Jmat.Real.erf

Average Error: 13.7 → 12.9

Time: 44.9s

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} - \sqrt{{\left(\frac{0.2548295919999999936678136691625695675611 + \left(-0.2844967359999999723108032867457950487733 + \left(\frac{1}{\frac{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}{-1.453152027000000012790792425221297889948 + 1.061405428999999900341322245367337018251 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}}} + 1.421413741000000063863240029604639858007\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \frac{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}{1}}\right)}^{3}} \cdot \sqrt{\sqrt[3]{{\left({\left(\frac{0.2548295919999999936678136691625695675611 + \left(-0.2844967359999999723108032867457950487733 + \left(\frac{1}{\frac{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}{-1.453152027000000012790792425221297889948 + 1.061405428999999900341322245367337018251 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}}} + 1.421413741000000063863240029604639858007\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}}{\frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699\right)}{1}}\right)}^{3}\right)}^{3}}}}{1 \cdot 1 + \left(1 + \frac{\left(\frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699} \cdot \left(1.421413741000000063863240029604639858007 + \frac{\left(-1.453152027000000012790792425221297889948 + 1.061405428999999900341322245367337018251 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}\right) \cdot 1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}\right) + -0.2844967359999999723108032867457950487733\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699} + 0.2548295919999999936678136691625695675611}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \frac{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}{1}}\right) \cdot \frac{\left(\frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699} \cdot \left(1.421413741000000063863240029604639858007 + \frac{\left(-1.453152027000000012790792425221297889948 + 1.061405428999999900341322245367337018251 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}\right) \cdot 1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}\right) + -0.2844967359999999723108032867457950487733\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699} + 0.2548295919999999936678136691625695675611}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \frac{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}{1}}}\]

C

double f(double x) {
        double r188510 = 1.0;
        double r188511 = 0.3275911;
        double r188512 = x;
        double r188513 = fabs(r188512);
        double r188514 = r188511 * r188513;
        double r188515 = r188510 + r188514;
        double r188516 = r188510 / r188515;
        double r188517 = 0.254829592;
        double r188518 = -0.284496736;
        double r188519 = 1.421413741;
        double r188520 = -1.453152027;
        double r188521 = 1.061405429;
        double r188522 = r188516 * r188521;
        double r188523 = r188520 + r188522;
        double r188524 = r188516 * r188523;
        double r188525 = r188519 + r188524;
        double r188526 = r188516 * r188525;
        double r188527 = r188518 + r188526;
        double r188528 = r188516 * r188527;
        double r188529 = r188517 + r188528;
        double r188530 = r188516 * r188529;
        double r188531 = r188513 * r188513;
        double r188532 = -r188531;
        double r188533 = exp(r188532);
        double r188534 = r188530 * r188533;
        double r188535 = r188510 - r188534;
        return r188535;
}

↓

double f(double x) {
        double r188536 = 1.0;
        double r188537 = 3.0;
        double r188538 = pow(r188536, r188537);
        double r188539 = 0.254829592;
        double r188540 = -0.284496736;
        double r188541 = x;
        double r188542 = fabs(r188541);
        double r188543 = 0.3275911;
        double r188544 = r188542 * r188543;
        double r188545 = r188536 + r188544;
        double r188546 = -1.453152027;
        double r188547 = 1.061405429;
        double r188548 = r188536 / r188545;
        double r188549 = r188547 * r188548;
        double r188550 = r188546 + r188549;
        double r188551 = r188545 / r188550;
        double r188552 = r188536 / r188551;
        double r188553 = 1.421413741;
        double r188554 = r188552 + r188553;
        double r188555 = r188554 * r188548;
        double r188556 = r188540 + r188555;
        double r188557 = r188556 * r188548;
        double r188558 = r188539 + r188557;
        double r188559 = 2.0;
        double r188560 = pow(r188542, r188559);
        double r188561 = exp(r188560);
        double r188562 = r188545 / r188536;
        double r188563 = r188561 * r188562;
        double r188564 = r188558 / r188563;
        double r188565 = pow(r188564, r188537);
        double r188566 = sqrt(r188565);
        double r188567 = r188561 * r188545;
        double r188568 = r188567 / r188536;
        double r188569 = r188558 / r188568;
        double r188570 = pow(r188569, r188537);
        double r188571 = pow(r188570, r188537);
        double r188572 = cbrt(r188571);
        double r188573 = sqrt(r188572);
        double r188574 = r188566 * r188573;
        double r188575 = r188538 - r188574;
        double r188576 = r188536 * r188536;
        double r188577 = r188550 * r188536;
        double r188578 = r188577 / r188545;
        double r188579 = r188553 + r188578;
        double r188580 = r188548 * r188579;
        double r188581 = r188580 + r188540;
        double r188582 = r188581 * r188548;
        double r188583 = r188582 + r188539;
        double r188584 = exp(r188542);
        double r188585 = pow(r188584, r188542);
        double r188586 = r188585 * r188562;
        double r188587 = r188583 / r188586;
        double r188588 = r188536 + r188587;
        double r188589 = r188588 * r188587;
        double r188590 = r188576 + r188589;
        double r188591 = r188575 / r188590;
        return r188591;
}

