\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
Test:
Jmat.Real.erf
Bits:
128 bits
Bits error versus x
Time: 51.1 s
Input Error: 13.8
Output Error: 13.8
Log: ⚲
Profile: 🕒
\(\frac{{1}^{3} - {\left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}^{3}}{\left(\frac{\frac{\frac{\left(\frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911} - 0.284496736\right) - \frac{1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2}}{1 + \left|x\right| \cdot 0.3275911} + \left(0.254829592 + \frac{\frac{1.061405429}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^3}}{1 + \left|x\right| \cdot 0.3275911}\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}} + 1\right) + \frac{{\left(\frac{\frac{\left(\frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911} - 0.284496736\right) - \frac{1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2}}{1 + \left|x\right| \cdot 0.3275911} + \left(0.254829592 + \frac{\frac{1.061405429}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^3}}{1 + \left|x\right| \cdot 0.3275911}\right)}{e^{\left|x\right| \cdot \left|x\right|}}\right)}^2}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2}}\)
  1. Started with
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.8
  2. Applied taylor to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.9
  3. Taylor expanded around 0 to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{red}{\left(\left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(\left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.9
  4. Using strategy rm
    13.9
  5. Applied flip3-- to get
    \[\color{red}{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(\left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \leadsto \color{blue}{\frac{{1}^{3} - {\left(\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(\left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}^{3}}{{1}^2 + \left({\left(\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(\left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}^2 + 1 \cdot \left(\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(\left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)\right)}}\]
    13.8
  6. Applied simplify to get
    \[\frac{{1}^{3} - {\left(\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(\left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}^{3}}{\color{red}{{1}^2 + \left({\left(\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(\left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}^2 + 1 \cdot \left(\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(\left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)\right)}} \leadsto \frac{{1}^{3} - {\left(\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(\left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}^{3}}{\color{blue}{\left(\frac{\frac{\frac{\left(\frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911} - 0.284496736\right) - \frac{1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2}}{1 + \left|x\right| \cdot 0.3275911} + \left(0.254829592 + \frac{\frac{1.061405429}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^3}}{1 + \left|x\right| \cdot 0.3275911}\right)}{1 + \left|x\right| \cdot 0.3275911}}{e^{\left|x\right| \cdot \left|x\right|}} + 1\right) + \frac{{\left(\frac{\frac{\left(\frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911} - 0.284496736\right) - \frac{1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2}}{1 + \left|x\right| \cdot 0.3275911} + \left(0.254829592 + \frac{\frac{1.061405429}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^3}}{1 + \left|x\right| \cdot 0.3275911}\right)}{e^{\left|x\right| \cdot \left|x\right|}}\right)}^2}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2}}}\]
    13.8

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)))))))