Jmat.Real.erf

Average Error: 13.5 → 12.7

Time: 1.7m

Precision: 64

Math

\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

↓

\[\left(\sqrt[3]{\log \left(e^{1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)} + \left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right)}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)} + \left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right)}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}\right) \cdot \sqrt[3]{1 - \log \left(e^{\frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)} + \left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right)}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}\]

C

double f(double x) {
        double r3846384 = 1.0;
        double r3846385 = 0.3275911;
        double r3846386 = x;
        double r3846387 = fabs(r3846386);
        double r3846388 = r3846385 * r3846387;
        double r3846389 = r3846384 + r3846388;
        double r3846390 = r3846384 / r3846389;
        double r3846391 = 0.254829592;
        double r3846392 = -0.284496736;
        double r3846393 = 1.421413741;
        double r3846394 = -1.453152027;
        double r3846395 = 1.061405429;
        double r3846396 = r3846390 * r3846395;
        double r3846397 = r3846394 + r3846396;
        double r3846398 = r3846390 * r3846397;
        double r3846399 = r3846393 + r3846398;
        double r3846400 = r3846390 * r3846399;
        double r3846401 = r3846392 + r3846400;
        double r3846402 = r3846390 * r3846401;
        double r3846403 = r3846391 + r3846402;
        double r3846404 = r3846390 * r3846403;
        double r3846405 = r3846387 * r3846387;
        double r3846406 = -r3846405;
        double r3846407 = exp(r3846406);
        double r3846408 = r3846404 * r3846407;
        double r3846409 = r3846384 - r3846408;
        return r3846409;
}

↓

double f(double x) {
        double r3846410 = 1.0;
        double r3846411 = 0.254829592;
        double r3846412 = -0.284496736;
        double r3846413 = 1.061405429;
        double r3846414 = x;
        double r3846415 = fabs(r3846414);
        double r3846416 = 0.3275911;
        double r3846417 = fma(r3846415, r3846416, r3846410);
        double r3846418 = r3846417 * r3846417;
        double r3846419 = r3846413 / r3846418;
        double r3846420 = 1.421413741;
        double r3846421 = 1.453152027;
        double r3846422 = r3846421 / r3846417;
        double r3846423 = r3846420 - r3846422;
        double r3846424 = r3846419 + r3846423;
        double r3846425 = r3846424 / r3846417;
        double r3846426 = r3846412 + r3846425;
        double r3846427 = r3846426 / r3846417;
        double r3846428 = r3846411 + r3846427;
        double r3846429 = r3846428 / r3846417;
        double r3846430 = r3846415 * r3846415;
        double r3846431 = exp(r3846430);
        double r3846432 = r3846429 / r3846431;
        double r3846433 = r3846410 - r3846432;
        double r3846434 = exp(r3846433);
        double r3846435 = log(r3846434);
        double r3846436 = cbrt(r3846435);
        double r3846437 = r3846436 * r3846436;
        double r3846438 = exp(r3846432);
        double r3846439 = log(r3846438);
        double r3846440 = r3846410 - r3846439;
        double r3846441 = cbrt(r3846440);
        double r3846442 = r3846437 * r3846441;
        return r3846442;
}

Error

Bits error versus x

Derivation

1. Initial program 13.5

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

2. Taylor expanded around inf 13.5

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \color{blue}{\frac{\left(1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}} + 1.421413741\right) - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}}{0.3275911 \cdot \left|x\right| + 1}}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

3. Simplified 13.5

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \color{blue}{\frac{\left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right) + \frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
4. Using strategy rm

5. Applied add-log-exp 13.5

   \[\leadsto 1 - \color{blue}{\log \left(e^{\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{\left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right) + \frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\]

6. Applied add-log-exp 13.5

   \[\leadsto \color{blue}{\log \left(e^{1}\right)} - \log \left(e^{\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{\left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right) + \frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)\]

7. Applied diff-log 14.2

   \[\leadsto \color{blue}{\log \left(\frac{e^{1}}{e^{\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{\left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right) + \frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}}\right)}\]

8. Simplified 13.5

   \[\leadsto \log \color{blue}{\left(e^{1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{\left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right) + \frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}\]
9. Using strategy rm

10. Applied add-cube-cbrt 13.5

    \[\leadsto \color{blue}{\left(\sqrt[3]{\log \left(e^{1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{\left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right) + \frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{\left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right) + \frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{\left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right) + \frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}}\]
11. Using strategy rm

12. Applied exp-diff 12.8

    \[\leadsto \left(\sqrt[3]{\log \left(e^{1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{\left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right) + \frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{\left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right) + \frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}\right) \cdot \sqrt[3]{\log \color{blue}{\left(\frac{e^{1}}{e^{\frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{\left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right) + \frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}}\right)}}\]

13. Applied log-div 12.7

    \[\leadsto \left(\sqrt[3]{\log \left(e^{1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{\left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right) + \frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{\left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right) + \frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}\right) \cdot \sqrt[3]{\color{blue}{\log \left(e^{1}\right) - \log \left(e^{\frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{\left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right) + \frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}}\]

14. Simplified 12.7

    \[\leadsto \left(\sqrt[3]{\log \left(e^{1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{\left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right) + \frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{\left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right) + \frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}\right) \cdot \sqrt[3]{\color{blue}{\log e} - \log \left(e^{\frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{\left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right) + \frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}\]

15. Final simplification 12.7

    \[\leadsto \left(\sqrt[3]{\log \left(e^{1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)} + \left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right)}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)} + \left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right)}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}\right) \cdot \sqrt[3]{1 - \log \left(e^{\frac{\frac{0.254829592 + \frac{-0.284496736 + \frac{\frac{1.061405429}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)} + \left(1.421413741 - \frac{1.453152027}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}\right)}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{\mathsf{fma}\left(\left(\left|x\right|\right), 0.3275911, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erf"
  (- 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)))))))