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. Simplified 13.7

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

4. Applied add-log-exp 13.7

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

5. Simplified 13.7

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

7. Applied flip3-- 13.7

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

8. Simplified 13.7

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

9. Simplified 13.7

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

11. Applied add-sqr-sqrt 12.9

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

12. Simplified 12.9

    \[\leadsto \frac{{1}^{3} - \color{blue}{\sqrt{{\left(\frac{0.2548295919999999936678136691625695675611 + \left(\left(1.421413741000000063863240029604639858007 + \frac{1}{\frac{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}{-1.453152027000000012790792425221297889948 + 1.061405428999999900341322245367337018251 \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}}\right) \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} + -0.2844967359999999723108032867457950487733\right) \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \frac{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}{1}}\right)}^{3}}} \cdot \sqrt{{\left(\frac{\left(-0.2844967359999999723108032867457950487733 + \left(1.421413741000000063863240029604639858007 + \frac{1 \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1} \cdot 1.061405428999999900341322245367337018251\right)}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right) \cdot \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right) \cdot \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1} + 0.2548295919999999936678136691625695675611}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \frac{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}{1}}\right)}^{3}}}{\frac{\left(-0.2844967359999999723108032867457950487733 + \left(1.421413741000000063863240029604639858007 + \frac{1 \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1} \cdot 1.061405428999999900341322245367337018251\right)}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right) \cdot \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right) \cdot \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1} + 0.2548295919999999936678136691625695675611}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \frac{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}{1}} \cdot \left(\frac{\left(-0.2844967359999999723108032867457950487733 + \left(1.421413741000000063863240029604639858007 + \frac{1 \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1} \cdot 1.061405428999999900341322245367337018251\right)}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right) \cdot \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right) \cdot \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1} + 0.2548295919999999936678136691625695675611}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \frac{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}{1}} + 1\right) + 1 \cdot 1}\]

13. Simplified 12.9

    \[\leadsto \frac{{1}^{3} - \sqrt{{\left(\frac{0.2548295919999999936678136691625695675611 + \left(\left(1.421413741000000063863240029604639858007 + \frac{1}{\frac{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}{-1.453152027000000012790792425221297889948 + 1.061405428999999900341322245367337018251 \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}}\right) \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} + -0.2844967359999999723108032867457950487733\right) \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \frac{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}{1}}\right)}^{3}} \cdot \color{blue}{\sqrt{{\left(\frac{0.2548295919999999936678136691625695675611 + \left(\left(1.421413741000000063863240029604639858007 + \frac{1}{\frac{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}{-1.453152027000000012790792425221297889948 + 1.061405428999999900341322245367337018251 \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}}\right) \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} + -0.2844967359999999723108032867457950487733\right) \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \frac{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}{1}}\right)}^{3}}}}{\frac{\left(-0.2844967359999999723108032867457950487733 + \left(1.421413741000000063863240029604639858007 + \frac{1 \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1} \cdot 1.061405428999999900341322245367337018251\right)}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right) \cdot \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right) \cdot \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1} + 0.2548295919999999936678136691625695675611}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \frac{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}{1}} \cdot \left(\frac{\left(-0.2844967359999999723108032867457950487733 + \left(1.421413741000000063863240029604639858007 + \frac{1 \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1} \cdot 1.061405428999999900341322245367337018251\right)}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right) \cdot \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right) \cdot \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1} + 0.2548295919999999936678136691625695675611}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \frac{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}{1}} + 1\right) + 1 \cdot 1}\]
14. Using strategy rm

