Jmat.Real.erf

Average Error: 13.8 → 13.7

Time: 48.8s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r11780511 = 1.0;
        double r11780512 = 0.3275911;
        double r11780513 = x;
        double r11780514 = fabs(r11780513);
        double r11780515 = r11780512 * r11780514;
        double r11780516 = r11780511 + r11780515;
        double r11780517 = r11780511 / r11780516;
        double r11780518 = 0.254829592;
        double r11780519 = -0.284496736;
        double r11780520 = 1.421413741;
        double r11780521 = -1.453152027;
        double r11780522 = 1.061405429;
        double r11780523 = r11780517 * r11780522;
        double r11780524 = r11780521 + r11780523;
        double r11780525 = r11780517 * r11780524;
        double r11780526 = r11780520 + r11780525;
        double r11780527 = r11780517 * r11780526;
        double r11780528 = r11780519 + r11780527;
        double r11780529 = r11780517 * r11780528;
        double r11780530 = r11780518 + r11780529;
        double r11780531 = r11780517 * r11780530;
        double r11780532 = r11780514 * r11780514;
        double r11780533 = -r11780532;
        double r11780534 = exp(r11780533);
        double r11780535 = r11780531 * r11780534;
        double r11780536 = r11780511 - r11780535;
        return r11780536;
}

↓

double f(double x) {
        double r11780537 = 1.0;
        double r11780538 = r11780537 * r11780537;
        double r11780539 = 3.0;
        double r11780540 = pow(r11780538, r11780539);
        double r11780541 = 0.3275911;
        double r11780542 = x;
        double r11780543 = fabs(r11780542);
        double r11780544 = r11780541 * r11780543;
        double r11780545 = r11780537 + r11780544;
        double r11780546 = r11780537 / r11780545;
        double r11780547 = 0.254829592;
        double r11780548 = -0.284496736;
        double r11780549 = 1.421413741;
        double r11780550 = -1.453152027;
        double r11780551 = r11780544 * r11780544;
        double r11780552 = r11780538 - r11780551;
        double r11780553 = r11780537 / r11780552;
        double r11780554 = r11780537 - r11780544;
        double r11780555 = r11780553 * r11780554;
        double r11780556 = 1.061405429;
        double r11780557 = r11780555 * r11780556;
        double r11780558 = r11780550 + r11780557;
        double r11780559 = r11780546 * r11780558;
        double r11780560 = r11780549 + r11780559;
        double r11780561 = r11780546 * r11780560;
        double r11780562 = r11780548 + r11780561;
        double r11780563 = r11780546 * r11780562;
        double r11780564 = r11780547 + r11780563;
        double r11780565 = r11780546 * r11780564;
        double r11780566 = r11780543 * r11780543;
        double r11780567 = -r11780566;
        double r11780568 = exp(r11780567);
        double r11780569 = r11780565 * r11780568;
        double r11780570 = r11780569 * r11780569;
        double r11780571 = pow(r11780570, r11780539);
        double r11780572 = r11780540 - r11780571;
        double r11780573 = r11780538 * r11780538;
        double r11780574 = r11780570 * r11780570;
        double r11780575 = r11780538 * r11780570;
        double r11780576 = r11780574 + r11780575;
        double r11780577 = r11780573 + r11780576;
        double r11780578 = r11780572 / r11780577;
        double r11780579 = r11780537 + r11780569;
        double r11780580 = r11780578 / r11780579;
        return r11780580;
}

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 flip-+ 13.8

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

4. Applied associate-/r/ 13.8

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

6. Applied flip-- 13.8

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

8. Applied flip3-- 13.7

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

9. Final simplification 13.7

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

Reproduce

herbie shell --seed 2019174 +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)))))))