Jmat.Real.erf

Average Error: 13.8 → 13.7

Time: 52.7s

Precision: 64

Math

\[1 - \left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

↓

\[\frac{\frac{{\left(1 \cdot 1\right)}^{3} - {\left(\left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right) \cdot \left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right)\right)}^{3}}{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(\left(\left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right) \cdot \left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right)\right) \cdot \left(\left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right) \cdot \left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right)\right) + \left(1 \cdot 1\right) \cdot \left(\left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right) \cdot \left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right)\right)\right)}}{1 + \frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}\]

C

double f(double x) {
        double r11881911 = 1.0;
        double r11881912 = 0.3275911;
        double r11881913 = x;
        double r11881914 = fabs(r11881913);
        double r11881915 = r11881912 * r11881914;
        double r11881916 = r11881911 + r11881915;
        double r11881917 = r11881911 / r11881916;
        double r11881918 = 0.254829592;
        double r11881919 = -0.284496736;
        double r11881920 = 1.421413741;
        double r11881921 = -1.453152027;
        double r11881922 = 1.061405429;
        double r11881923 = r11881917 * r11881922;
        double r11881924 = r11881921 + r11881923;
        double r11881925 = r11881917 * r11881924;
        double r11881926 = r11881920 + r11881925;
        double r11881927 = r11881917 * r11881926;
        double r11881928 = r11881919 + r11881927;
        double r11881929 = r11881917 * r11881928;
        double r11881930 = r11881918 + r11881929;
        double r11881931 = r11881917 * r11881930;
        double r11881932 = r11881914 * r11881914;
        double r11881933 = -r11881932;
        double r11881934 = exp(r11881933);
        double r11881935 = r11881931 * r11881934;
        double r11881936 = r11881911 - r11881935;
        return r11881936;
}

↓

double f(double x) {
        double r11881937 = 1.0;
        double r11881938 = r11881937 * r11881937;
        double r11881939 = 3.0;
        double r11881940 = pow(r11881938, r11881939);
        double r11881941 = 0.254829592;
        double r11881942 = 0.3275911;
        double r11881943 = x;
        double r11881944 = fabs(r11881943);
        double r11881945 = r11881942 * r11881944;
        double r11881946 = r11881937 + r11881945;
        double r11881947 = r11881937 / r11881946;
        double r11881948 = -0.284496736;
        double r11881949 = 1.421413741;
        double r11881950 = -1.453152027;
        double r11881951 = r11881945 * r11881945;
        double r11881952 = r11881938 - r11881951;
        double r11881953 = r11881937 / r11881952;
        double r11881954 = r11881937 - r11881945;
        double r11881955 = r11881953 * r11881954;
        double r11881956 = 1.061405429;
        double r11881957 = r11881955 * r11881956;
        double r11881958 = r11881950 + r11881957;
        double r11881959 = r11881947 * r11881958;
        double r11881960 = r11881949 + r11881959;
        double r11881961 = r11881947 * r11881960;
        double r11881962 = r11881948 + r11881961;
        double r11881963 = r11881947 * r11881962;
        double r11881964 = r11881941 + r11881963;
        double r11881965 = r11881944 * r11881944;
        double r11881966 = exp(r11881965);
        double r11881967 = r11881964 / r11881966;
        double r11881968 = r11881967 * r11881947;
        double r11881969 = r11881968 * r11881968;
        double r11881970 = pow(r11881969, r11881939);
        double r11881971 = r11881940 - r11881970;
        double r11881972 = r11881938 * r11881938;
        double r11881973 = r11881969 * r11881969;
        double r11881974 = r11881938 * r11881969;
        double r11881975 = r11881973 + r11881974;
        double r11881976 = r11881972 + r11881975;
        double r11881977 = r11881971 / r11881976;
        double r11881978 = r11881937 + r11881968;
        double r11881979 = r11881977 / r11881978;
        return r11881979;
}

Error

Bits error versus x

Derivation

1. Initial program 13.8

   \[1 - \left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

2. Simplified 13.8

   \[\leadsto \color{blue}{1 - \frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}\]
3. Using strategy rm

4. Applied flip-+ 13.8

   \[\leadsto 1 - \frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{\color{blue}{\frac{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)}{1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\]

5. Applied associate-/r/ 13.8

   \[\leadsto 1 - \frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \color{blue}{\left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right)} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\]
6. Using strategy rm

7. Applied flip-- 13.8

   \[\leadsto \color{blue}{\frac{1 \cdot 1 - \left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right) \cdot \left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right)}{1 + \frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}}\]
8. Using strategy rm

9. Applied flip3-- 13.7

   \[\leadsto \frac{\color{blue}{\frac{{\left(1 \cdot 1\right)}^{3} - {\left(\left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right) \cdot \left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right)\right)}^{3}}{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(\left(\left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right) \cdot \left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right)\right) \cdot \left(\left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right) \cdot \left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right)\right) + \left(1 \cdot 1\right) \cdot \left(\left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right) \cdot \left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right)\right)\right)}}}{1 + \frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}\]

10. Final simplification 13.7

    \[\leadsto \frac{\frac{{\left(1 \cdot 1\right)}^{3} - {\left(\left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right) \cdot \left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right)\right)}^{3}}{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(\left(\left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right) \cdot \left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right)\right) \cdot \left(\left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right) \cdot \left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right)\right) + \left(1 \cdot 1\right) \cdot \left(\left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right) \cdot \left(\frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right)\right)\right)}}{1 + \frac{0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \left(\frac{1}{1 \cdot 1 - \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right) \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911000000000239396058532292954623699 \cdot \left|x\right|\right)\right) \cdot 1.061405428999999900341322245367337018251\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "Jmat.Real.erf"
  (- 1.0 (* (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))