Average Error: 53.1 → 0.2
Time: 19.2s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.002814715336328044159586170280817896128:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9017023301953626113203199565759859979153:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + x\right) + x\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.002814715336328044159586170280817896128:\\
\;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\

\mathbf{elif}\;x \le 0.9017023301953626113203199565759859979153:\\
\;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + x\right) + x\right)\\

\end{array}
double f(double x) {
        double r117782 = x;
        double r117783 = r117782 * r117782;
        double r117784 = 1.0;
        double r117785 = r117783 + r117784;
        double r117786 = sqrt(r117785);
        double r117787 = r117782 + r117786;
        double r117788 = log(r117787);
        return r117788;
}
↓
double f(double x) {
        double r117789 = x;
        double r117790 = -1.002814715336328;
        bool r117791 = r117789 <= r117790;
        double r117792 = 0.125;
        double r117793 = 3.0;
        double r117794 = pow(r117789, r117793);
        double r117795 = r117792 / r117794;
        double r117796 = 0.0625;
        double r117797 = 5.0;
        double r117798 = pow(r117789, r117797);
        double r117799 = r117796 / r117798;
        double r117800 = 0.5;
        double r117801 = r117800 / r117789;
        double r117802 = r117799 + r117801;
        double r117803 = r117795 - r117802;
        double r117804 = log(r117803);
        double r117805 = 0.9017023301953626;
        bool r117806 = r117789 <= r117805;
        double r117807 = 1.0;
        double r117808 = sqrt(r117807);
        double r117809 = pow(r117808, r117793);
        double r117810 = r117794 / r117809;
        double r117811 = -0.16666666666666666;
        double r117812 = log(r117808);
        double r117813 = r117789 / r117808;
        double r117814 = r117812 + r117813;
        double r117815 = fma(r117810, r117811, r117814);
        double r117816 = r117801 - r117795;
        double r117817 = r117816 + r117789;
        double r117818 = r117817 + r117789;
        double r117819 = log(r117818);
        double r117820 = r117806 ? r117815 : r117819;
        double r117821 = r117791 ? r117804 : r117820;
        return r117821;
}

Error

Bits error versus x

Target

Original53.1
Target45.2
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 < -1.002814715336328

    1. Initial program 63.1

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

    2. Simplified63.1

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

    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

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

    if -1.002814715336328 < x < 0.9017023301953626

    1. Initial program 58.7

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

    2. Simplified58.7

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

    3. Taylor expanded around 0 0.2

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

    4. Simplified0.2

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

    if 0.9017023301953626 < x

    1. Initial program 32.2

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

    2. Simplified32.2

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

    3. Taylor expanded around inf 0.1

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

    4. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.002814715336328044159586170280817896128:\\ \;\;\;\;\log \left(\frac{0.125}{{x}^{3}} - \left(\frac{0.0625}{{x}^{5}} + \frac{0.5}{x}\right)\right)\\ \mathbf{elif}\;x \le 0.9017023301953626113203199565759859979153:\\ \;\;\;\;\mathsf{fma}\left(\frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}, \frac{-1}{6}, \log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\left(\left(\frac{0.5}{x} - \frac{0.125}{{x}^{3}}\right) + x\right) + x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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)))))