Jmat.Real.erf

Average Error: 13.8 → 13.0

Time: 13.0s

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{e^{\log \left(\mathsf{fma}\left(-{\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right)}^{3} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)\right)}^{3}\right)\right), {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right), 1\right), \frac{\frac{\left(\left(\sqrt[3]{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}\right) \cdot 1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}}{e^{{\left(\left|x\right|\right)}^{2}}}, 1 \cdot 1\right)}\]

C

double f(double x) {
        double r316402 = 1.0;
        double r316403 = 0.3275911;
        double r316404 = x;
        double r316405 = fabs(r316404);
        double r316406 = r316403 * r316405;
        double r316407 = r316402 + r316406;
        double r316408 = r316402 / r316407;
        double r316409 = 0.254829592;
        double r316410 = -0.284496736;
        double r316411 = 1.421413741;
        double r316412 = -1.453152027;
        double r316413 = 1.061405429;
        double r316414 = r316408 * r316413;
        double r316415 = r316412 + r316414;
        double r316416 = r316408 * r316415;
        double r316417 = r316411 + r316416;
        double r316418 = r316408 * r316417;
        double r316419 = r316410 + r316418;
        double r316420 = r316408 * r316419;
        double r316421 = r316409 + r316420;
        double r316422 = r316408 * r316421;
        double r316423 = r316405 * r316405;
        double r316424 = -r316423;
        double r316425 = exp(r316424);
        double r316426 = r316422 * r316425;
        double r316427 = r316402 - r316426;
        return r316427;
}

↓

double f(double x) {
        double r316428 = 1.0;
        double r316429 = 0.3275911;
        double r316430 = x;
        double r316431 = fabs(r316430);
        double r316432 = r316429 * r316431;
        double r316433 = r316428 + r316432;
        double r316434 = r316428 / r316433;
        double r316435 = 3.0;
        double r316436 = pow(r316434, r316435);
        double r316437 = 1.061405429;
        double r316438 = -1.453152027;
        double r316439 = fma(r316434, r316437, r316438);
        double r316440 = 1.421413741;
        double r316441 = fma(r316434, r316439, r316440);
        double r316442 = -0.284496736;
        double r316443 = fma(r316434, r316441, r316442);
        double r316444 = 0.254829592;
        double r316445 = fma(r316434, r316443, r316444);
        double r316446 = pow(r316445, r316435);
        double r316447 = expm1(r316446);
        double r316448 = log1p(r316447);
        double r316449 = r316436 * r316448;
        double r316450 = -r316449;
        double r316451 = 1.0;
        double r316452 = 2.0;
        double r316453 = pow(r316431, r316452);
        double r316454 = exp(r316453);
        double r316455 = r316451 / r316454;
        double r316456 = pow(r316455, r316435);
        double r316457 = pow(r316428, r316435);
        double r316458 = fma(r316450, r316456, r316457);
        double r316459 = log(r316458);
        double r316460 = exp(r316459);
        double r316461 = r316455 * r316434;
        double r316462 = fma(r316461, r316445, r316428);
        double r316463 = cbrt(r316445);
        double r316464 = r316463 * r316463;
        double r316465 = r316464 * r316463;
        double r316466 = r316465 * r316428;
        double r316467 = fma(r316429, r316431, r316428);
        double r316468 = r316466 / r316467;
        double r316469 = r316468 / r316454;
        double r316470 = r316428 * r316428;
        double r316471 = fma(r316462, r316469, r316470);
        double r316472 = r316460 / r316471;
        return r316472;
}

Error

Bits error versus x

Derivation

1. Initial program 13.8

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

3. Applied flip3-- 13.8

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

4. Simplified 13.8

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

5. Simplified 13.8

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

7. Applied log1p-expm1-u 13.0

   \[\leadsto \frac{\mathsf{fma}\left(-{\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right)}^{3} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)\right)}^{3}\right)\right)}, {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right), 1\right), \frac{\frac{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right) \cdot 1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}}{e^{{\left(\left|x\right|\right)}^{2}}}, 1 \cdot 1\right)}\]
8. Using strategy rm

9. Applied add-cube-cbrt 13.0

   \[\leadsto \frac{\mathsf{fma}\left(-{\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right)}^{3} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)\right)}^{3}\right)\right), {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right), 1\right), \frac{\frac{\color{blue}{\left(\left(\sqrt[3]{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}\right)} \cdot 1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}}{e^{{\left(\left|x\right|\right)}^{2}}}, 1 \cdot 1\right)}\]
10. Using strategy rm

11. Applied add-exp-log 13.0

    \[\leadsto \frac{\color{blue}{e^{\log \left(\mathsf{fma}\left(-{\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right)}^{3} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)\right)}^{3}\right)\right), {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)\right)}}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right), 1\right), \frac{\frac{\left(\left(\sqrt[3]{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}\right) \cdot 1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}}{e^{{\left(\left|x\right|\right)}^{2}}}, 1 \cdot 1\right)}\]

12. Final simplification 13.0

    \[\leadsto \frac{e^{\log \left(\mathsf{fma}\left(-{\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}\right)}^{3} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\left(\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)\right)}^{3}\right)\right), {\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right)}^{3}, {1}^{3}\right)\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right), 1\right), \frac{\frac{\left(\left(\sqrt[3]{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}\right) \cdot 1}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)}}{e^{{\left(\left|x\right|\right)}^{2}}}, 1 \cdot 1\right)}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 1 (* (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))