Average Error: 52.9 → 0.3
Time: 14.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.020049338560968577027665560308378189802:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.8895740307587358675078803571523167192936:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{{x}^{3}}{1}}{\sqrt{1}}, \frac{-1}{6}, \frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.020049338560968577027665560308378189802:\\
\;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\

\mathbf{elif}\;x \le 0.8895740307587358675078803571523167192936:\\
\;\;\;\;\mathsf{fma}\left(\frac{\frac{{x}^{3}}{1}}{\sqrt{1}}, \frac{-1}{6}, \frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\\

\end{array}
double f(double x) {
        double r141832 = x;
        double r141833 = r141832 * r141832;
        double r141834 = 1.0;
        double r141835 = r141833 + r141834;
        double r141836 = sqrt(r141835);
        double r141837 = r141832 + r141836;
        double r141838 = log(r141837);
        return r141838;
}
↓
double f(double x) {
        double r141839 = x;
        double r141840 = -1.0200493385609686;
        bool r141841 = r141839 <= r141840;
        double r141842 = 0.125;
        double r141843 = 3.0;
        double r141844 = pow(r141839, r141843);
        double r141845 = r141842 / r141844;
        double r141846 = 0.5;
        double r141847 = r141846 / r141839;
        double r141848 = r141845 - r141847;
        double r141849 = 0.0625;
        double r141850 = 5.0;
        double r141851 = pow(r141839, r141850);
        double r141852 = r141849 / r141851;
        double r141853 = r141848 - r141852;
        double r141854 = log(r141853);
        double r141855 = 0.8895740307587359;
        bool r141856 = r141839 <= r141855;
        double r141857 = 1.0;
        double r141858 = r141844 / r141857;
        double r141859 = sqrt(r141857);
        double r141860 = r141858 / r141859;
        double r141861 = -0.16666666666666666;
        double r141862 = r141839 / r141859;
        double r141863 = log(r141859);
        double r141864 = r141862 + r141863;
        double r141865 = fma(r141860, r141861, r141864);
        double r141866 = 2.0;
        double r141867 = fma(r141839, r141866, r141847);
        double r141868 = r141867 - r141845;
        double r141869 = log(r141868);
        double r141870 = r141856 ? r141865 : r141869;
        double r141871 = r141841 ? r141854 : r141870;
        return r141871;
}

Error

Bits error versus x

Target

Original52.9
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 < -1.0200493385609686

    1. Initial program 62.7

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

    2. Simplified62.7

      \[\leadsto \color{blue}{\log \left(x + \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)}\]

    3. Taylor expanded around -inf 0.2

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

    4. Simplified0.2

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

    if -1.0200493385609686 < x < 0.8895740307587359

    1. Initial program 58.6

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

    2. Simplified58.6

      \[\leadsto \color{blue}{\log \left(x + \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)}\]

    3. Taylor expanded around 0 0.3

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

    4. Simplified0.3

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

    if 0.8895740307587359 < x

    1. Initial program 31.1

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

    2. Simplified31.1

      \[\leadsto \color{blue}{\log \left(x + \sqrt{\mathsf{fma}\left(x, x, 1\right)}\right)}\]

    3. Taylor expanded around inf 0.3

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

    4. Simplified0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.020049338560968577027665560308378189802:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \frac{0.5}{x}\right) - \frac{0.0625}{{x}^{5}}\right)\\ \mathbf{elif}\;x \le 0.8895740307587358675078803571523167192936:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{{x}^{3}}{1}}{\sqrt{1}}, \frac{-1}{6}, \frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x}\right) - \frac{0.125}{{x}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic arcsine"

  :herbie-target
  (if (< x 0.0) (log (/ -1.0 (- x (sqrt (+ (* x x) 1.0))))) (log (+ x (sqrt (+ (* x x) 1.0)))))

  (log (+ x (sqrt (+ (* x x) 1.0)))))