Average Error: 53.1 → 0.2
Time: 12.8s
Precision: 64
\[\log \left(x + \sqrt{x \cdot x + 1}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.02484362312584243:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{x}} \cdot \sqrt[3]{\frac{0.5}{x}}, \sqrt[3]{\frac{0.5}{x}}, -\sqrt[3]{\frac{-0.0625}{{x}^{5}}} \cdot \left(\sqrt[3]{\frac{-0.0625}{{x}^{5}}} \cdot \sqrt[3]{\frac{-0.0625}{{x}^{5}}}\right)\right)\right) - \mathsf{fma}\left(-\sqrt[3]{\frac{-0.0625}{{x}^{5}}}, \sqrt[3]{\frac{-0.0625}{{x}^{5}}} \cdot \sqrt[3]{\frac{-0.0625}{{x}^{5}}}, \sqrt[3]{\frac{-0.0625}{{x}^{5}}} \cdot \left(\sqrt[3]{\frac{-0.0625}{{x}^{5}}} \cdot \sqrt[3]{\frac{-0.0625}{{x}^{5}}}\right)\right)\right)\\ \mathbf{elif}\;x \le 9.28347963773585075 \cdot 10^{-4}:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(x, \sqrt{1}\right)\right)\\ \end{array}\]
\log \left(x + \sqrt{x \cdot x + 1}\right)
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.02484362312584243:\\
\;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{x}} \cdot \sqrt[3]{\frac{0.5}{x}}, \sqrt[3]{\frac{0.5}{x}}, -\sqrt[3]{\frac{-0.0625}{{x}^{5}}} \cdot \left(\sqrt[3]{\frac{-0.0625}{{x}^{5}}} \cdot \sqrt[3]{\frac{-0.0625}{{x}^{5}}}\right)\right)\right) - \mathsf{fma}\left(-\sqrt[3]{\frac{-0.0625}{{x}^{5}}}, \sqrt[3]{\frac{-0.0625}{{x}^{5}}} \cdot \sqrt[3]{\frac{-0.0625}{{x}^{5}}}, \sqrt[3]{\frac{-0.0625}{{x}^{5}}} \cdot \left(\sqrt[3]{\frac{-0.0625}{{x}^{5}}} \cdot \sqrt[3]{\frac{-0.0625}{{x}^{5}}}\right)\right)\right)\\

\mathbf{elif}\;x \le 9.28347963773585075 \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(x + \mathsf{hypot}\left(x, \sqrt{1}\right)\right)\\

\end{array}
double f(double x) {
        double r201891 = x;
        double r201892 = r201891 * r201891;
        double r201893 = 1.0;
        double r201894 = r201892 + r201893;
        double r201895 = sqrt(r201894);
        double r201896 = r201891 + r201895;
        double r201897 = log(r201896);
        return r201897;
}
↓
double f(double x) {
        double r201898 = x;
        double r201899 = -1.0248436231258424;
        bool r201900 = r201898 <= r201899;
        double r201901 = 0.125;
        double r201902 = 3.0;
        double r201903 = pow(r201898, r201902);
        double r201904 = r201901 / r201903;
        double r201905 = 0.5;
        double r201906 = r201905 / r201898;
        double r201907 = cbrt(r201906);
        double r201908 = r201907 * r201907;
        double r201909 = 0.0625;
        double r201910 = -r201909;
        double r201911 = 5.0;
        double r201912 = pow(r201898, r201911);
        double r201913 = r201910 / r201912;
        double r201914 = cbrt(r201913);
        double r201915 = r201914 * r201914;
        double r201916 = r201914 * r201915;
        double r201917 = -r201916;
        double r201918 = fma(r201908, r201907, r201917);
        double r201919 = r201904 - r201918;
        double r201920 = -r201914;
        double r201921 = fma(r201920, r201915, r201916);
        double r201922 = r201919 - r201921;
        double r201923 = log(r201922);
        double r201924 = 0.0009283479637735851;
        bool r201925 = r201898 <= r201924;
        double r201926 = 1.0;
        double r201927 = sqrt(r201926);
        double r201928 = log(r201927);
        double r201929 = r201898 / r201927;
        double r201930 = r201928 + r201929;
        double r201931 = 0.16666666666666666;
        double r201932 = pow(r201927, r201902);
        double r201933 = r201903 / r201932;
        double r201934 = r201931 * r201933;
        double r201935 = r201930 - r201934;
        double r201936 = hypot(r201898, r201927);
        double r201937 = r201898 + r201936;
        double r201938 = log(r201937);
        double r201939 = r201925 ? r201935 : r201938;
        double r201940 = r201900 ? r201923 : r201939;
        return r201940;
}

Error

Bits error versus x

Target

Original53.1
Target45.1
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 < -1.0248436231258424

    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

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

    5. Applied add-cube-cbrt0.2

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

    6. Applied add-cube-cbrt0.2

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

    7. Applied prod-diff0.2

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

    8. Applied associate--r+0.2

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

    if -1.0248436231258424 < x < 0.0009283479637735851

    1. Initial program 58.8

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

    2. Taylor expanded around 0 0.2

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

    if 0.0009283479637735851 < x

    1. Initial program 31.7

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

    3. Applied add-sqr-sqrt31.7

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

    4. Applied hypot-def0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.02484362312584243:\\ \;\;\;\;\log \left(\left(\frac{0.125}{{x}^{3}} - \mathsf{fma}\left(\sqrt[3]{\frac{0.5}{x}} \cdot \sqrt[3]{\frac{0.5}{x}}, \sqrt[3]{\frac{0.5}{x}}, -\sqrt[3]{\frac{-0.0625}{{x}^{5}}} \cdot \left(\sqrt[3]{\frac{-0.0625}{{x}^{5}}} \cdot \sqrt[3]{\frac{-0.0625}{{x}^{5}}}\right)\right)\right) - \mathsf{fma}\left(-\sqrt[3]{\frac{-0.0625}{{x}^{5}}}, \sqrt[3]{\frac{-0.0625}{{x}^{5}}} \cdot \sqrt[3]{\frac{-0.0625}{{x}^{5}}}, \sqrt[3]{\frac{-0.0625}{{x}^{5}}} \cdot \left(\sqrt[3]{\frac{-0.0625}{{x}^{5}}} \cdot \sqrt[3]{\frac{-0.0625}{{x}^{5}}}\right)\right)\right)\\ \mathbf{elif}\;x \le 9.28347963773585075 \cdot 10^{-4}:\\ \;\;\;\;\left(\log \left(\sqrt{1}\right) + \frac{x}{\sqrt{1}}\right) - \frac{1}{6} \cdot \frac{{x}^{3}}{{\left(\sqrt{1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + \mathsf{hypot}\left(x, \sqrt{1}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 +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)))))