15. Applied add-cbrt-cube 12.9

    \[\leadsto \frac{{1}^{3} - \sqrt{\color{blue}{\sqrt[3]{\left({\left(\frac{0.2548295919999999936678136691625695675611 + \left(\left(1.421413741000000063863240029604639858007 + \frac{1}{\frac{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}{-1.453152027000000012790792425221297889948 + 1.061405428999999900341322245367337018251 \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}}\right) \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} + -0.2844967359999999723108032867457950487733\right) \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \frac{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}{1}}\right)}^{3} \cdot {\left(\frac{0.2548295919999999936678136691625695675611 + \left(\left(1.421413741000000063863240029604639858007 + \frac{1}{\frac{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}{-1.453152027000000012790792425221297889948 + 1.061405428999999900341322245367337018251 \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}}\right) \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} + -0.2844967359999999723108032867457950487733\right) \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \frac{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}{1}}\right)}^{3}\right) \cdot {\left(\frac{0.2548295919999999936678136691625695675611 + \left(\left(1.421413741000000063863240029604639858007 + \frac{1}{\frac{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}{-1.453152027000000012790792425221297889948 + 1.061405428999999900341322245367337018251 \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}}\right) \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} + -0.2844967359999999723108032867457950487733\right) \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \frac{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}{1}}\right)}^{3}}}} \cdot \sqrt{{\left(\frac{0.2548295919999999936678136691625695675611 + \left(\left(1.421413741000000063863240029604639858007 + \frac{1}{\frac{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}{-1.453152027000000012790792425221297889948 + 1.061405428999999900341322245367337018251 \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}}\right) \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} + -0.2844967359999999723108032867457950487733\right) \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \frac{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}{1}}\right)}^{3}}}{\frac{\left(-0.2844967359999999723108032867457950487733 + \left(1.421413741000000063863240029604639858007 + \frac{1 \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1} \cdot 1.061405428999999900341322245367337018251\right)}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right) \cdot \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right) \cdot \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1} + 0.2548295919999999936678136691625695675611}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \frac{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}{1}} \cdot \left(\frac{\left(-0.2844967359999999723108032867457950487733 + \left(1.421413741000000063863240029604639858007 + \frac{1 \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1} \cdot 1.061405428999999900341322245367337018251\right)}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right) \cdot \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right) \cdot \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1} + 0.2548295919999999936678136691625695675611}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \frac{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}{1}} + 1\right) + 1 \cdot 1}\]

16. Simplified 12.9

    \[\leadsto \frac{{1}^{3} - \sqrt{\sqrt[3]{\color{blue}{{\left({\left(\frac{0.2548295919999999936678136691625695675611 + \left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(\frac{1}{\frac{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}{\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251 + -1.453152027000000012790792425221297889948}} + 1.421413741000000063863240029604639858007\right) + -0.2844967359999999723108032867457950487733\right) \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}{\frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1\right)}{1}}\right)}^{3}\right)}^{3}}}} \cdot \sqrt{{\left(\frac{0.2548295919999999936678136691625695675611 + \left(\left(1.421413741000000063863240029604639858007 + \frac{1}{\frac{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}{-1.453152027000000012790792425221297889948 + 1.061405428999999900341322245367337018251 \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}}\right) \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} + -0.2844967359999999723108032867457950487733\right) \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \frac{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}{1}}\right)}^{3}}}{\frac{\left(-0.2844967359999999723108032867457950487733 + \left(1.421413741000000063863240029604639858007 + \frac{1 \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1} \cdot 1.061405428999999900341322245367337018251\right)}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right) \cdot \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right) \cdot \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1} + 0.2548295919999999936678136691625695675611}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \frac{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}{1}} \cdot \left(\frac{\left(-0.2844967359999999723108032867457950487733 + \left(1.421413741000000063863240029604639858007 + \frac{1 \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1} \cdot 1.061405428999999900341322245367337018251\right)}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right) \cdot \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}\right) \cdot \frac{1}{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1} + 0.2548295919999999936678136691625695675611}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \frac{0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1}{1}} + 1\right) + 1 \cdot 1}\]

17. Final simplification 12.9

    \[\leadsto \frac{{1}^{3} - \sqrt{{\left(\frac{0.2548295919999999936678136691625695675611 + \left(-0.2844967359999999723108032867457950487733 + \left(\frac{1}{\frac{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}{-1.453152027000000012790792425221297889948 + 1.061405428999999900341322245367337018251 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}}} + 1.421413741000000063863240029604639858007\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \frac{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}{1}}\right)}^{3}} \cdot \sqrt{\sqrt[3]{{\left({\left(\frac{0.2548295919999999936678136691625695675611 + \left(-0.2844967359999999723108032867457950487733 + \left(\frac{1}{\frac{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}{-1.453152027000000012790792425221297889948 + 1.061405428999999900341322245367337018251 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}}} + 1.421413741000000063863240029604639858007\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}}{\frac{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699\right)}{1}}\right)}^{3}\right)}^{3}}}}{1 \cdot 1 + \left(1 + \frac{\left(\frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699} \cdot \left(1.421413741000000063863240029604639858007 + \frac{\left(-1.453152027000000012790792425221297889948 + 1.061405428999999900341322245367337018251 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}\right) \cdot 1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}\right) + -0.2844967359999999723108032867457950487733\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699} + 0.2548295919999999936678136691625695675611}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \frac{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}{1}}\right) \cdot \frac{\left(\frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699} \cdot \left(1.421413741000000063863240029604639858007 + \frac{\left(-1.453152027000000012790792425221297889948 + 1.061405428999999900341322245367337018251 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}\right) \cdot 1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}\right) + -0.2844967359999999723108032867457950487733\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699} + 0.2548295919999999936678136691625695675611}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right|\right)} \cdot \frac{1 + \left|x\right| \cdot 0.3275911000000000239396058532292954623699}{1}}}\]

Reproduce

herbie shell --seed 2019196 
(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)))))))