Jmat.Real.erf

Average Error: 14.0 → 14.0

Time: 13.6s

Precision: 64

Math

\[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

↓

\[{\left(e^{\sqrt[3]{\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \log \left(e^{\frac{1}{0.32759110000000002 \cdot \left|x\right| + 1}}\right) \cdot \left(1.42141374100000006 + \left(\frac{-1.45315202700000001 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} + \frac{\frac{1.0614054289999999 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)} \cdot \sqrt[3]{\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \log \left(e^{\frac{1}{0.32759110000000002 \cdot \left|x\right| + 1}}\right) \cdot \left(1.42141374100000006 + \left(\frac{-1.45315202700000001 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} + \frac{\frac{1.0614054289999999 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}}\right)}^{\left(\sqrt[3]{\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \log \left(e^{\frac{1}{0.32759110000000002 \cdot \left|x\right| + 1}}\right) \cdot \left(1.42141374100000006 + \left(\frac{-1.45315202700000001 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} + \frac{\frac{1.0614054289999999 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}\right)}\]

C

double f(double x) {
        double r206881 = 1.0;
        double r206882 = 0.3275911;
        double r206883 = x;
        double r206884 = fabs(r206883);
        double r206885 = r206882 * r206884;
        double r206886 = r206881 + r206885;
        double r206887 = r206881 / r206886;
        double r206888 = 0.254829592;
        double r206889 = -0.284496736;
        double r206890 = 1.421413741;
        double r206891 = -1.453152027;
        double r206892 = 1.061405429;
        double r206893 = r206887 * r206892;
        double r206894 = r206891 + r206893;
        double r206895 = r206887 * r206894;
        double r206896 = r206890 + r206895;
        double r206897 = r206887 * r206896;
        double r206898 = r206889 + r206897;
        double r206899 = r206887 * r206898;
        double r206900 = r206888 + r206899;
        double r206901 = r206887 * r206900;
        double r206902 = r206884 * r206884;
        double r206903 = -r206902;
        double r206904 = exp(r206903);
        double r206905 = r206901 * r206904;
        double r206906 = r206881 - r206905;
        return r206906;
}

↓

double f(double x) {
        double r206907 = 1.0;
        double r206908 = 0.3275911;
        double r206909 = x;
        double r206910 = fabs(r206909);
        double r206911 = r206908 * r206910;
        double r206912 = r206907 + r206911;
        double r206913 = r206907 / r206912;
        double r206914 = 0.254829592;
        double r206915 = -0.284496736;
        double r206916 = r206911 + r206907;
        double r206917 = r206907 / r206916;
        double r206918 = exp(r206917);
        double r206919 = log(r206918);
        double r206920 = 1.421413741;
        double r206921 = -1.453152027;
        double r206922 = r206921 * r206907;
        double r206923 = fma(r206908, r206910, r206907);
        double r206924 = r206922 / r206923;
        double r206925 = 1.061405429;
        double r206926 = r206925 * r206907;
        double r206927 = r206926 / r206923;
        double r206928 = r206927 * r206907;
        double r206929 = r206928 / r206923;
        double r206930 = r206924 + r206929;
        double r206931 = r206920 + r206930;
        double r206932 = r206919 * r206931;
        double r206933 = r206915 + r206932;
        double r206934 = r206913 * r206933;
        double r206935 = r206914 + r206934;
        double r206936 = r206913 * r206935;
        double r206937 = r206910 * r206910;
        double r206938 = -r206937;
        double r206939 = exp(r206938);
        double r206940 = r206936 * r206939;
        double r206941 = r206907 - r206940;
        double r206942 = log(r206941);
        double r206943 = cbrt(r206942);
        double r206944 = r206943 * r206943;
        double r206945 = exp(r206944);
        double r206946 = pow(r206945, r206943);
        return r206946;
}

Error

Bits error versus x

Derivation

1. Initial program 14.0

   \[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
2. Using strategy rm

3. Applied distribute-lft-in 14.0

   \[\leadsto 1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \color{blue}{\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

