Jmat.Real.erf

Average Error: 13.9 → 13.0

Time: 5.2m

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r15074739 = 1.0;
        double r15074740 = 0.3275911;
        double r15074741 = x;
        double r15074742 = fabs(r15074741);
        double r15074743 = r15074740 * r15074742;
        double r15074744 = r15074739 + r15074743;
        double r15074745 = r15074739 / r15074744;
        double r15074746 = 0.254829592;
        double r15074747 = -0.284496736;
        double r15074748 = 1.421413741;
        double r15074749 = -1.453152027;
        double r15074750 = 1.061405429;
        double r15074751 = r15074745 * r15074750;
        double r15074752 = r15074749 + r15074751;
        double r15074753 = r15074745 * r15074752;
        double r15074754 = r15074748 + r15074753;
        double r15074755 = r15074745 * r15074754;
        double r15074756 = r15074747 + r15074755;
        double r15074757 = r15074745 * r15074756;
        double r15074758 = r15074746 + r15074757;
        double r15074759 = r15074745 * r15074758;
        double r15074760 = r15074742 * r15074742;
        double r15074761 = -r15074760;
        double r15074762 = exp(r15074761);
        double r15074763 = r15074759 * r15074762;
        double r15074764 = r15074739 - r15074763;
        return r15074764;
}

↓

double f(double x) {
        double r15074765 = 1.0;
        double r15074766 = x;
        double r15074767 = fabs(r15074766);
        double r15074768 = r15074767 * r15074767;
        double r15074769 = -r15074768;
        double r15074770 = exp(r15074769);
        double r15074771 = 0.3275911;
        double r15074772 = r15074767 * r15074771;
        double r15074773 = r15074765 + r15074772;
        double r15074774 = r15074765 / r15074773;
        double r15074775 = sqrt(r15074774);
        double r15074776 = r15074775 * r15074775;
        double r15074777 = 1.061405429;
        double r15074778 = r15074776 * r15074777;
        double r15074779 = -1.453152027;
        double r15074780 = r15074778 + r15074779;
        double r15074781 = r15074774 * r15074780;
        double r15074782 = 1.421413741;
        double r15074783 = r15074781 + r15074782;
        double r15074784 = r15074783 * r15074774;
        double r15074785 = -0.284496736;
        double r15074786 = r15074784 + r15074785;
        double r15074787 = r15074786 * r15074774;
        double r15074788 = 0.254829592;
        double r15074789 = r15074787 + r15074788;
        double r15074790 = r15074774 * r15074789;
        double r15074791 = r15074770 * r15074790;
        double r15074792 = 3.0;
        double r15074793 = pow(r15074791, r15074792);
        double r15074794 = sqrt(r15074793);
        double r15074795 = r15074794 * r15074794;
        double r15074796 = pow(r15074795, r15074792);
        double r15074797 = r15074765 - r15074796;
        double r15074798 = r15074795 * r15074795;
        double r15074799 = r15074798 + r15074795;
        double r15074800 = r15074799 + r15074765;
        double r15074801 = r15074797 / r15074800;
        double r15074802 = r15074791 * r15074791;
        double r15074803 = r15074802 + r15074791;
        double r15074804 = r15074803 + r15074765;
        double r15074805 = r15074801 / r15074804;
        return r15074805;
}

Error

Bits error versus x

Derivation

1. Initial program 13.9

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

3. Applied add-sqr-sqrt 13.9

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

5. Applied flip3-- 13.9

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

7. Applied add-sqr-sqrt 13.1

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

9. Applied flip3-- 13.0

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

10. Final simplification 13.0

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(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)))))))