Jmat.Real.erf

Average Error: 13.7 → 13.0

Time: 11.0s

Precision: 64

Math

\[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

↓

\[\left(\sqrt[3]{\left(\left(\mathsf{fma}\left(0.284496735999999972, \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}, 1\right) - \frac{1.42141374100000006}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}} \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right) - \mathsf{fma}\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{5}}, 1.0614054289999999, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} \cdot 0.25482959199999999\right)\right) + \frac{\frac{1.45315202700000001}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}}}{e^{{\left(\left|x\right|\right)}^{2}}}} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)}\right) \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left({\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3}, \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, -0.284496735999999972\right), 0.25482959199999999\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}\right)}\]

C

double f(double x) {
        double r263395 = 1.0;
        double r263396 = 0.3275911;
        double r263397 = x;
        double r263398 = fabs(r263397);
        double r263399 = r263396 * r263398;
        double r263400 = r263395 + r263399;
        double r263401 = r263395 / r263400;
        double r263402 = 0.254829592;
        double r263403 = -0.284496736;
        double r263404 = 1.421413741;
        double r263405 = -1.453152027;
        double r263406 = 1.061405429;
        double r263407 = r263401 * r263406;
        double r263408 = r263405 + r263407;
        double r263409 = r263401 * r263408;
        double r263410 = r263404 + r263409;
        double r263411 = r263401 * r263410;
        double r263412 = r263403 + r263411;
        double r263413 = r263401 * r263412;
        double r263414 = r263402 + r263413;
        double r263415 = r263401 * r263414;
        double r263416 = r263398 * r263398;
        double r263417 = -r263416;
        double r263418 = exp(r263417);
        double r263419 = r263415 * r263418;
        double r263420 = r263395 - r263419;
        return r263420;
}

↓

double f(double x) {
        double r263421 = 0.284496736;
        double r263422 = 1.0;
        double r263423 = x;
        double r263424 = fabs(r263423);
        double r263425 = 2.0;
        double r263426 = pow(r263424, r263425);
        double r263427 = exp(r263426);
        double r263428 = 0.3275911;
        double r263429 = r263428 * r263424;
        double r263430 = 1.0;
        double r263431 = r263429 + r263430;
        double r263432 = pow(r263431, r263425);
        double r263433 = r263427 * r263432;
        double r263434 = r263422 / r263433;
        double r263435 = fma(r263421, r263434, r263430);
        double r263436 = 1.421413741;
        double r263437 = fma(r263428, r263424, r263430);
        double r263438 = 3.0;
        double r263439 = pow(r263437, r263438);
        double r263440 = r263436 / r263439;
        double r263441 = r263422 / r263427;
        double r263442 = r263440 * r263441;
        double r263443 = r263435 - r263442;
        double r263444 = 5.0;
        double r263445 = pow(r263431, r263444);
        double r263446 = r263427 * r263445;
        double r263447 = r263422 / r263446;
        double r263448 = 1.061405429;
        double r263449 = r263441 / r263437;
        double r263450 = 0.254829592;
        double r263451 = r263449 * r263450;
        double r263452 = fma(r263447, r263448, r263451);
        double r263453 = r263443 - r263452;
        double r263454 = 1.453152027;
        double r263455 = 4.0;
        double r263456 = pow(r263437, r263455);
        double r263457 = r263454 / r263456;
        double r263458 = r263457 / r263427;
        double r263459 = r263453 + r263458;
        double r263460 = cbrt(r263459);
        double r263461 = r263430 + r263429;
        double r263462 = r263430 / r263461;
        double r263463 = cbrt(r263462);
        double r263464 = r263463 * r263463;
        double r263465 = r263464 * r263463;
        double r263466 = -1.453152027;
        double r263467 = fma(r263462, r263448, r263466);
        double r263468 = fma(r263465, r263467, r263436);
        double r263469 = -0.284496736;
        double r263470 = fma(r263462, r263468, r263469);
        double r263471 = fma(r263462, r263470, r263450);
        double r263472 = r263424 * r263424;
        double r263473 = exp(r263472);
        double r263474 = r263471 / r263473;
        double r263475 = -r263430;
        double r263476 = fma(r263424, r263428, r263430);
        double r263477 = r263475 / r263476;
        double r263478 = fma(r263474, r263477, r263430);
        double r263479 = cbrt(r263478);
        double r263480 = r263460 * r263479;
        double r263481 = r263430 / r263476;
        double r263482 = pow(r263463, r263438);
        double r263483 = fma(r263481, r263448, r263466);
        double r263484 = fma(r263482, r263483, r263436);
        double r263485 = fma(r263484, r263481, r263469);
        double r263486 = fma(r263481, r263485, r263450);
        double r263487 = r263481 * r263486;
        double r263488 = r263487 / r263427;
        double r263489 = r263430 - r263488;
        double r263490 = exp(r263489);
        double r263491 = log(r263490);
        double r263492 = cbrt(r263491);
        double r263493 = r263480 * r263492;
        return r263493;
}

