Average Error: 13.9 → 13.9
Time: 16.9s
Precision: 64
\[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[{e}^{\left(\log \left(\left(1 + \left(0.284496735999999972 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + 1.45315202700000001 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}}\right)\right) - \left(1.0614054289999999 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{5}} + \left(1.42141374100000006 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}} + 0.25482959199999999 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{0.32759110000000002 \cdot \left|x\right| + 1}\right)\right)\right)\right)}\]
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|}
↓
{e}^{\left(\log \left(\left(1 + \left(0.284496735999999972 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + 1.45315202700000001 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}}\right)\right) - \left(1.0614054289999999 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{5}} + \left(1.42141374100000006 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}} + 0.25482959199999999 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{0.32759110000000002 \cdot \left|x\right| + 1}\right)\right)\right)\right)}
double f(double x) {
        double r175595 = 1.0;
        double r175596 = 0.3275911;
        double r175597 = x;
        double r175598 = fabs(r175597);
        double r175599 = r175596 * r175598;
        double r175600 = r175595 + r175599;
        double r175601 = r175595 / r175600;
        double r175602 = 0.254829592;
        double r175603 = -0.284496736;
        double r175604 = 1.421413741;
        double r175605 = -1.453152027;
        double r175606 = 1.061405429;
        double r175607 = r175601 * r175606;
        double r175608 = r175605 + r175607;
        double r175609 = r175601 * r175608;
        double r175610 = r175604 + r175609;
        double r175611 = r175601 * r175610;
        double r175612 = r175603 + r175611;
        double r175613 = r175601 * r175612;
        double r175614 = r175602 + r175613;
        double r175615 = r175601 * r175614;
        double r175616 = r175598 * r175598;
        double r175617 = -r175616;
        double r175618 = exp(r175617);
        double r175619 = r175615 * r175618;
        double r175620 = r175595 - r175619;
        return r175620;
}
↓
double f(double x) {
        double r175621 = exp(1.0);
        double r175622 = 1.0;
        double r175623 = 0.284496736;
        double r175624 = x;
        double r175625 = fabs(r175624);
        double r175626 = 2.0;
        double r175627 = pow(r175625, r175626);
        double r175628 = -r175627;
        double r175629 = exp(r175628);
        double r175630 = 0.3275911;
        double r175631 = r175630 * r175625;
        double r175632 = r175631 + r175622;
        double r175633 = pow(r175632, r175626);
        double r175634 = r175629 / r175633;
        double r175635 = r175623 * r175634;
        double r175636 = 1.453152027;
        double r175637 = 4.0;
        double r175638 = pow(r175632, r175637);
        double r175639 = r175629 / r175638;
        double r175640 = r175636 * r175639;
        double r175641 = r175635 + r175640;
        double r175642 = r175622 + r175641;
        double r175643 = 1.061405429;
        double r175644 = 5.0;
        double r175645 = pow(r175632, r175644);
        double r175646 = r175629 / r175645;
        double r175647 = r175643 * r175646;
        double r175648 = 1.421413741;
        double r175649 = 3.0;
        double r175650 = pow(r175632, r175649);
        double r175651 = r175629 / r175650;
        double r175652 = r175648 * r175651;
        double r175653 = 0.254829592;
        double r175654 = r175629 / r175632;
        double r175655 = r175653 * r175654;
        double r175656 = r175652 + r175655;
        double r175657 = r175647 + r175656;
        double r175658 = r175642 - r175657;
        double r175659 = log(r175658);
        double r175660 = pow(r175621, r175659);
        return r175660;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.9

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

  2. Taylor expanded around 0 13.9

    \[\leadsto \color{blue}{\left(1 + \left(0.284496735999999972 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + 1.45315202700000001 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}}\right)\right) - \left(1.0614054289999999 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{5}} + \left(1.42141374100000006 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}} + 0.25482959199999999 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{0.32759110000000002 \cdot \left|x\right| + 1}\right)\right)}\]
  3. Using strategy rm

  4. Applied add-exp-log13.9

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

  6. Applied pow113.9

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

  7. Applied log-pow13.9

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

  8. Applied exp-prod13.9

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

  9. Simplified13.9

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

  10. Final simplification13.9

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

Reproduce

herbie shell --seed 2020027 
(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)))))))