4. Simplified 14.0

   \[\leadsto 1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\color{blue}{\frac{-1.45315202700000001 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}} + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

5. Simplified 14.0

   \[\leadsto 1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{-1.45315202700000001 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} + \color{blue}{\frac{\frac{1.0614054289999999 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
6. Using strategy rm

7. Applied add-exp-log 14.0

   \[\leadsto \color{blue}{e^{\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{-1.45315202700000001 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} + \frac{\frac{1.0614054289999999 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}}\]
8. Using strategy rm

9. Applied add-log-exp 14.0

   \[\leadsto e^{\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \color{blue}{\log \left(e^{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)} \cdot \left(1.42141374100000006 + \left(\frac{-1.45315202700000001 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} + \frac{\frac{1.0614054289999999 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}\]

10. Simplified 14.0

    \[\leadsto e^{\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \log \color{blue}{\left(e^{\frac{1}{0.32759110000000002 \cdot \left|x\right| + 1}}\right)} \cdot \left(1.42141374100000006 + \left(\frac{-1.45315202700000001 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} + \frac{\frac{1.0614054289999999 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}\]
11. Using strategy rm

12. Applied add-cube-cbrt 14.0

    \[\leadsto e^{\color{blue}{\left(\sqrt[3]{\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \log \left(e^{\frac{1}{0.32759110000000002 \cdot \left|x\right| + 1}}\right) \cdot \left(1.42141374100000006 + \left(\frac{-1.45315202700000001 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} + \frac{\frac{1.0614054289999999 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)} \cdot \sqrt[3]{\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \log \left(e^{\frac{1}{0.32759110000000002 \cdot \left|x\right| + 1}}\right) \cdot \left(1.42141374100000006 + \left(\frac{-1.45315202700000001 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} + \frac{\frac{1.0614054289999999 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}\right) \cdot \sqrt[3]{\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \log \left(e^{\frac{1}{0.32759110000000002 \cdot \left|x\right| + 1}}\right) \cdot \left(1.42141374100000006 + \left(\frac{-1.45315202700000001 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} + \frac{\frac{1.0614054289999999 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}}}\]

13. Applied exp-prod 14.0

    \[\leadsto \color{blue}{{\left(e^{\sqrt[3]{\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \log \left(e^{\frac{1}{0.32759110000000002 \cdot \left|x\right| + 1}}\right) \cdot \left(1.42141374100000006 + \left(\frac{-1.45315202700000001 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} + \frac{\frac{1.0614054289999999 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)} \cdot \sqrt[3]{\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \log \left(e^{\frac{1}{0.32759110000000002 \cdot \left|x\right| + 1}}\right) \cdot \left(1.42141374100000006 + \left(\frac{-1.45315202700000001 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} + \frac{\frac{1.0614054289999999 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}}\right)}^{\left(\sqrt[3]{\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \log \left(e^{\frac{1}{0.32759110000000002 \cdot \left|x\right| + 1}}\right) \cdot \left(1.42141374100000006 + \left(\frac{-1.45315202700000001 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} + \frac{\frac{1.0614054289999999 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}\right)}}\]

14. Final simplification 14.0

    \[\leadsto {\left(e^{\sqrt[3]{\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \log \left(e^{\frac{1}{0.32759110000000002 \cdot \left|x\right| + 1}}\right) \cdot \left(1.42141374100000006 + \left(\frac{-1.45315202700000001 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} + \frac{\frac{1.0614054289999999 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)} \cdot \sqrt[3]{\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \log \left(e^{\frac{1}{0.32759110000000002 \cdot \left|x\right| + 1}}\right) \cdot \left(1.42141374100000006 + \left(\frac{-1.45315202700000001 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} + \frac{\frac{1.0614054289999999 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}}\right)}^{\left(\sqrt[3]{\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \log \left(e^{\frac{1}{0.32759110000000002 \cdot \left|x\right| + 1}}\right) \cdot \left(1.42141374100000006 + \left(\frac{-1.45315202700000001 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} + \frac{\frac{1.0614054289999999 \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} \cdot 1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}\right)}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 1 (* (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))