Jmat.Real.erf

Average Error: 13.8 → 3.2

Time: 3.9m

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]{1 - \frac{\left(\left(\left(\frac{1.421413741}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} - \left(\sqrt[3]{\frac{1.453152027}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} \cdot \left(\frac{1.453152027}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} \cdot \frac{1.453152027}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}\right)} + 0.284496736\right) \cdot \frac{1}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}\right) + \frac{1.061405429}{\left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right) \cdot \left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right)}\right) + 0.254829592\right) \cdot \frac{1}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}} \cdot \left(\sqrt[3]{\mathsf{fma}\left(\left(\frac{\frac{0.284496736}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}\right), \left(e^{\left(-\left|x\right|\right) \cdot \left|x\right|}\right), \left(\mathsf{fma}\left(\left(\frac{1.453152027}{\left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right) \cdot \left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right)}\right), \left(e^{\left(-\left|x\right|\right) \cdot \left|x\right|}\right), \left(1 - \mathsf{fma}\left(\left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right)}^{5}}\right), \left(e^{\left(-\left|x\right|\right) \cdot \left|x\right|}\right), \left(\left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} + \frac{\frac{1.421413741}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}\right) \cdot e^{\left(-\left|x\right|\right) \cdot \left|x\right|}\right)\right)\right)\right)\right)\right)} \cdot \sqrt[3]{1 - \frac{\left(\left(\left(\frac{1.421413741}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} - \frac{1}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} \cdot \left(0.284496736 + \frac{1.453152027}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}\right)\right) + \frac{1.061405429}{\left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right) \cdot \left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right)}\right) + 0.254829592\right) \cdot \frac{1}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)\]

C

double f(double x) {
        double r44652773 = 1.0;
        double r44652774 = 0.3275911;
        double r44652775 = x;
        double r44652776 = fabs(r44652775);
        double r44652777 = r44652774 * r44652776;
        double r44652778 = r44652773 + r44652777;
        double r44652779 = r44652773 / r44652778;
        double r44652780 = 0.254829592;
        double r44652781 = -0.284496736;
        double r44652782 = 1.421413741;
        double r44652783 = -1.453152027;
        double r44652784 = 1.061405429;
        double r44652785 = r44652779 * r44652784;
        double r44652786 = r44652783 + r44652785;
        double r44652787 = r44652779 * r44652786;
        double r44652788 = r44652782 + r44652787;
        double r44652789 = r44652779 * r44652788;
        double r44652790 = r44652781 + r44652789;
        double r44652791 = r44652779 * r44652790;
        double r44652792 = r44652780 + r44652791;
        double r44652793 = r44652779 * r44652792;
        double r44652794 = r44652776 * r44652776;
        double r44652795 = -r44652794;
        double r44652796 = exp(r44652795);
        double r44652797 = r44652793 * r44652796;
        double r44652798 = r44652773 - r44652797;
        return r44652798;
}

↓

double f(double x) {
        double r44652799 = 1.0;
        double r44652800 = 1.421413741;
        double r44652801 = 0.3275911;
        double r44652802 = x;
        double r44652803 = fabs(r44652802);
        double r44652804 = fma(r44652801, r44652803, r44652799);
        double r44652805 = r44652804 * r44652804;
        double r44652806 = r44652800 / r44652805;
        double r44652807 = 1.453152027;
        double r44652808 = r44652807 / r44652805;
        double r44652809 = r44652808 * r44652808;
        double r44652810 = r44652808 * r44652809;
        double r44652811 = cbrt(r44652810);
        double r44652812 = 0.284496736;
        double r44652813 = r44652811 + r44652812;
        double r44652814 = r44652799 / r44652804;
        double r44652815 = r44652813 * r44652814;
        double r44652816 = r44652806 - r44652815;
        double r44652817 = 1.061405429;
        double r44652818 = r44652805 * r44652805;
        double r44652819 = r44652817 / r44652818;
        double r44652820 = r44652816 + r44652819;
        double r44652821 = 0.254829592;
        double r44652822 = r44652820 + r44652821;
        double r44652823 = r44652822 * r44652814;
        double r44652824 = r44652803 * r44652803;
        double r44652825 = exp(r44652824);
        double r44652826 = r44652823 / r44652825;
        double r44652827 = r44652799 - r44652826;
        double r44652828 = cbrt(r44652827);
        double r44652829 = r44652812 / r44652804;
        double r44652830 = r44652829 / r44652804;
        double r44652831 = -r44652803;
        double r44652832 = r44652831 * r44652803;
        double r44652833 = exp(r44652832);
        double r44652834 = r44652807 / r44652818;
        double r44652835 = 5.0;
        double r44652836 = pow(r44652804, r44652835);
        double r44652837 = r44652817 / r44652836;
        double r44652838 = r44652821 / r44652804;
        double r44652839 = r44652806 / r44652804;
        double r44652840 = r44652838 + r44652839;
        double r44652841 = r44652840 * r44652833;
        double r44652842 = fma(r44652837, r44652833, r44652841);
        double r44652843 = r44652799 - r44652842;
        double r44652844 = fma(r44652834, r44652833, r44652843);
        double r44652845 = fma(r44652830, r44652833, r44652844);
        double r44652846 = cbrt(r44652845);
        double r44652847 = r44652812 + r44652808;
        double r44652848 = r44652814 * r44652847;
        double r44652849 = r44652806 - r44652848;
        double r44652850 = r44652849 + r44652819;
        double r44652851 = r44652850 + r44652821;
        double r44652852 = r44652851 * r44652814;
        double r44652853 = r44652852 / r44652825;
        double r44652854 = r44652799 - r44652853;
        double r44652855 = cbrt(r44652854);
        double r44652856 = r44652846 * r44652855;
        double r44652857 = r44652828 * r44652856;
        return r44652857;
}

