Jmat.Real.erf

Average Error: 13.7 → 12.9

Time: 1.1m

Precision: 64

Math

\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

↓

\[\sqrt[3]{\left(\left(\log \left(\frac{1}{\sqrt[3]{e^{\frac{\frac{\left(0.254829592 + \frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911}\right) + \frac{\frac{\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}} \cdot \sqrt[3]{e^{\frac{\frac{\left(0.254829592 + \frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911}\right) + \frac{\frac{\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}}}\right) + \log \left(\frac{e}{\sqrt[3]{e^{\frac{\frac{\left(0.254829592 + \frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911}\right) + \frac{\frac{\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}}}\right)\right) \cdot \left(1 - \frac{\frac{\left(0.254829592 + \frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911}\right) + \frac{\frac{\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)\right) \cdot \left(1 - \frac{\frac{\left(0.254829592 + \frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911}\right) + \frac{\frac{\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)}\]

C

double f(double x) {
        double r5828976 = 1.0;
        double r5828977 = 0.3275911;
        double r5828978 = x;
        double r5828979 = fabs(r5828978);
        double r5828980 = r5828977 * r5828979;
        double r5828981 = r5828976 + r5828980;
        double r5828982 = r5828976 / r5828981;
        double r5828983 = 0.254829592;
        double r5828984 = -0.284496736;
        double r5828985 = 1.421413741;
        double r5828986 = -1.453152027;
        double r5828987 = 1.061405429;
        double r5828988 = r5828982 * r5828987;
        double r5828989 = r5828986 + r5828988;
        double r5828990 = r5828982 * r5828989;
        double r5828991 = r5828985 + r5828990;
        double r5828992 = r5828982 * r5828991;
        double r5828993 = r5828984 + r5828992;
        double r5828994 = r5828982 * r5828993;
        double r5828995 = r5828983 + r5828994;
        double r5828996 = r5828982 * r5828995;
        double r5828997 = r5828979 * r5828979;
        double r5828998 = -r5828997;
        double r5828999 = exp(r5828998);
        double r5829000 = r5828996 * r5828999;
        double r5829001 = r5828976 - r5829000;
        return r5829001;
}

↓

double f(double x) {
        double r5829002 = 1.0;
        double r5829003 = 0.254829592;
        double r5829004 = -0.284496736;
        double r5829005 = x;
        double r5829006 = fabs(r5829005);
        double r5829007 = 0.3275911;
        double r5829008 = r5829006 * r5829007;
        double r5829009 = r5829002 + r5829008;
        double r5829010 = r5829004 / r5829009;
        double r5829011 = r5829003 + r5829010;
        double r5829012 = 1.061405429;
        double r5829013 = r5829012 / r5829009;
        double r5829014 = -1.453152027;
        double r5829015 = r5829013 + r5829014;
        double r5829016 = r5829015 / r5829009;
        double r5829017 = 1.421413741;
        double r5829018 = r5829016 + r5829017;
        double r5829019 = r5829018 / r5829009;
        double r5829020 = r5829019 / r5829009;
        double r5829021 = r5829011 + r5829020;
        double r5829022 = r5829021 / r5829009;
        double r5829023 = r5829006 * r5829006;
        double r5829024 = exp(r5829023);
        double r5829025 = r5829022 / r5829024;
        double r5829026 = exp(r5829025);
        double r5829027 = cbrt(r5829026);
        double r5829028 = r5829027 * r5829027;
        double r5829029 = r5829002 / r5829028;
        double r5829030 = log(r5829029);
        double r5829031 = exp(1.0);
        double r5829032 = r5829031 / r5829027;
        double r5829033 = log(r5829032);
        double r5829034 = r5829030 + r5829033;
        double r5829035 = r5829002 - r5829025;
        double r5829036 = r5829034 * r5829035;
        double r5829037 = r5829036 * r5829035;
        double r5829038 = cbrt(r5829037);
        return r5829038;
}

