199076 = r199075 / r199068;
        double r199077 = 0.0625;
        double r199078 = -r199077;
        double r199079 = 5.0;
        double r199080 = pow(r199068, r199079);
        double r199081 = r199078 / r199080;
        double r199082 = r199076 - r199081;
        double r199083 = r199074 - r199082;
        double r199084 = log(r199083);
        double r199085 = 0.0009520350065117032;
        bool r199086 = r199068 <= r199085;
        double r199087 = 1.0;
        double r199088 = sqrt(r199087);
        double r199089 = log(r199088);
        double r199090 = r199068 / r199088;
        double r199091 = r199089 + r199090;
        double r199092 = 0.16666666666666666;
        double r199093 = pow(r199088, r199072);
        double r199094 = r199073 / r199093;
        double r199095 = r199092 * r199094;
        double r199096 = r199091 - r199095;
        double r199097 = hypot(r199068, r199088);
        double r199098 = r199097 + r199068;
        double r199099 = log(r199098);
        double r199100 = r199086 ? r199096 : r199099;
        double r199101 = r199070 ? r199084 : r199100;
        return r199101;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original53.0
Target45.1
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 < -1.020862079458779

    1. Initial program 63.1

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

    2. Taylor expanded around -inf 0.1

      \[\leadsto \log \color{blue}{\left(0.125 \cdot \frac{1}{{x}^{3}} - \left(0.5 \cdot \frac{1}{x} + 0.0625 \cdot \frac{1}{{x}^{5}}\right)\right)}\]

    3. Simplified0.1

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

    if -1.020862079458779 < x < 0.0009520350065117032

    1. Initial program 59.0

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

    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 0.0009520350065117032 < x

    1. Initial program 32.2

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

    3. Applied add-log-exp32.2

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.02086207945877905:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.5}{x} - \frac{-0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 9.5203500651170323 \cdot 10^{-4}:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{hypot}\left(x, \sqrt{1}\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +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)))))