Jmat.Real.erf

Average Error: 13.8 → 13.1

Time: 37.5s

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

C

double f(double x) {
        double r140512 = 1.0;
        double r140513 = 0.3275911;
        double r140514 = x;
        double r140515 = fabs(r140514);
        double r140516 = r140513 * r140515;
        double r140517 = r140512 + r140516;
        double r140518 = r140512 / r140517;
        double r140519 = 0.254829592;
        double r140520 = -0.284496736;
        double r140521 = 1.421413741;
        double r140522 = -1.453152027;
        double r140523 = 1.061405429;
        double r140524 = r140518 * r140523;
        double r140525 = r140522 + r140524;
        double r140526 = r140518 * r140525;
        double r140527 = r140521 + r140526;
        double r140528 = r140518 * r140527;
        double r140529 = r140520 + r140528;
        double r140530 = r140518 * r140529;
        double r140531 = r140519 + r140530;
        double r140532 = r140518 * r140531;
        double r140533 = r140515 * r140515;
        double r140534 = -r140533;
        double r140535 = exp(r140534);
        double r140536 = r140532 * r140535;
        double r140537 = r140512 - r140536;
        return r140537;
}

↓

double f(double x) {
        double r140538 = 1.0;
        double r140539 = 3.0;
        double r140540 = pow(r140538, r140539);
        double r140541 = 0.254829592;
        double r140542 = -0.284496736;
        double r140543 = 0.3275911;
        double r140544 = x;
        double r140545 = fabs(r140544);
        double r140546 = r140543 * r140545;
        double r140547 = r140546 + r140538;
        double r140548 = 1.421413741;
        double r140549 = r140538 + r140546;
        double r140550 = r140538 / r140549;
        double r140551 = -1.453152027;
        double r140552 = 1.061405429;
        double r140553 = r140550 * r140552;
        double r140554 = r140551 + r140553;
        double r140555 = r140550 * r140554;
        double r140556 = r140548 + r140555;
        double r140557 = r140547 / r140556;
        double r140558 = r140538 / r140557;
        double r140559 = r140542 + r140558;
        double r140560 = r140538 * r140559;
        double r140561 = r140560 / r140547;
        double r140562 = r140541 + r140561;
        double r140563 = 2.0;
        double r140564 = pow(r140545, r140563);
        double r140565 = exp(r140564);
        double r140566 = r140565 / r140538;
        double r140567 = r140562 / r140566;
        double r140568 = r140567 / r140547;
        double r140569 = pow(r140568, r140539);
        double r140570 = sqrt(r140569);
        double r140571 = r140570 * r140570;
        double r140572 = r140540 - r140571;
        double r140573 = r140538 * r140538;
        double r140574 = r140538 + r140568;
        double r140575 = r140574 * r140550;
        double r140576 = r140562 / r140565;
        double r140577 = r140575 * r140576;
        double r140578 = r140573 + r140577;
        double r140579 = r140572 / r140578;
        return r140579;
}

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. Simplified 13.8

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

4. Applied distribute-rgt-in 13.8

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

5. Applied associate-+r+ 13.8

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

7. Applied add-cube-cbrt 13.8

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

8. Applied associate-/r* 13.8

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

10. Applied flip3-- 13.9

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

11. Simplified 13.8

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

12. Simplified 13.8

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

14. Applied add-sqr-sqrt 13.1

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

15. Final simplification 13.1

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

Reproduce

herbie shell --seed 2019303 
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 1 (* (* (/ 1 (+ 1 (* 0.32759110000000002 (fabs x)))) (+ 0.25482959199999999 (* (/ 1 (+ 1 (* 0.32759110000000002 (fabs x)))) (+ -0.284496735999999972 (* (/ 1 (+ 1 (* 0.32759110000000002 (fabs x)))) (+ 1.42141374100000006 (* (/ 1 (+ 1 (* 0.32759110000000002 (fabs x)))) (+ -1.45315202700000001 (* (/ 1 (+ 1 (* 0.32759110000000002 (fabs x)))) 1.0614054289999999))))))))) (exp (- (* (fabs x) (fabs x)))))))