Average Error: 13.7 → 1.9
Time: 9.8s
Precision: 64
\[1 - \left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(0.2548295919999999936678136691625695675611 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-0.2844967359999999723108032867457950487733 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(1.421413741000000063863240029604639858007 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot \left(-1.453152027000000012790792425221297889948 + \frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|} \cdot 1.061405428999999900341322245367337018251\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[\sqrt{\left(\left(\mathsf{fma}\left(0.2844967359999999723108032867457950487733, \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1\right)}^{2}}, 1\right) - \frac{1.421413741000000063863240029604639858007}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)\right)}^{3}} \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right) - \mathsf{fma}\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1\right)}^{5}}, 1.061405428999999900341322245367337018251, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)} \cdot 0.2548295919999999936678136691625695675611\right)\right) + \frac{\frac{1.453152027000000012790792425221297889948}{{\left(\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)\right)}^{4}}}{e^{{\left(\left|x\right|\right)}^{2}}}} \cdot \sqrt{e^{\log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\sqrt[3]{{\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}\right)}^{3}}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 1\right)\right)}}\]
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|}
↓
\sqrt{\left(\left(\mathsf{fma}\left(0.2844967359999999723108032867457950487733, \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1\right)}^{2}}, 1\right) - \frac{1.421413741000000063863240029604639858007}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 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(0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1\right)}^{5}}, 1.061405428999999900341322245367337018251, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)} \cdot 0.2548295919999999936678136691625695675611\right)\right) + \frac{\frac{1.453152027000000012790792425221297889948}{{\left(\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)\right)}^{4}}}{e^{{\left(\left|x\right|\right)}^{2}}}} \cdot \sqrt{e^{\log \left(\mathsf{fma}\left(\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(\sqrt[3]{{\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}\right)}^{3}}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 1\right)\right)}}
double f(double x) {
        double r200373 = 1.0;
        double r200374 = 0.3275911;
        double r200375 = x;
        double r200376 = fabs(r200375);
        double r200377 = r200374 * r200376;
        double r200378 = r200373 + r200377;
        double r200379 = r200373 / r200378;
        double r200380 = 0.254829592;
        double r200381 = -0.284496736;
        double r200382 = 1.421413741;
        double r200383 = -1.453152027;
        double r200384 = 1.061405429;
        double r200385 = r200379 * r200384;
        double r200386 = r200383 + r200385;
        double r200387 = r200379 * r200386;
        double r200388 = r200382 + r200387;
        double r200389 = r200379 * r200388;
        double r200390 = r200381 + r200389;
        double r200391 = r200379 * r200390;
        double r200392 = r200380 + r200391;
        double r200393 = r200379 * r200392;
        double r200394 = r200376 * r200376;
        double r200395 = -r200394;
        double r200396 = exp(r200395);
        double r200397 = r200393 * r200396;
        double r200398 = r200373 - r200397;
        return r200398;
}
↓
double f(double x) {
        double r200399 = 0.284496736;
        double r200400 = 1.0;
        double r200401 = x;
        double r200402 = fabs(r200401);
        double r200403 = 2.0;
        double r200404 = pow(r200402, r200403);
        double r200405 = exp(r200404);
        double r200406 = 0.3275911;
        double r200407 = r200406 * r200402;
        double r200408 = 1.0;
        double r200409 = r200407 + r200408;
        double r200410 = pow(r200409, r200403);
        double r200411 = r200405 * r200410;
        double r200412 = r200400 / r200411;
        double r200413 = fma(r200399, r200412, r200408);
        double r200414 = 1.421413741;
        double r200415 = fma(r200402, r200406, r200408);
        double r200416 = 3.0;
        double r200417 = pow(r200415, r200416);
        double r200418 = r200414 / r200417;
        double r200419 = r200400 / r200405;
        double r200420 = r200418 * r200419;
        double r200421 = r200413 - r200420;
        double r200422 = 5.0;
        double r200423 = pow(r200409, r200422);
        double r200424 = r200405 * r200423;
        double r200425 = r200400 / r200424;
        double r200426 = 1.061405429;
        double r200427 = fma(r200406, r200402, r200408);
        double r200428 = r200419 / r200427;
        double r200429 = 0.254829592;
        double r200430 = r200428 * r200429;
        double r200431 = fma(r200425, r200426, r200430);
        double r200432 = r200421 - r200431;
        double r200433 = 1.453152027;
        double r200434 = 4.0;
        double r200435 = pow(r200427, r200434);
        double r200436 = r200433 / r200435;
        double r200437 = r200436 / r200405;
        double r200438 = r200432 + r200437;
        double r200439 = sqrt(r200438);
        double r200440 = r200408 + r200407;
        double r200441 = r200408 / r200440;
        double r200442 = r200408 / r200415;
        double r200443 = pow(r200442, r200416);
        double r200444 = cbrt(r200443);
        double r200445 = -1.453152027;
        double r200446 = fma(r200444, r200426, r200445);
        double r200447 = fma(r200441, r200446, r200414);
        double r200448 = -0.284496736;
        double r200449 = fma(r200441, r200447, r200448);
        double r200450 = fma(r200441, r200449, r200429);
        double r200451 = r200402 * r200402;
        double r200452 = exp(r200451);
        double r200453 = r200450 / r200452;
        double r200454 = -r200408;
        double r200455 = r200454 / r200415;
        double r200456 = fma(r200453, r200455, r200408);
        double r200457 = log(r200456);
        double r200458 = exp(r200457);
        double r200459 = sqrt(r200458);
        double r200460 = r200439 * r200459;
        return r200460;
}

