Average Error: 52.7 → 0.2
Time: 31.1s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.014477909773237040980120582389645278454:\\ \;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.8909481582438814051272402139147743582726:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x \cdot x}{1} \cdot x}{\sqrt{1}}, \frac{-1}{6}, \frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x} - \frac{\frac{0.125}{x}}{x \cdot x}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.014477909773237040980120582389645278454:\\
\;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\

\mathbf{elif}\;x \le 0.8909481582438814051272402139147743582726:\\
\;\;\;\;\mathsf{fma}\left(\frac{\frac{x \cdot x}{1} \cdot x}{\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} - \frac{\frac{0.125}{x}}{x \cdot x}\right)\right)\\

\end{array}
double f(double x) {
        double r5867837 = x;
        double r5867838 = r5867837 * r5867837;
        double r5867839 = 1.0;
        double r5867840 = r5867838 + r5867839;
        double r5867841 = sqrt(r5867840);
        double r5867842 = r5867837 + r5867841;
        double r5867843 = log(r5867842);
        return r5867843;
}
↓
double f(double x) {
        double r5867844 = x;
        double r5867845 = -1.014477909773237;
        bool r5867846 = r5867844 <= r5867845;
        double r5867847 = 0.125;
        double r5867848 = r5867847 / r5867844;
        double r5867849 = r5867844 * r5867844;
        double r5867850 = r5867848 / r5867849;
        double r5867851 = 0.5;
        double r5867852 = r5867851 / r5867844;
        double r5867853 = 0.0625;
        double r5867854 = 5.0;
        double r5867855 = pow(r5867844, r5867854);
        double r5867856 = r5867853 / r5867855;
        double r5867857 = r5867852 + r5867856;
        double r5867858 = r5867850 - r5867857;
        double r5867859 = log(r5867858);
        double r5867860 = 0.8909481582438814;
        bool r5867861 = r5867844 <= r5867860;
        double r5867862 = 1.0;
        double r5867863 = r5867849 / r5867862;
        double r5867864 = r5867863 * r5867844;
        double r5867865 = sqrt(r5867862);
        double r5867866 = r5867864 / r5867865;
        double r5867867 = -0.16666666666666666;
        double r5867868 = r5867844 / r5867865;
        double r5867869 = log(r5867865);
        double r5867870 = r5867868 + r5867869;
        double r5867871 = fma(r5867866, r5867867, r5867870);
        double r5867872 = 2.0;
        double r5867873 = r5867852 - r5867850;
        double r5867874 = fma(r5867844, r5867872, r5867873);
        double r5867875 = log(r5867874);
        double r5867876 = r5867861 ? r5867871 : r5867875;
        double r5867877 = r5867846 ? r5867859 : r5867876;
        return r5867877;
}

Error

Bits error versus x

Target

Original52.7
Target44.9
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 0.0:\\ \;\;\;\;\log \left(\frac{-1}{x - \sqrt{x \cdot x + 1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \sqrt{x \cdot x + 1}\right)\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.014477909773237

    1. Initial program 63.0

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

    2. Simplified63.0

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

    3. Taylor expanded around -inf 0.2

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

    4. Simplified0.2

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

    if -1.014477909773237 < x < 0.8909481582438814

    1. Initial program 58.4

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

    2. Simplified58.4

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

    3. Taylor expanded around 0 0.3

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

    4. Simplified0.3

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

    if 0.8909481582438814 < x

    1. Initial program 31.4

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

    2. Simplified31.4

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

    3. Taylor expanded around inf 0.1

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

    4. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.014477909773237040980120582389645278454:\\ \;\;\;\;\log \left(\frac{\frac{0.125}{x}}{x \cdot x} - \left(\frac{0.5}{x} + \frac{0.0625}{{x}^{5}}\right)\right)\\ \mathbf{elif}\;x \le 0.8909481582438814051272402139147743582726:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x \cdot x}{1} \cdot x}{\sqrt{1}}, \frac{-1}{6}, \frac{x}{\sqrt{1}} + \log \left(\sqrt{1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\mathsf{fma}\left(x, 2, \frac{0.5}{x} - \frac{\frac{0.125}{x}}{x \cdot x}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 +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)))))