\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
Test:
Jmat.Real.erf
Bits:
128 bits
Bits error versus x
Time: 44.0 s
Input Error: 13.8
Output Error: 13.8
Log: ⚲
Profile: 🕒
\(\frac{{1}^2 - {\left(\frac{0.254829592 + \sqrt[3]{\frac{1}{{\left(\left|x\right| \cdot 0.3275911 + 1\right)}^3} \cdot {\left(\left(-0.284496736 + \frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1}\right) + {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^2 \cdot \left(-1.453152027 + \frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}\right)\right)}^3}}{\left|x\right| \cdot 0.3275911 + 1} \cdot \frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right)}^2}{1 - \frac{\sqrt[3]{\frac{{\left(\frac{\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|} + -1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2} + \left(\frac{1.421413741}{1 + 0.3275911 \cdot \left|x\right|} + -0.284496736\right)\right)}^3}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^3}} + 0.254829592}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}}\)
  1. Started with
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.8
  2. Using strategy rm
    13.8
  3. Applied add-cbrt-cube to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{red}{\left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\sqrt[3]{{\left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)}^3}}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.8
  4. Applied add-cbrt-cube to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \color{red}{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \cdot \sqrt[3]{{\left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)}^3}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \color{blue}{\sqrt[3]{{\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right)}^3}} \cdot \sqrt[3]{{\left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)}^3}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.8
  5. Applied cbrt-unprod to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \color{red}{\sqrt[3]{{\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right)}^3} \cdot \sqrt[3]{{\left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)}^3}}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \color{blue}{\sqrt[3]{{\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right)}^3 \cdot {\left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)}^3}}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.8
  6. Applied simplify to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \sqrt[3]{\color{red}{{\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right)}^3 \cdot {\left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)}^3}}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \sqrt[3]{\color{blue}{{\left(\left(\frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1} + -0.284496736\right) + {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^2 \cdot \left(\frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1} + -1.453152027\right)\right)}^3 \cdot {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^3}}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.8
  7. Using strategy rm
    13.8
  8. Applied sub-neg to get
    \[\color{red}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \sqrt[3]{{\left(\left(\frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1} + -0.284496736\right) + {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^2 \cdot \left(\frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1} + -1.453152027\right)\right)}^3 \cdot {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^3}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \leadsto \color{blue}{1 + \left(-\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \sqrt[3]{{\left(\left(\frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1} + -0.284496736\right) + {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^2 \cdot \left(\frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1} + -1.453152027\right)\right)}^3 \cdot {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^3}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}\]
    13.8
  9. Applied simplify to get
    \[1 + \color{red}{\left(-\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \sqrt[3]{{\left(\left(\frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1} + -0.284496736\right) + {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^2 \cdot \left(\frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1} + -1.453152027\right)\right)}^3 \cdot {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^3}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)} \leadsto 1 + \color{blue}{\frac{0.254829592 + \sqrt[3]{\frac{1}{{\left(\left|x\right| \cdot 0.3275911 + 1\right)}^3} \cdot {\left(\left(-0.284496736 + \frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1}\right) + {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^2 \cdot \left(-1.453152027 + \frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}\right)\right)}^3}}{\left|x\right| \cdot 0.3275911 + 1} \cdot \frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}}\]
    13.8
  10. Using strategy rm
    13.8
  11. Applied flip-+ to get
    \[\color{red}{1 + \frac{0.254829592 + \sqrt[3]{\frac{1}{{\left(\left|x\right| \cdot 0.3275911 + 1\right)}^3} \cdot {\left(\left(-0.284496736 + \frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1}\right) + {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^2 \cdot \left(-1.453152027 + \frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}\right)\right)}^3}}{\left|x\right| \cdot 0.3275911 + 1} \cdot \frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}} \leadsto \color{blue}{\frac{{1}^2 - {\left(\frac{0.254829592 + \sqrt[3]{\frac{1}{{\left(\left|x\right| \cdot 0.3275911 + 1\right)}^3} \cdot {\left(\left(-0.284496736 + \frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1}\right) + {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^2 \cdot \left(-1.453152027 + \frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}\right)\right)}^3}}{\left|x\right| \cdot 0.3275911 + 1} \cdot \frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right)}^2}{1 - \frac{0.254829592 + \sqrt[3]{\frac{1}{{\left(\left|x\right| \cdot 0.3275911 + 1\right)}^3} \cdot {\left(\left(-0.284496736 + \frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1}\right) + {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^2 \cdot \left(-1.453152027 + \frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}\right)\right)}^3}}{\left|x\right| \cdot 0.3275911 + 1} \cdot \frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}}}\]
    13.8
  12. Applied simplify to get
    \[\frac{{1}^2 - {\left(\frac{0.254829592 + \sqrt[3]{\frac{1}{{\left(\left|x\right| \cdot 0.3275911 + 1\right)}^3} \cdot {\left(\left(-0.284496736 + \frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1}\right) + {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^2 \cdot \left(-1.453152027 + \frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}\right)\right)}^3}}{\left|x\right| \cdot 0.3275911 + 1} \cdot \frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right)}^2}{\color{red}{1 - \frac{0.254829592 + \sqrt[3]{\frac{1}{{\left(\left|x\right| \cdot 0.3275911 + 1\right)}^3} \cdot {\left(\left(-0.284496736 + \frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1}\right) + {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^2 \cdot \left(-1.453152027 + \frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}\right)\right)}^3}}{\left|x\right| \cdot 0.3275911 + 1} \cdot \frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}}} \leadsto \frac{{1}^2 - {\left(\frac{0.254829592 + \sqrt[3]{\frac{1}{{\left(\left|x\right| \cdot 0.3275911 + 1\right)}^3} \cdot {\left(\left(-0.284496736 + \frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1}\right) + {\left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}^2 \cdot \left(-1.453152027 + \frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}\right)\right)}^3}}{\left|x\right| \cdot 0.3275911 + 1} \cdot \frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right)}^2}{\color{blue}{1 - \frac{\sqrt[3]{\frac{{\left(\frac{\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|} + -1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2} + \left(\frac{1.421413741}{1 + 0.3275911 \cdot \left|x\right|} + -0.284496736\right)\right)}^3}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^3}} + 0.254829592}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}}}\]
    13.8
  13. 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)))))))