Error

Bits error versus x

Derivation

1. Initial program 13.8

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

2. Simplified 13.8

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

3. Taylor expanded around -inf 14.5

   \[\leadsto 1 - \frac{\frac{1}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} \cdot \color{blue}{\left(\left(1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{4}} + \left(1.421413741 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}} + 0.254829592\right)\right) - \left(0.284496736 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1} + 1.453152027 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{3}}\right)\right)}}{e^{\left|x\right| \cdot \left|x\right|}}\]

4. Simplified 13.8

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

6. Applied add-cube-cbrt 13.8

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

8. Applied add-cbrt-cube 13.0

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

9. Taylor expanded around inf 14.1

   \[\leadsto \left(\sqrt[3]{\color{blue}{\left(0.284496736 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2} \cdot e^{{\left(\left|x\right|\right)}^{2}}} + \left(1.453152027 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{4} \cdot e^{{\left(\left|x\right|\right)}^{2}}} + 1\right)\right) - \left(0.254829592 \cdot \frac{1}{\left(0.3275911 \cdot \left|x\right| + 1\right) \cdot e^{{\left(\left|x\right|\right)}^{2}}} + \left(1.421413741 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{3} \cdot e^{{\left(\left|x\right|\right)}^{2}}} + 1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{5} \cdot e^{{\left(\left|x\right|\right)}^{2}}}\right)\right)}} \cdot \sqrt[3]{1 - \frac{\frac{1}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} \cdot \left(0.254829592 + \left(\left(\frac{1.421413741}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} - \frac{1}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} \cdot \left(\frac{1.453152027}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} + 0.284496736\right)\right) + \frac{1.061405429}{\left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right) \cdot \left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right)}\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}}}\right) \cdot \sqrt[3]{1 - \frac{\frac{1}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} \cdot \left(0.254829592 + \left(\left(\frac{1.421413741}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} - \frac{1}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} \cdot \left(\sqrt[3]{\left(\frac{1.453152027}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} \cdot \frac{1.453152027}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}\right) \cdot \frac{1.453152027}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}} + 0.284496736\right)\right) + \frac{1.061405429}{\left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right) \cdot \left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right)}\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}}}\]

10. Simplified 3.2

    \[\leadsto \left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(\left(\frac{\frac{0.284496736}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}\right), \left(e^{-\left|x\right| \cdot \left|x\right|}\right), \left(\mathsf{fma}\left(\left(\frac{1.453152027}{\left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right) \cdot \left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right)}\right), \left(e^{-\left|x\right| \cdot \left|x\right|}\right), \left(1 - \mathsf{fma}\left(\left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right)}^{5}}\right), \left(e^{-\left|x\right| \cdot \left|x\right|}\right), \left(e^{-\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} + \frac{\frac{1.421413741}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}\right)\right)\right)\right)\right)\right)\right)}} \cdot \sqrt[3]{1 - \frac{\frac{1}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} \cdot \left(0.254829592 + \left(\left(\frac{1.421413741}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} - \frac{1}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} \cdot \left(\frac{1.453152027}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} + 0.284496736\right)\right) + \frac{1.061405429}{\left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right) \cdot \left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right)}\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}}}\right) \cdot \sqrt[3]{1 - \frac{\frac{1}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} \cdot \left(0.254829592 + \left(\left(\frac{1.421413741}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} - \frac{1}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} \cdot \left(\sqrt[3]{\left(\frac{1.453152027}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} \cdot \frac{1.453152027}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}\right) \cdot \frac{1.453152027}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}} + 0.284496736\right)\right) + \frac{1.061405429}{\left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right) \cdot \left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right)}\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}}}\]

11. Final simplification 3.2

    \[\leadsto \sqrt[3]{1 - \frac{\left(\left(\left(\frac{1.421413741}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} - \left(\sqrt[3]{\frac{1.453152027}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} \cdot \left(\frac{1.453152027}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} \cdot \frac{1.453152027}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}\right)} + 0.284496736\right) \cdot \frac{1}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}\right) + \frac{1.061405429}{\left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right) \cdot \left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right)}\right) + 0.254829592\right) \cdot \frac{1}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}} \cdot \left(\sqrt[3]{\mathsf{fma}\left(\left(\frac{\frac{0.284496736}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}\right), \left(e^{\left(-\left|x\right|\right) \cdot \left|x\right|}\right), \left(\mathsf{fma}\left(\left(\frac{1.453152027}{\left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right) \cdot \left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right)}\right), \left(e^{\left(-\left|x\right|\right) \cdot \left|x\right|}\right), \left(1 - \mathsf{fma}\left(\left(\frac{1.061405429}{{\left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right)}^{5}}\right), \left(e^{\left(-\left|x\right|\right) \cdot \left|x\right|}\right), \left(\left(\frac{0.254829592}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} + \frac{\frac{1.421413741}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}\right) \cdot e^{\left(-\left|x\right|\right) \cdot \left|x\right|}\right)\right)\right)\right)\right)\right)} \cdot \sqrt[3]{1 - \frac{\left(\left(\left(\frac{1.421413741}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} - \frac{1}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)} \cdot \left(0.284496736 + \frac{1.453152027}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}\right)\right) + \frac{1.061405429}{\left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right) \cdot \left(\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right) \cdot \mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)\right)}\right) + 0.254829592\right) \cdot \frac{1}{\mathsf{fma}\left(0.3275911, \left(\left|x\right|\right), 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)\]

Reproduce

herbie shell --seed 2019124 +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)))))))