Jmat.Real.erf

Average Error: 13.8 → 1.9

Time: 15.9s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r268682 = 1.0;
        double r268683 = 0.3275911;
        double r268684 = x;
        double r268685 = fabs(r268684);
        double r268686 = r268683 * r268685;
        double r268687 = r268682 + r268686;
        double r268688 = r268682 / r268687;
        double r268689 = 0.254829592;
        double r268690 = -0.284496736;
        double r268691 = 1.421413741;
        double r268692 = -1.453152027;
        double r268693 = 1.061405429;
        double r268694 = r268688 * r268693;
        double r268695 = r268692 + r268694;
        double r268696 = r268688 * r268695;
        double r268697 = r268691 + r268696;
        double r268698 = r268688 * r268697;
        double r268699 = r268690 + r268698;
        double r268700 = r268688 * r268699;
        double r268701 = r268689 + r268700;
        double r268702 = r268688 * r268701;
        double r268703 = r268685 * r268685;
        double r268704 = -r268703;
        double r268705 = exp(r268704);
        double r268706 = r268702 * r268705;
        double r268707 = r268682 - r268706;
        return r268707;
}

↓

double f(double x) {
        double r268708 = 1.0;
        double r268709 = cbrt(r268708);
        double r268710 = r268709 * r268709;
        double r268711 = x;
        double r268712 = fabs(r268711);
        double r268713 = 2.0;
        double r268714 = pow(r268712, r268713);
        double r268715 = -r268714;
        double r268716 = exp(r268715);
        double r268717 = sqrt(r268708);
        double r268718 = 0.3275911;
        double r268719 = r268718 * r268712;
        double r268720 = r268708 + r268719;
        double r268721 = sqrt(r268720);
        double r268722 = r268717 / r268721;
        double r268723 = r268722 * r268722;
        double r268724 = r268716 * r268723;
        double r268725 = -r268724;
        double r268726 = fma(r268718, r268712, r268708);
        double r268727 = r268708 / r268726;
        double r268728 = 1.061405429;
        double r268729 = -1.453152027;
        double r268730 = fma(r268727, r268728, r268729);
        double r268731 = 1.421413741;
        double r268732 = fma(r268727, r268730, r268731);
        double r268733 = -0.284496736;
        double r268734 = fma(r268727, r268732, r268733);
        double r268735 = 0.254829592;
        double r268736 = fma(r268727, r268734, r268735);
        double r268737 = r268725 * r268736;
        double r268738 = fma(r268710, r268709, r268737);
        double r268739 = log(r268738);
        double r268740 = exp(r268739);
        double r268741 = sqrt(r268740);
        double r268742 = r268708 / r268720;
        double r268743 = r268742 * r268728;
        double r268744 = r268729 + r268743;
        double r268745 = r268742 * r268744;
        double r268746 = r268731 + r268745;
        double r268747 = r268742 * r268746;
        double r268748 = r268733 + r268747;
        double r268749 = r268742 * r268748;
        double r268750 = r268735 + r268749;
        double r268751 = sqrt(r268750);
        double r268752 = r268722 * r268751;
        double r268753 = r268752 * r268752;
        double r268754 = r268712 * r268712;
        double r268755 = -r268754;
        double r268756 = exp(r268755);
        double r268757 = r268753 * r268756;
        double r268758 = r268708 - r268757;
        double r268759 = sqrt(r268758);
        double r268760 = r268741 * r268759;
        return r268760;
}

Error

Bits error versus x

Derivation

1. Initial program 13.8

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

3. Applied add-sqr-sqrt 13.8

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

4. Applied add-sqr-sqrt 13.8

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

5. Applied add-sqr-sqrt 13.8

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

6. Applied times-frac 13.8

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

7. Applied unswap-sqr 13.8

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

9. Applied add-sqr-sqrt 13.8

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

11. Applied add-cube-cbrt 13.8

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

12. Applied fma-neg 13.8

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

13. Simplified 2.2

    \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1}, \color{blue}{\left(-e^{-{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)\right) \cdot \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}, \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}, \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}, \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}\right)} \cdot \sqrt{1 - \left(\left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt{0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)}\right) \cdot \left(\frac{\sqrt{1}}{\sqrt{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt{0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\]
14. Using strategy rm

15. Applied add-exp-log 1.9

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

16. Final simplification 1.9

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 1 (* (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))