\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
Test:
Jmat.Real.erf
Bits:
128 bits
Bits error versus x
Time: 3.4 m
Input Error: 13.8
Output Error: 13.0
Log: ⚲
Profile: 🕒
\(\frac{{1}^{3} - {\left(\sqrt{{\left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot {\left(\sqrt[3]{-0.284496736 + \left(\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)}\right)}^3\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}^{3}}\right)}^2}{{1}^2 + \left({\left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot {\left(\sqrt[3]{-0.284496736 + \left(\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)}\right)}^3\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}^2 + 1 \cdot \left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot {\left(\sqrt[3]{-0.284496736 + \left(\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)}\right)}^3\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)\right)}\)
  1. Started with
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.8
  2. Using strategy rm
    13.8
  3. Applied add-cube-cbrt to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{red}{\left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{{\left(\sqrt[3]{-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)}\right)}^3}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.8
  4. Applied simplify to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot {\color{red}{\left(\sqrt[3]{-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)}\right)}}^3\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot {\color{blue}{\left(\sqrt[3]{-0.284496736 + \left(\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)}\right)}}^3\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.8
  5. Using strategy rm
    13.8
  6. Applied flip3-- to get
    \[\color{red}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot {\left(\sqrt[3]{-0.284496736 + \left(\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)}\right)}^3\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \leadsto \color{blue}{\frac{{1}^{3} - {\left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot {\left(\sqrt[3]{-0.284496736 + \left(\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)}\right)}^3\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}^{3}}{{1}^2 + \left({\left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot {\left(\sqrt[3]{-0.284496736 + \left(\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)}\right)}^3\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}^2 + 1 \cdot \left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot {\left(\sqrt[3]{-0.284496736 + \left(\frac{\frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^2} + \frac{1.421413741}{1 + \left|x\right| \cdot 0.3275911}\right)}\right)}^3\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)\right)}}\]
    13.8
  7. Using strategy rm
    13.8
  8. Applied add-sqr-sqrt to get
    \[\frac{{1}^{3} - \color{red}{{\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(\sqrt[3]{-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)}^3\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(\sqrt[3]{-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)}^3\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(\sqrt[3]{-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)}^3\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)\right)} \leadsto \frac{{1}^{3} - \color{blue}{{\left(\sqrt{{\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(\sqrt[3]{-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)}^3\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}^{3}}\right)}^2}}{{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(\sqrt[3]{-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)}^3\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(\sqrt[3]{-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)}^3\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)\right)}\]
    13.0
  9. Removed slow pow expressions

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)))))))