\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
Test:
Jmat.Real.erf
Bits:
128 bits
Bits error versus x
Time: 42.6 s
Input Error: 14.3
Output Error: 14.1
Log:
Profile: 🕒
\(\frac{\left(\left(1 - \frac{\frac{0.6434981745697937}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right| + \left|x\right|\right)}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}}\right) + \left(\frac{\frac{0.568993472}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)} + \frac{\frac{2.906304054}{e^{\left|x\right| \cdot \left|x\right|}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}}\right)\right) - \left(\left(\frac{\frac{0.06493812095888646}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right| + \left|x\right|\right)}}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)} + \frac{\frac{0.9057417095309908}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right| + \left|x\right|\right)}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{8}}\right) + \left(\frac{\frac{1.1265814847106739}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right| + \left|x\right|\right)}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{10}} + \frac{\frac{1.734538030754357}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right| + \left|x\right|\right)}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{6}}\right)\right)}{\left(\left(\frac{\frac{1.061405429}{e^{\left|x\right| \cdot \left|x\right|}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + \frac{\frac{1.421413741}{e^{\left|x\right| \cdot \left|x\right|}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^3}\right) + \frac{0.254829592}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{\left|x\right| \cdot \left|x\right|}}\right) + \left(\frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)} + \left(\frac{\frac{1.453152027}{e^{\left|x\right| \cdot \left|x\right|}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1\right)\right)}\)
  1. Started with
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    14.3
  2. Applied taylor to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto \left(1 + \left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right) - \left(0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + 1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)\]
    13.3
  3. Taylor expanded around 0 to get
    \[\color{red}{\left(1 + \left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right) - \left(0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + 1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)} \leadsto \color{blue}{\left(1 + \left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right) - \left(0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + 1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}\]
    13.3
  4. Using strategy rm
    13.3
  5. Applied flip-- to get
    \[\color{red}{\left(1 + \left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right) - \left(0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + 1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)} \leadsto \color{blue}{\frac{{\left(1 + \left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right)}^2 - {\left(0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + 1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}^2}{\left(1 + \left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right) + \left(0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + 1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}}\]
    13.6
  6. Applied taylor to get
    \[\frac{{\left(1 + \left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right)}^2 - {\left(0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + 1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}^2}{\left(1 + \left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right) + \left(0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + 1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)} \leadsto \frac{\left(0.568993472 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2} + \left(2.906304054 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1\right)\right) - \left(0.6434981745697937 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + \left(0.06493812095888646 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2} + \left(0.9057417095309908 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{8}} + \left(1.734538030754357 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{6}} + 1.1265814847106739 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{10}}\right)\right)\right)\right)}{\left(1 + \left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right) + \left(0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + 1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}\]
    13.6
  7. Taylor expanded around 0 to get
    \[\frac{\color{red}{\left(0.568993472 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2} + \left(2.906304054 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1\right)\right) - \left(0.6434981745697937 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + \left(0.06493812095888646 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2} + \left(0.9057417095309908 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{8}} + \left(1.734538030754357 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{6}} + 1.1265814847106739 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{10}}\right)\right)\right)\right)}}{\left(1 + \left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right) + \left(0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + 1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)} \leadsto \frac{\color{blue}{\left(0.568993472 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2} + \left(2.906304054 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1\right)\right) - \left(0.6434981745697937 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + \left(0.06493812095888646 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2} + \left(0.9057417095309908 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{8}} + \left(1.734538030754357 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{6}} + 1.1265814847106739 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{10}}\right)\right)\right)\right)}}{\left(1 + \left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right) + \left(0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + 1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}\]
