Jmat.Real.erf

Average Error: 13.8 → 13.1

Time: 10.1s

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(\sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)}\right) \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left({\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3}, \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, -0.284496735999999972\right), 0.25482959199999999\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}\right)}\]

C

double f(double x) {
        double r204833 = 1.0;
        double r204834 = 0.3275911;
        double r204835 = x;
        double r204836 = fabs(r204835);
        double r204837 = r204834 * r204836;
        double r204838 = r204833 + r204837;
        double r204839 = r204833 / r204838;
        double r204840 = 0.254829592;
        double r204841 = -0.284496736;
        double r204842 = 1.421413741;
        double r204843 = -1.453152027;
        double r204844 = 1.061405429;
        double r204845 = r204839 * r204844;
        double r204846 = r204843 + r204845;
        double r204847 = r204839 * r204846;
        double r204848 = r204842 + r204847;
        double r204849 = r204839 * r204848;
        double r204850 = r204841 + r204849;
        double r204851 = r204839 * r204850;
        double r204852 = r204840 + r204851;
        double r204853 = r204839 * r204852;
        double r204854 = r204836 * r204836;
        double r204855 = -r204854;
        double r204856 = exp(r204855);
        double r204857 = r204853 * r204856;
        double r204858 = r204833 - r204857;
        return r204858;
}

↓

double f(double x) {
        double r204859 = 1.0;
        double r204860 = 0.3275911;
        double r204861 = x;
        double r204862 = fabs(r204861);
        double r204863 = r204860 * r204862;
        double r204864 = r204859 + r204863;
        double r204865 = r204859 / r204864;
        double r204866 = cbrt(r204865);
        double r204867 = r204866 * r204866;
        double r204868 = r204867 * r204866;
        double r204869 = 1.061405429;
        double r204870 = -1.453152027;
        double r204871 = fma(r204865, r204869, r204870);
        double r204872 = 1.421413741;
        double r204873 = fma(r204868, r204871, r204872);
        double r204874 = -0.284496736;
        double r204875 = fma(r204865, r204873, r204874);
        double r204876 = 0.254829592;
        double r204877 = fma(r204865, r204875, r204876);
        double r204878 = r204862 * r204862;
        double r204879 = exp(r204878);
        double r204880 = r204877 / r204879;
        double r204881 = -r204859;
        double r204882 = fma(r204862, r204860, r204859);
        double r204883 = r204881 / r204882;
        double r204884 = fma(r204880, r204883, r204859);
        double r204885 = cbrt(r204884);
        double r204886 = r204885 * r204885;
        double r204887 = r204859 / r204882;
        double r204888 = 3.0;
        double r204889 = pow(r204866, r204888);
        double r204890 = fma(r204887, r204869, r204870);
        double r204891 = fma(r204889, r204890, r204872);
        double r204892 = fma(r204891, r204887, r204874);
        double r204893 = fma(r204887, r204892, r204876);
        double r204894 = r204887 * r204893;
        double r204895 = 2.0;
        double r204896 = pow(r204862, r204895);
        double r204897 = exp(r204896);
        double r204898 = r204894 / r204897;
        double r204899 = r204859 - r204898;
        double r204900 = exp(r204899);
        double r204901 = log(r204900);
        double r204902 = cbrt(r204901);
        double r204903 = r204886 * r204902;
        return r204903;
}

Error

Bits error versus x

Derivation

1. Initial program 13.8

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

2. Simplified 13.8

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

4. Applied add-cube-cbrt 13.8

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

6. Applied add-cube-cbrt 13.8

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

8. Applied add-log-exp 13.1

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

9. Simplified 13.1

   \[\leadsto \left(\sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)}\right) \cdot \sqrt[3]{\log \color{blue}{\left(e^{1 - \frac{\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left({\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3}, \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, -0.284496735999999972\right), 0.25482959199999999\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}\right)}}\]

10. Final simplification 13.1

    \[\leadsto \left(\sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)}\right) \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left({\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3}, \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, -0.284496735999999972\right), 0.25482959199999999\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}\right)}\]

Reproduce

herbie shell --seed 2020039 +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)))))))