Error

Bits error versus x

Derivation

  1. Initial program 13.7

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

  2. Simplified13.7

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

  4. Applied add-cbrt-cube13.7

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

  5. Applied add-cbrt-cube13.7

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

  6. Applied cbrt-undiv13.7

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

  7. Simplified13.7

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

  9. Applied add-sqr-sqrt13.7

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

  10. Taylor expanded around 0 2.2

    \[\leadsto \sqrt{\color{blue}{\left(1 + \left(1.453152027000000012790792425221297889948 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1\right)}^{4}} + 0.2844967359999999723108032867457950487733 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1\right)}^{2}}\right)\right) - \left(1.421413741000000063863240029604639858007 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1\right)}^{3}} + \left(1.061405428999999900341322245367337018251 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1\right)}^{5}} + 0.2548295919999999936678136691625695675611 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1\right)}\right)\right)}} \cdot \sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\sqrt[3]{{\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}\right)}^{3}}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 1\right)}\]

  11. Simplified2.2

    \[\leadsto \sqrt{\color{blue}{\left(\left(\mathsf{fma}\left(0.2844967359999999723108032867457950487733, \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1\right)}^{2}}, 1\right) - \frac{1.421413741000000063863240029604639858007}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)\right)}^{3}} \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right) - \mathsf{fma}\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1\right)}^{5}}, 1.061405428999999900341322245367337018251, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)} \cdot 0.2548295919999999936678136691625695675611\right)\right) + \frac{\frac{1.453152027000000012790792425221297889948}{{\left(\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)\right)}^{4}}}{e^{{\left(\left|x\right|\right)}^{2}}}}} \cdot \sqrt{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\sqrt[3]{{\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}\right)}^{3}}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 1\right)}\]
  12. Using strategy rm

  13. Applied add-exp-log1.9

    \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(0.2844967359999999723108032867457950487733, \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1\right)}^{2}}, 1\right) - \frac{1.421413741000000063863240029604639858007}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)\right)}^{3}} \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right) - \mathsf{fma}\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1\right)}^{5}}, 1.061405428999999900341322245367337018251, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)} \cdot 0.2548295919999999936678136691625695675611\right)\right) + \frac{\frac{1.453152027000000012790792425221297889948}{{\left(\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)\right)}^{4}}}{e^{{\left(\left|x\right|\right)}^{2}}}} \cdot \sqrt{\color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\sqrt[3]{{\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}\right)}^{3}}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 1\right)\right)}}}\]

  14. Final simplification1.9

    \[\leadsto \sqrt{\left(\left(\mathsf{fma}\left(0.2844967359999999723108032867457950487733, \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1\right)}^{2}}, 1\right) - \frac{1.421413741000000063863240029604639858007}{{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)\right)}^{3}} \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}\right) - \mathsf{fma}\left(\frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911000000000239396058532292954623699 \cdot \left|x\right| + 1\right)}^{5}}, 1.061405428999999900341322245367337018251, \frac{\frac{1}{e^{{\left(\left|x\right|\right)}^{2}}}}{\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)} \cdot 0.2548295919999999936678136691625695675611\right)\right) + \frac{\frac{1.453152027000000012790792425221297889948}{{\left(\mathsf{fma}\left(0.3275911000000000239396058532292954623699, \left|x\right|, 1\right)\right)}^{4}}}{e^{{\left(\left|x\right|\right)}^{2}}}} \cdot \sqrt{e^{\log \left(\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\frac{1}{1 + 0.3275911000000000239396058532292954623699 \cdot \left|x\right|}, \mathsf{fma}\left(\sqrt[3]{{\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}\right)}^{3}}, 1.061405428999999900341322245367337018251, -1.453152027000000012790792425221297889948\right), 1.421413741000000063863240029604639858007\right), -0.2844967359999999723108032867457950487733\right), 0.2548295919999999936678136691625695675611\right)}{e^{\left|x\right| \cdot \left|x\right|}}, \frac{-1}{\mathsf{fma}\left(\left|x\right|, 0.3275911000000000239396058532292954623699, 1\right)}, 1\right)\right)}}\]

Reproduce

herbie shell --seed 2019353 +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)))))))