    13.6
  8. Applied simplify to get
    \[\frac{\left(0.568993472 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2} + \left(2.906304054 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1\right)\right) - \left(0.6434981745697937 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + \left(0.06493812095888646 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2} + \left(0.9057417095309908 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{8}} + \left(1.734538030754357 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{6}} + 1.1265814847106739 \cdot \frac{{\left(e^{-{\left(\left|x\right|\right)}^2}\right)}^2}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{10}}\right)\right)\right)\right)}{\left(1 + \left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right) + \left(0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + 1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)} \leadsto \frac{\left(\left(\frac{\frac{0.568993472}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)} + \frac{\frac{2.906304054}{e^{\left|x\right| \cdot \left|x\right|}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}}\right) + \left(1 - \frac{0.6434981745697937 \cdot e^{\left(-\left|x\right|\right) \cdot \left(\left|x\right| + \left|x\right|\right)}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}}\right)\right) - \left(\left(\frac{e^{\left(-\left|x\right|\right) \cdot \left(\left|x\right| + \left|x\right|\right)}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{10}} \cdot 1.1265814847106739 + \frac{e^{\left(-\left|x\right|\right) \cdot \left(\left|x\right| + \left|x\right|\right)}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{6}} \cdot 1.734538030754357\right) + \left(\frac{e^{\left(-\left|x\right|\right) \cdot \left(\left|x\right| + \left|x\right|\right)}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{8}} \cdot 0.9057417095309908 + 0.06493812095888646 \cdot \frac{e^{\left(-\left|x\right|\right) \cdot \left(\left|x\right| + \left|x\right|\right)}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)\right)}{\left(\left(1 + \frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right) + \frac{\frac{1.453152027}{e^{\left|x\right| \cdot \left|x\right|}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}}\right) + \left(\frac{\frac{1.421413741}{e^{\left|x\right| \cdot \left|x\right|}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^3} + \left(\frac{0.254829592}{e^{\left|x\right| \cdot \left|x\right|} \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)} + \frac{\frac{1.061405429}{e^{\left|x\right| \cdot \left|x\right|}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}}\right)\right)}\]
    14.1

  9. Applied final simplification
  10. Applied simplify to get
    \[\color{red}{\frac{\left(\left(\frac{\frac{0.568993472}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)} + \frac{\frac{2.906304054}{e^{\left|x\right| \cdot \left|x\right|}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}}\right) + \left(1 - \frac{0.6434981745697937 \cdot e^{\left(-\left|x\right|\right) \cdot \left(\left|x\right| + \left|x\right|\right)}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}}\right)\right) - \left(\left(\frac{e^{\left(-\left|x\right|\right) \cdot \left(\left|x\right| + \left|x\right|\right)}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{10}} \cdot 1.1265814847106739 + \frac{e^{\left(-\left|x\right|\right) \cdot \left(\left|x\right| + \left|x\right|\right)}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{6}} \cdot 1.734538030754357\right) + \left(\frac{e^{\left(-\left|x\right|\right) \cdot \left(\left|x\right| + \left|x\right|\right)}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{8}} \cdot 0.9057417095309908 + 0.06493812095888646 \cdot \frac{e^{\left(-\left|x\right|\right) \cdot \left(\left|x\right| + \left|x\right|\right)}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right)\right)}{\left(\left(1 + \frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)}\right) + \frac{\frac{1.453152027}{e^{\left|x\right| \cdot \left|x\right|}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}}\right) + \left(\frac{\frac{1.421413741}{e^{\left|x\right| \cdot \left|x\right|}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^3} + \left(\frac{0.254829592}{e^{\left|x\right| \cdot \left|x\right|} \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)} + \frac{\frac{1.061405429}{e^{\left|x\right| \cdot \left|x\right|}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}}\right)\right)}} \leadsto \color{blue}{\frac{\left(\left(1 - \frac{\frac{0.6434981745697937}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right| + \left|x\right|\right)}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}}\right) + \left(\frac{\frac{0.568993472}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)} + \frac{\frac{2.906304054}{e^{\left|x\right| \cdot \left|x\right|}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}}\right)\right) - \left(\left(\frac{\frac{0.06493812095888646}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right| + \left|x\right|\right)}}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)} + \frac{\frac{0.9057417095309908}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right| + \left|x\right|\right)}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{8}}\right) + \left(\frac{\frac{1.1265814847106739}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right| + \left|x\right|\right)}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{10}} + \frac{\frac{1.734538030754357}{{\left(e^{\left|x\right|}\right)}^{\left(\left|x\right| + \left|x\right|\right)}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{6}}\right)\right)}{\left(\left(\frac{\frac{1.061405429}{e^{\left|x\right| \cdot \left|x\right|}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + \frac{\frac{1.421413741}{e^{\left|x\right| \cdot \left|x\right|}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^3}\right) + \frac{0.254829592}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{\left|x\right| \cdot \left|x\right|}}\right) + \left(\frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)} + \left(\frac{\frac{1.453152027}{e^{\left|x\right| \cdot \left|x\right|}}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 1\right)\right)}}\]
    14.1

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)))))))