\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
Test:
Jmat.Real.erf
Bits:
128 bits
Bits error versus x
Time: 43.9 s
Input Error: 11.9
Output Error: 10.2
Log: ⚲
Profile: 🕒
\(1 - \frac{\frac{1}{\left|x\right| \cdot 0.3275911 + 1} \cdot \left(\left(0.254829592 + \frac{-0.284496736}{\left|x\right| \cdot 0.3275911 + 1}\right) + {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^2 \cdot \left(1.421413741 + \frac{\frac{1}{\left|x\right| \cdot 0.3275911 + 1}}{-1.453152027 \cdot -1.453152027 + \frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1} \cdot \frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}} \cdot \frac{{\left(-1.453152027 \cdot -1.453152027\right)}^2 - \frac{1.2691858416929063}{{\left(\left|x\right| \cdot 0.3275911 + 1\right)}^{4}}}{-1.453152027 - \frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}}\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}}\)
  1. Started with
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    11.9
  2. Using strategy rm
    11.9
  3. Applied flip-+ to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{red}{\left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\frac{{-1.453152027}^2 - {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)}^2}{-1.453152027 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429}}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    10.7
  4. Applied simplify to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{{-1.453152027}^2 - {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)}^2}{\color{red}{-1.453152027 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429}}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{{-1.453152027}^2 - {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)}^2}{\color{blue}{-1.453152027 - \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    10.7
  5. Using strategy rm
    10.7
  6. Applied flip-- to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{\color{red}{{-1.453152027}^2 - {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)}^2}}{-1.453152027 - \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{\color{blue}{\frac{{\left({-1.453152027}^2\right)}^2 - {\left({\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)}^2\right)}^2}{{-1.453152027}^2 + {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)}^2}}}{-1.453152027 - \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    10.7
  7. Applied taylor to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{\frac{{\left({-1.453152027}^2\right)}^2 - {\left({\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)}^2\right)}^2}{{-1.453152027}^2 + {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)}^2}}{-1.453152027 - \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{\frac{{\left({-1.453152027}^2\right)}^2 - \frac{1.2691858416929063}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}}}{{-1.453152027}^2 + {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)}^2}}{-1.453152027 - \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    10.2
  8. Taylor expanded around 0 to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{\frac{{\left({-1.453152027}^2\right)}^2 - \color{red}{\frac{1.2691858416929063}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}}}}{{-1.453152027}^2 + {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)}^2}}{-1.453152027 - \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{\frac{{\left({-1.453152027}^2\right)}^2 - \color{blue}{\frac{1.2691858416929063}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}}}}{{-1.453152027}^2 + {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)}^2}}{-1.453152027 - \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    10.2
  9. Applied simplify to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{\frac{{\left({-1.453152027}^2\right)}^2 - \frac{1.2691858416929063}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}}}{{-1.453152027}^2 + {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)}^2}}{-1.453152027 - \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \frac{\frac{1}{\left|x\right| \cdot 0.3275911 + 1} \cdot \left(\left(0.254829592 + \frac{-0.284496736}{\left|x\right| \cdot 0.3275911 + 1}\right) + {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^2 \cdot \left(1.421413741 + \frac{\frac{1}{\left|x\right| \cdot 0.3275911 + 1}}{-1.453152027 \cdot -1.453152027 + \frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1} \cdot \frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}} \cdot \frac{{\left(-1.453152027 \cdot -1.453152027\right)}^2 - \frac{1.2691858416929063}{{\left(\left|x\right| \cdot 0.3275911 + 1\right)}^{4}}}{-1.453152027 - \frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}}\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}}\]
    10.2
  10. 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)))))))