Error

Bits error versus x

Derivation

1. Initial program 13.7

   \[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

2. Simplified 13.7

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)}\]
3. Using strategy rm

4. Applied add-cube-cbrt 13.7

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\color{blue}{\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)\]
5. Using strategy rm

6. Applied add-cube-cbrt 13.7

   \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)}}\]
7. Using strategy rm

8. Applied add-log-exp 13.0

   \[\leadsto \left(\sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)}\right) \cdot \sqrt[3]{\color{blue}{\log \left(e^{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)}\right)}}\]

9. Simplified 13.0

   \[\leadsto \left(\sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)}\right) \cdot \sqrt[3]{\log \color{blue}{\left(e^{1 - \frac{\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left({\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3}, \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, -0.284496735999999972\right), 0.25482959199999999\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}\right)}}\]

10. Taylor expanded around 0 13.0

    \[\leadsto \left(\sqrt[3]{\color{blue}{\left(1 + \left(1.45315202700000001 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}} + 0.284496735999999972 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}\right)\right) - \left(1.42141374100000006 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}} + \left(1.0614054289999999 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{5}} + 0.25482959199999999 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(0.32759110000000002 \cdot \left|x\right| + 1\right)}\right)\right)}} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)}\right) \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left({\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3}, \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, -0.284496735999999972\right), 0.25482959199999999\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}\right)}\]

11. Simplified 13.0

    \[\leadsto \left(\sqrt[3]{\color{blue}{\left(\left(\mathsf{fma}\left(0.284496735999999972, \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}, 1\right) - \frac{1.42141374100000006}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}} \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right) - \mathsf{fma}\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{5}}, 1.0614054289999999, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} \cdot 0.25482959199999999\right)\right) + \frac{\frac{1.45315202700000001}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}}}{e^{{\left(\left|x\right|\right)}^{2}}}}} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)}\right) \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left({\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3}, \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, -0.284496735999999972\right), 0.25482959199999999\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}\right)}\]

12. Final simplification 13.0

    \[\leadsto \left(\sqrt[3]{\left(\left(\mathsf{fma}\left(0.284496735999999972, \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}, 1\right) - \frac{1.42141374100000006}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{3}} \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right) - \mathsf{fma}\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{5}}, 1.0614054289999999, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)} \cdot 0.25482959199999999\right)\right) + \frac{\frac{1.45315202700000001}{{\left(\mathsf{fma}\left(0.32759110000000002, \left|x\right|, 1\right)\right)}^{4}}}{e^{{\left(\left|x\right|\right)}^{2}}}} \cdot \sqrt[3]{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, \mathsf{fma}\left(\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}} \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}, \mathsf{fma}\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), -0.284496735999999972\right), 0.25482959199999999\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1\right)}\right) \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)} \cdot \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, \mathsf{fma}\left(\mathsf{fma}\left({\left(\sqrt[3]{\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|}}\right)}^{3}, \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, 1.0614054289999999, -1.45315202700000001\right), 1.42141374100000006\right), \frac{1}{\mathsf{fma}\left(\left|x\right|, 0.32759110000000002, 1\right)}, -0.284496735999999972\right), 0.25482959199999999\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}\right)}\]

Reproduce

herbie shell --seed 2020024 +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)))))))