Jmat.Real.erf

Average Error: 13.8 → 13.0

Time: 3.7m

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r14275588 = 1.0;
        double r14275589 = 0.3275911;
        double r14275590 = x;
        double r14275591 = fabs(r14275590);
        double r14275592 = r14275589 * r14275591;
        double r14275593 = r14275588 + r14275592;
        double r14275594 = r14275588 / r14275593;
        double r14275595 = 0.254829592;
        double r14275596 = -0.284496736;
        double r14275597 = 1.421413741;
        double r14275598 = -1.453152027;
        double r14275599 = 1.061405429;
        double r14275600 = r14275594 * r14275599;
        double r14275601 = r14275598 + r14275600;
        double r14275602 = r14275594 * r14275601;
        double r14275603 = r14275597 + r14275602;
        double r14275604 = r14275594 * r14275603;
        double r14275605 = r14275596 + r14275604;
        double r14275606 = r14275594 * r14275605;
        double r14275607 = r14275595 + r14275606;
        double r14275608 = r14275594 * r14275607;
        double r14275609 = r14275591 * r14275591;
        double r14275610 = -r14275609;
        double r14275611 = exp(r14275610);
        double r14275612 = r14275608 * r14275611;
        double r14275613 = r14275588 - r14275612;
        return r14275613;
}

↓

double f(double x) {
        double r14275614 = 1.0;
        double r14275615 = 3.0;
        double r14275616 = pow(r14275614, r14275615);
        double r14275617 = 0.254829592;
        double r14275618 = r14275617 * r14275617;
        double r14275619 = 1.421413741;
        double r14275620 = 1.061405429;
        double r14275621 = x;
        double r14275622 = fabs(r14275621);
        double r14275623 = 0.3275911;
        double r14275624 = r14275622 * r14275623;
        double r14275625 = r14275614 + r14275624;
        double r14275626 = r14275614 / r14275625;
        double r14275627 = r14275620 * r14275626;
        double r14275628 = -1.453152027;
        double r14275629 = r14275627 + r14275628;
        double r14275630 = r14275629 * r14275626;
        double r14275631 = r14275619 + r14275630;
        double r14275632 = r14275631 * r14275626;
        double r14275633 = -0.284496736;
        double r14275634 = r14275632 + r14275633;
        double r14275635 = r14275634 * r14275626;
        double r14275636 = r14275635 * r14275635;
        double r14275637 = r14275618 - r14275636;
        double r14275638 = r14275637 * r14275614;
        double r14275639 = -r14275622;
        double r14275640 = r14275639 * r14275622;
        double r14275641 = exp(r14275640);
        double r14275642 = r14275638 * r14275641;
        double r14275643 = pow(r14275642, r14275615);
        double r14275644 = r14275617 - r14275635;
        double r14275645 = r14275625 * r14275644;
        double r14275646 = pow(r14275645, r14275615);
        double r14275647 = r14275643 / r14275646;
        double r14275648 = r14275616 - r14275647;
        double r14275649 = r14275617 + r14275635;
        double r14275650 = r14275649 * r14275626;
        double r14275651 = r14275641 * r14275650;
        double r14275652 = r14275651 * r14275651;
        double r14275653 = r14275651 * r14275614;
        double r14275654 = r14275652 + r14275653;
        double r14275655 = r14275614 * r14275614;
        double r14275656 = r14275654 + r14275655;
        double r14275657 = r14275648 / r14275656;
        return r14275657;
}

Error

Bits error versus x

Derivation

1. Initial program 13.8

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

3. Applied flip3-- 13.8

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

5. Applied flip-+ 13.8

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

6. Applied frac-times 13.8

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

7. Applied associate-*l/ 13.8

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

8. Applied cube-div 13.0

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

9. Final simplification 13.0

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erf"
  (- 1.0 (* (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))