Jmat.Real.erf

Average Error: 13.6 → 10.4

Time: 24.6s

Precision: 64

Math

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

↓

\[\frac{1 \cdot 1 + \left(-\left(\left(\sqrt{\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\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 \sqrt{\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\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)}\right) \cdot \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\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)\right)\right) \cdot \left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)\right)}{1 + \left(1 \cdot \frac{\left(\left(1.0614054289999999 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}} + \left(1.42141374100000006 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + 0.25482959199999999\right)\right) - \left(1.45315202700000001 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}} + 0.284496735999999972 \cdot \frac{1}{0.32759110000000002 \cdot \left|x\right| + 1}\right)\right) \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)}{1 - 0.107315928799210022 \cdot {\left(\left|x\right|\right)}^{2}}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\]

C

double f(double x) {
        double r318916 = 1.0;
        double r318917 = 0.3275911;
        double r318918 = x;
        double r318919 = fabs(r318918);
        double r318920 = r318917 * r318919;
        double r318921 = r318916 + r318920;
        double r318922 = r318916 / r318921;
        double r318923 = 0.254829592;
        double r318924 = -0.284496736;
        double r318925 = 1.421413741;
        double r318926 = -1.453152027;
        double r318927 = 1.061405429;
        double r318928 = r318922 * r318927;
        double r318929 = r318926 + r318928;
        double r318930 = r318922 * r318929;
        double r318931 = r318925 + r318930;
        double r318932 = r318922 * r318931;
        double r318933 = r318924 + r318932;
        double r318934 = r318922 * r318933;
        double r318935 = r318923 + r318934;
        double r318936 = r318922 * r318935;
        double r318937 = r318919 * r318919;
        double r318938 = -r318937;
        double r318939 = exp(r318938);
        double r318940 = r318936 * r318939;
        double r318941 = r318916 - r318940;
        return r318941;
}

↓

double f(double x) {
        double r318942 = 1.0;
        double r318943 = r318942 * r318942;
        double r318944 = 0.3275911;
        double r318945 = x;
        double r318946 = fabs(r318945);
        double r318947 = r318944 * r318946;
        double r318948 = r318947 * r318947;
        double r318949 = r318943 - r318948;
        double r318950 = r318942 / r318949;
        double r318951 = r318942 - r318947;
        double r318952 = 0.254829592;
        double r318953 = r318942 + r318947;
        double r318954 = r318942 / r318953;
        double r318955 = -0.284496736;
        double r318956 = 1.421413741;
        double r318957 = -1.453152027;
        double r318958 = 1.061405429;
        double r318959 = r318954 * r318958;
        double r318960 = r318957 + r318959;
        double r318961 = r318954 * r318960;
        double r318962 = r318956 + r318961;
        double r318963 = r318954 * r318962;
        double r318964 = r318955 + r318963;
        double r318965 = r318954 * r318964;
        double r318966 = r318952 + r318965;
        double r318967 = r318951 * r318966;
        double r318968 = r318950 * r318967;
        double r318969 = sqrt(r318968);
        double r318970 = r318969 * r318969;
        double r318971 = r318970 * r318968;
        double r318972 = 1.0;
        double r318973 = 2.0;
        double r318974 = pow(r318946, r318973);
        double r318975 = exp(r318974);
        double r318976 = r318972 / r318975;
        double r318977 = r318976 * r318976;
        double r318978 = r318971 * r318977;
        double r318979 = -r318978;
        double r318980 = r318943 + r318979;
        double r318981 = r318947 + r318942;
        double r318982 = 4.0;
        double r318983 = pow(r318981, r318982);
        double r318984 = r318972 / r318983;
        double r318985 = r318958 * r318984;
        double r318986 = pow(r318981, r318973);
        double r318987 = r318972 / r318986;
        double r318988 = r318956 * r318987;
        double r318989 = r318988 + r318952;
        double r318990 = r318985 + r318989;
        double r318991 = 1.453152027;
        double r318992 = 3.0;
        double r318993 = pow(r318981, r318992);
        double r318994 = r318972 / r318993;
        double r318995 = r318991 * r318994;
        double r318996 = 0.284496736;
        double r318997 = r318972 / r318981;
        double r318998 = r318996 * r318997;
        double r318999 = r318995 + r318998;
        double r319000 = r318990 - r318999;
        double r319001 = r319000 * r318951;
        double r319002 = 0.10731592879921002;
        double r319003 = r319002 * r318974;
        double r319004 = r318942 - r319003;
        double r319005 = r319001 / r319004;
        double r319006 = r318942 * r319005;
        double r319007 = r318946 * r318946;
        double r319008 = -r319007;
        double r319009 = exp(r319008);
        double r319010 = r319006 * r319009;
        double r319011 = r318942 + r319010;
        double r319012 = r318980 / r319011;
        return r319012;
}

Error

Bits error versus x

Derivation

1. Initial program 13.6

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

3. Applied flip-+ 13.6

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

4. Applied associate-/r/ 13.6

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

5. Applied associate-*l* 13.6

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

7. Applied flip-- 13.6

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

8. Simplified 13.6

   \[\leadsto \frac{\color{blue}{1 \cdot 1 + \left(-\left(\left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\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)\right) \cdot \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\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)\right)\right) \cdot \left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)\right)}}{1 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\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)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\]
9. Using strategy rm

10. Applied add-sqr-sqrt 10.4

    \[\leadsto \frac{1 \cdot 1 + \left(-\left(\color{blue}{\left(\sqrt{\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\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 \sqrt{\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\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)}\right)} \cdot \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\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)\right)\right) \cdot \left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)\right)}{1 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\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)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\]

11. Taylor expanded around 0 10.4

    \[\leadsto \frac{1 \cdot 1 + \left(-\left(\left(\sqrt{\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\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 \sqrt{\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\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)}\right) \cdot \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\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)\right)\right) \cdot \left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)\right)}{1 + \color{blue}{\left(1 \cdot \frac{\left(\left(1.0614054289999999 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}} + \left(1.42141374100000006 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + 0.25482959199999999\right)\right) - \left(1.45315202700000001 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}} + 0.284496735999999972 \cdot \frac{1}{0.32759110000000002 \cdot \left|x\right| + 1}\right)\right) \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)}{1 - 0.107315928799210022 \cdot {\left(\left|x\right|\right)}^{2}}\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|}}\]

12. Final simplification 10.4

    \[\leadsto \frac{1 \cdot 1 + \left(-\left(\left(\sqrt{\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\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 \sqrt{\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\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)}\right) \cdot \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot \left|x\right|\right) \cdot \left(0.32759110000000002 \cdot \left|x\right|\right)} \cdot \left(\left(1 - 0.32759110000000002 \cdot \left|x\right|\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)\right)\right) \cdot \left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)\right)}{1 + \left(1 \cdot \frac{\left(\left(1.0614054289999999 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}} + \left(1.42141374100000006 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + 0.25482959199999999\right)\right) - \left(1.45315202700000001 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}} + 0.284496735999999972 \cdot \frac{1}{0.32759110000000002 \cdot \left|x\right| + 1}\right)\right) \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)}{1 - 0.107315928799210022 \cdot {\left(\left|x\right|\right)}^{2}}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\]

Reproduce

herbie shell --seed 2020049 
(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)))))))