Error

Bits error versus x

Derivation

1. Initial program 13.7

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
2. Using strategy rm

3. Applied distribute-lft-in 13.7

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \color{blue}{\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot -0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

4. Applied associate-+r+ 13.7

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot -0.284496736\right) + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
5. Using strategy rm

6. Applied add-cbrt-cube 13.7

   \[\leadsto \color{blue}{\sqrt[3]{\left(\left(1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot -0.284496736\right) + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) \cdot \left(1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot -0.284496736\right) + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)\right) \cdot \left(1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot -0.284496736\right) + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}}\]

7. Simplified 13.7

   \[\leadsto \sqrt[3]{\color{blue}{\left(\left(1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left(1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)}}\]
8. Using strategy rm

9. Applied add-log-exp 13.7

   \[\leadsto \sqrt[3]{\left(\left(1 - \color{blue}{\log \left(e^{\frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}\right) \cdot \left(1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)}\]

10. Applied add-log-exp 13.7

    \[\leadsto \sqrt[3]{\left(\left(\color{blue}{\log \left(e^{1}\right)} - \log \left(e^{\frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)}\]

11. Applied diff-log 13.7

    \[\leadsto \sqrt[3]{\left(\color{blue}{\log \left(\frac{e^{1}}{e^{\frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}}\right)} \cdot \left(1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)}\]
12. Using strategy rm

13. Applied add-cube-cbrt 12.9

    \[\leadsto \sqrt[3]{\left(\log \left(\frac{e^{1}}{\color{blue}{\left(\sqrt[3]{e^{\frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}} \cdot \sqrt[3]{e^{\frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}}\right) \cdot \sqrt[3]{e^{\frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}}}}\right) \cdot \left(1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)}\]

14. Applied *-un-lft-identity 12.9

    \[\leadsto \sqrt[3]{\left(\log \left(\frac{\color{blue}{1 \cdot e^{1}}}{\left(\sqrt[3]{e^{\frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}} \cdot \sqrt[3]{e^{\frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}}\right) \cdot \sqrt[3]{e^{\frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}}}\right) \cdot \left(1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)}\]

15. Applied times-frac 12.9

    \[\leadsto \sqrt[3]{\left(\log \color{blue}{\left(\frac{1}{\sqrt[3]{e^{\frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}} \cdot \sqrt[3]{e^{\frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}}} \cdot \frac{e^{1}}{\sqrt[3]{e^{\frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}}}\right)} \cdot \left(1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)}\]

16. Applied log-prod 12.9

    \[\leadsto \sqrt[3]{\left(\color{blue}{\left(\log \left(\frac{1}{\sqrt[3]{e^{\frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}} \cdot \sqrt[3]{e^{\frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}}}\right) + \log \left(\frac{e^{1}}{\sqrt[3]{e^{\frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}}}\right)\right)} \cdot \left(1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)\right) \cdot \left(1 - \frac{\frac{\frac{\frac{1.421413741 + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911} + 0.254829592\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)}\]

17. Final simplification 12.9

    \[\leadsto \sqrt[3]{\left(\left(\log \left(\frac{1}{\sqrt[3]{e^{\frac{\frac{\left(0.254829592 + \frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911}\right) + \frac{\frac{\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}} \cdot \sqrt[3]{e^{\frac{\frac{\left(0.254829592 + \frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911}\right) + \frac{\frac{\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}}}\right) + \log \left(\frac{e}{\sqrt[3]{e^{\frac{\frac{\left(0.254829592 + \frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911}\right) + \frac{\frac{\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}}}}\right)\right) \cdot \left(1 - \frac{\frac{\left(0.254829592 + \frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911}\right) + \frac{\frac{\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)\right) \cdot \left(1 - \frac{\frac{\left(0.254829592 + \frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911}\right) + \frac{\frac{\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}}\right)}\]

Reproduce

herbie shell --seed 2019135 
(FPCore (x)
  :name "Jmat.Real.erf"
  (- 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)))))))