\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
Test:
Jmat.Real.erf
Bits:
128 bits
Bits error versus x
Time: 51.6 s
Input Error: 12.3
Output Error: 9.6
Log:
Profile: 🕒
\(1 - \frac{0.254829592 \cdot 0.254829592 - \frac{\left(-0.284496736 \cdot -0.284496736\right) \cdot \left(-0.284496736 \cdot -0.284496736\right) - {\left(\frac{\left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right) + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} \cdot \frac{\left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right) + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}\right)}^2}{\left(\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(\left(-0.284496736 - \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\right) + \left(\frac{1.453152027}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)} - \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right) \cdot \left(-0.284496736 \cdot -0.284496736 + {\left(\frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911} + \frac{-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\right)}^2\right)} \cdot \frac{\left(-0.284496736 \cdot -0.284496736\right) \cdot \left(-0.284496736 \cdot -0.284496736\right) - {\left(\frac{\left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right) + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} \cdot \frac{\left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right) + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}\right)}^2}{\left(\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(\left(-0.284496736 - \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\right) + \left(\frac{1.453152027}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)} - \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right) \cdot \left(-0.284496736 \cdot -0.284496736 + {\left(\frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911} + \frac{-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\right)}^2\right)}}{\frac{\left(\left(0.254829592 - \frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911}\right) - {\left(\frac{1}{1 + \left|x\right| \cdot 0.3275911}\right)}^2 \cdot \left(\left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right) + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}\right)\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}{{\left(e^{\left|x\right|}\right)}^{\left(-\left|x\right|\right)}}}\)
  1. Started with
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    12.3
  2. Using strategy rm
    12.3
  3. Applied flip-+ to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{red}{\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\frac{{0.254829592}^2 - {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)}^2}{0.254829592 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)}}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    10.7
  4. Applied frac-times to get
    \[1 - \color{red}{\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{{0.254829592}^2 - {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)}^2}{0.254829592 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)}\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \color{blue}{\frac{1 \cdot \left({0.254829592}^2 - {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(0.254829592 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)}} \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    10.7
  5. Using strategy rm
    10.7
  6. Applied flip-+ to get
    \[1 - \frac{1 \cdot \left({0.254829592}^2 - {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{red}{\left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)}\right)}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(0.254829592 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \frac{1 \cdot \left({0.254829592}^2 - {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\frac{{-0.284496736}^2 - {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)}^2}{-0.284496736 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)}}\right)}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(0.254829592 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)} \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    10.3
  7. Applied associate-*r/ to get
    \[1 - \frac{1 \cdot \left({0.254829592}^2 - {\color{red}{\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{{-0.284496736}^2 - {\left(\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)}^2}{-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)}}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\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)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \frac{1 \cdot \left({0.254829592}^2 - {\color{blue}{\left(\frac{\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left({-0.284496736}^2 - {\left(\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)}^2\right)}{-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)}}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\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)} \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    10.3
  8. Using strategy rm
    10.3
  9. Applied flip-- to get
    \[1 - \frac{1 \cdot \left({0.254829592}^2 - {\left(\frac{\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{red}{\left({-0.284496736}^2 - {\left(\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)}^2\right)}}{-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)}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\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)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \frac{1 \cdot \left({0.254829592}^2 - {\left(\frac{\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\frac{{\left({-0.284496736}^2\right)}^2 - {\left({\left(\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)}^2\right)}^2}{{-0.284496736}^2 + {\left(\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)}^2}}}{-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)}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\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)} \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    9.6
  10. Applied frac-times to get
    \[1 - \frac{1 \cdot \left({0.254829592}^2 - {\left(\frac{\color{red}{\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{{\left({-0.284496736}^2\right)}^2 - {\left({\left(\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)}^2\right)}^2}{{-0.284496736}^2 + {\left(\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)}^2}}}{-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)}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\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)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \frac{1 \cdot \left({0.254829592}^2 - {\left(\frac{\color{blue}{\frac{1 \cdot \left({\left({-0.284496736}^2\right)}^2 - {\left({\left(\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)}^2\right)}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left({-0.284496736}^2 + {\left(\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)}^2\right)}}}{-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)}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\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)} \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    9.6
  11. Applied associate-/l/ to get
