Jmat.Real.erf

Average Error: 13.7 → 13.0

Time: 40.8s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r8033622 = 1.0;
        double r8033623 = 0.3275911;
        double r8033624 = x;
        double r8033625 = fabs(r8033624);
        double r8033626 = r8033623 * r8033625;
        double r8033627 = r8033622 + r8033626;
        double r8033628 = r8033622 / r8033627;
        double r8033629 = 0.254829592;
        double r8033630 = -0.284496736;
        double r8033631 = 1.421413741;
        double r8033632 = -1.453152027;
        double r8033633 = 1.061405429;
        double r8033634 = r8033628 * r8033633;
        double r8033635 = r8033632 + r8033634;
        double r8033636 = r8033628 * r8033635;
        double r8033637 = r8033631 + r8033636;
        double r8033638 = r8033628 * r8033637;
        double r8033639 = r8033630 + r8033638;
        double r8033640 = r8033628 * r8033639;
        double r8033641 = r8033629 + r8033640;
        double r8033642 = r8033628 * r8033641;
        double r8033643 = r8033625 * r8033625;
        double r8033644 = -r8033643;
        double r8033645 = exp(r8033644);
        double r8033646 = r8033642 * r8033645;
        double r8033647 = r8033622 - r8033646;
        return r8033647;
}

↓

double f(double x) {
        double r8033648 = 1.0;
        double r8033649 = x;
        double r8033650 = fabs(r8033649);
        double r8033651 = 0.3275911;
        double r8033652 = r8033650 * r8033651;
        double r8033653 = r8033648 + r8033652;
        double r8033654 = r8033648 / r8033653;
        double r8033655 = r8033650 * r8033650;
        double r8033656 = exp(r8033655);
        double r8033657 = r8033654 / r8033656;
        double r8033658 = 0.254829592;
        double r8033659 = -0.284496736;
        double r8033660 = 1.421413741;
        double r8033661 = r8033654 * r8033660;
        double r8033662 = r8033659 + r8033661;
        double r8033663 = r8033654 * r8033654;
        double r8033664 = 1.061405429;
        double r8033665 = r8033664 * r8033648;
        double r8033666 = r8033665 / r8033653;
        double r8033667 = -1.453152027;
        double r8033668 = r8033666 + r8033667;
        double r8033669 = r8033663 * r8033668;
        double r8033670 = r8033662 + r8033669;
        double r8033671 = r8033654 * r8033670;
        double r8033672 = r8033658 + r8033671;
        double r8033673 = r8033657 * r8033672;
        double r8033674 = r8033648 - r8033673;
        double r8033675 = exp(r8033674);
        double r8033676 = log(r8033675);
        double r8033677 = cbrt(r8033676);
        double r8033678 = cbrt(r8033674);
        double r8033679 = r8033678 * r8033678;
        double r8033680 = exp(r8033679);
        double r8033681 = log(r8033680);
        double r8033682 = r8033681 * r8033678;
        double r8033683 = cbrt(r8033682);
        double r8033684 = r8033677 * r8033683;
        double r8033685 = r8033677 * r8033684;
        return r8033685;
}

Error

Bits error versus x

Derivation

1. Initial program 13.7

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

3. Applied add-log-exp 13.7

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

4. Applied add-log-exp 13.7

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

5. Applied diff-log 14.5

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

6. Simplified 13.7

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

8. Applied add-log-exp 13.7

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

9. Simplified 13.7

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

11. Applied add-cube-cbrt 13.7

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

13. Applied add-cube-cbrt 13.7

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

14. Applied exp-prod 13.7

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

15. Applied log-pow 13.0

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

16. Final simplification 13.0

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

Reproduce

herbie shell --seed 2019169 
(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)))))))