Jmat.Real.erf

Average Error: 13.7 → 9.9

Time: 31.4s

Precision: 64

Math

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

↓

\[\left(\log \left(\sqrt{\sqrt{e^{1 - \mathsf{fma}\left(\mathsf{fma}\left(1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, -0.284496736\right), \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 0.254829592\right) \cdot \frac{\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}}}\right) + \log \left(\sqrt{\sqrt{e^{1 - \mathsf{fma}\left(\mathsf{fma}\left(1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, -0.284496736\right), \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 0.254829592\right) \cdot \frac{\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}}}\right)\right) + \frac{1 - \mathsf{fma}\left(\mathsf{fma}\left(1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, -0.284496736\right), \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 0.254829592\right) \cdot \frac{\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}{2}\]

C

double f(double x) {
        double r2335434 = 1.0;
        double r2335435 = 0.3275911;
        double r2335436 = x;
        double r2335437 = fabs(r2335436);
        double r2335438 = r2335435 * r2335437;
        double r2335439 = r2335434 + r2335438;
        double r2335440 = r2335434 / r2335439;
        double r2335441 = 0.254829592;
        double r2335442 = -0.284496736;
        double r2335443 = 1.421413741;
        double r2335444 = -1.453152027;
        double r2335445 = 1.061405429;
        double r2335446 = r2335440 * r2335445;
        double r2335447 = r2335444 + r2335446;
        double r2335448 = r2335440 * r2335447;
        double r2335449 = r2335443 + r2335448;
        double r2335450 = r2335440 * r2335449;
        double r2335451 = r2335442 + r2335450;
        double r2335452 = r2335440 * r2335451;
        double r2335453 = r2335441 + r2335452;
        double r2335454 = r2335440 * r2335453;
        double r2335455 = r2335437 * r2335437;
        double r2335456 = -r2335455;
        double r2335457 = exp(r2335456);
        double r2335458 = r2335454 * r2335457;
        double r2335459 = r2335434 - r2335458;
        return r2335459;
}

↓

double f(double x) {
        double r2335460 = 1.0;
        double r2335461 = 1.421413741;
        double r2335462 = 1.061405429;
        double r2335463 = 0.3275911;
        double r2335464 = x;
        double r2335465 = fabs(r2335464);
        double r2335466 = fma(r2335463, r2335465, r2335460);
        double r2335467 = r2335462 / r2335466;
        double r2335468 = -1.453152027;
        double r2335469 = r2335467 + r2335468;
        double r2335470 = r2335469 / r2335466;
        double r2335471 = r2335461 + r2335470;
        double r2335472 = r2335460 / r2335466;
        double r2335473 = -0.284496736;
        double r2335474 = fma(r2335471, r2335472, r2335473);
        double r2335475 = 0.254829592;
        double r2335476 = fma(r2335474, r2335472, r2335475);
        double r2335477 = r2335465 * r2335465;
        double r2335478 = exp(r2335477);
        double r2335479 = r2335472 / r2335478;
        double r2335480 = r2335476 * r2335479;
        double r2335481 = r2335460 - r2335480;
        double r2335482 = exp(r2335481);
        double r2335483 = sqrt(r2335482);
        double r2335484 = sqrt(r2335483);
        double r2335485 = log(r2335484);
        double r2335486 = r2335485 + r2335485;
        double r2335487 = 2.0;
        double r2335488 = r2335481 / r2335487;
        double r2335489 = r2335486 + r2335488;
        return r2335489;
}

Error

Bits error versus x

Derivation

1. Initial program 13.7

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

2. Simplified 13.6

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

4. Applied add-log-exp 13.6

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

5. Applied add-log-exp 13.6

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

6. Applied diff-log 14.4

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

7. Simplified 13.6

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

9. Applied add-sqr-sqrt 13.6

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

10. Applied log-prod 13.6

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

12. Applied *-un-lft-identity 13.6

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

13. Applied exp-prod 13.6

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

14. Applied sqrt-pow1 13.6

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

15. Applied log-pow 9.6

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

16. Simplified 9.6

    \[\leadsto \frac{1 - \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741, \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, -0.284496736\right), \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 0.254829592\right) \cdot \frac{\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}{2} \cdot \color{blue}{1} + \log \left(\sqrt{e^{1 - \mathsf{fma}\left(\mathsf{fma}\left(\frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + 1.421413741, \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, -0.284496736\right), \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 0.254829592\right) \cdot \frac{\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}}\right)\]
17. Using strategy rm

18. Applied add-sqr-sqrt 10.1

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

19. Applied log-prod 9.9

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

20. Final simplification 9.9

    \[\leadsto \left(\log \left(\sqrt{\sqrt{e^{1 - \mathsf{fma}\left(\mathsf{fma}\left(1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, -0.284496736\right), \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 0.254829592\right) \cdot \frac{\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}}}\right) + \log \left(\sqrt{\sqrt{e^{1 - \mathsf{fma}\left(\mathsf{fma}\left(1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, -0.284496736\right), \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 0.254829592\right) \cdot \frac{\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}}}\right)\right) + \frac{1 - \mathsf{fma}\left(\mathsf{fma}\left(1.421413741 + \frac{\frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -1.453152027}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, -0.284496736\right), \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}, 0.254829592\right) \cdot \frac{\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{e^{\left|x\right| \cdot \left|x\right|}}}{2}\]

Reproduce

herbie shell --seed 2019152 +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)))))))