    \[1 - \frac{1 \cdot \left({0.254829592}^2 - {\color{red}{\left(\frac{\frac{1 \cdot \left({\left({-0.284496736}^2\right)}^2 - {\left({\left(\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)}^2\right)}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left({-0.284496736}^2 + {\left(\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)}^2\right)}}{-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)}}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\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)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \frac{1 \cdot \left({0.254829592}^2 - {\color{blue}{\left(\frac{1 \cdot \left({\left({-0.284496736}^2\right)}^2 - {\left({\left(\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)}^2\right)}^2\right)}{\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) \cdot \left(\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left({-0.284496736}^2 + {\left(\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)}^2\right)\right)}\right)}}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\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)} \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    9.6
  12. Applied simplify to get
    \[1 - \frac{1 \cdot \left({0.254829592}^2 - {\left(\frac{1 \cdot \left({\left({-0.284496736}^2\right)}^2 - {\left({\left(\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)}^2\right)}^2\right)}{\color{red}{\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) \cdot \left(\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left({-0.284496736}^2 + {\left(\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)}^2\right)\right)}}\right)}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\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)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \frac{1 \cdot \left({0.254829592}^2 - {\left(\frac{1 \cdot \left({\left({-0.284496736}^2\right)}^2 - {\left({\left(\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)}^2\right)}^2\right)}{\color{blue}{\left({\left(\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)}^2 + {-0.284496736}^2\right) \cdot \left(\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(-0.284496736 - \left(\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right)}}\right)}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\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)} \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    9.6
  13. Applied taylor to get
    \[1 - \frac{1 \cdot \left({0.254829592}^2 - {\left(\frac{1 \cdot \left({\left({-0.284496736}^2\right)}^2 - {\left({\left(\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)}^2\right)}^2\right)}{\left({\left(\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)}^2 + {-0.284496736}^2\right) \cdot \left(\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(-0.284496736 - \left(\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right)}\right)}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\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)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \frac{1 \cdot \left({0.254829592}^2 - {\left(\frac{1 \cdot \left({\left({-0.284496736}^2\right)}^2 - {\left({\left(\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)}^2\right)}^2\right)}{\left({\left(\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)}^2 + {-0.284496736}^2\right) \cdot \left(\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(-0.284496736 - \left(\frac{1.061405429 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} - 1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right)}\right)}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\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)} \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    9.6
  14. Taylor expanded around 0 to get
    \[1 - \frac{1 \cdot \left({0.254829592}^2 - {\left(\frac{1 \cdot \left({\left({-0.284496736}^2\right)}^2 - {\left({\left(\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)}^2\right)}^2\right)}{\left({\left(\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)}^2 + {-0.284496736}^2\right) \cdot \left(\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(-0.284496736 - \left(\color{red}{\frac{1.061405429 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} - 1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right)}\right)}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\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)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \frac{1 \cdot \left({0.254829592}^2 - {\left(\frac{1 \cdot \left({\left({-0.284496736}^2\right)}^2 - {\left({\left(\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)}^2\right)}^2\right)}{\left({\left(\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)}^2 + {-0.284496736}^2\right) \cdot \left(\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(-0.284496736 - \left(\color{blue}{\frac{1.061405429 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} - 1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right)}\right)}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\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)} \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    9.6
  15. Applied simplify to get
    \[1 - \frac{1 \cdot \left({0.254829592}^2 - {\left(\frac{1 \cdot \left({\left({-0.284496736}^2\right)}^2 - {\left({\left(\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)}^2\right)}^2\right)}{\left({\left(\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)}^2 + {-0.284496736}^2\right) \cdot \left(\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(-0.284496736 - \left(\frac{1.061405429 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} - 1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right)}\right)}^2\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\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)} \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \frac{0.254829592 \cdot 0.254829592 - \frac{\left(-0.284496736 \cdot -0.284496736\right) \cdot \left(-0.284496736 \cdot -0.284496736\right) - {\left(\frac{\left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right) + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} \cdot \frac{\left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right) + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}\right)}^2}{\left(\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(\left(-0.284496736 - \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\right) + \left(\frac{1.453152027}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)} - \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right) \cdot \left(-0.284496736 \cdot -0.284496736 + {\left(\frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911} + \frac{-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\right)}^2\right)} \cdot \frac{\left(-0.284496736 \cdot -0.284496736\right) \cdot \left(-0.284496736 \cdot -0.284496736\right) - {\left(\frac{\left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right) + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} \cdot \frac{\left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right) + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}\right)}^2}{\left(\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(\left(-0.284496736 - \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\right) + \left(\frac{1.453152027}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)} - \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right) \cdot \left(-0.284496736 \cdot -0.284496736 + {\left(\frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911} + \frac{-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\right)}^2\right)}}{\frac{\left(\left(0.254829592 - \frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911}\right) - {\left(\frac{1}{1 + \left|x\right| \cdot 0.3275911}\right)}^2 \cdot \left(\left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right) + \frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911}\right)\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}{{\left(e^{\left|x\right|}\right)}^{\left(-\left|x\right|\right)}}}\]
    9.6

  16. Applied final simplification

Original test:


(lambda ((x default))
  #: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)))))))