\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
Test:
Jmat.Real.erf
Bits:
128 bits
Bits error versus x
Time: 17.2 s
Input Error: 13.9
Output Error: 13.9
Log:
Profile: 🕒
\(1 - \frac{\left(\left(\frac{-1.453152027}{1 + 0.3275911 \cdot \left|x\right|} + 1.421413741\right) + \frac{\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}\right) \cdot \frac{\frac{1 - 0.3275911 \cdot \left|x\right|}{1 + 0.3275911 \cdot \left|x\right|}}{1 - \left(\left|x\right| \cdot \left|x\right|\right) \cdot 0.10731592879921002} + \left(0.254829592 + \frac{-0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)}{e^{\left|x\right| \cdot \left|x\right|} \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)}\)
  1. Started with
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.9
  2. Using strategy rm
    13.9
  3. Applied add-sqr-sqrt to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \color{red}{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \color{blue}{{\left(\sqrt{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right)}^2} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.9
  4. Using strategy rm
    13.9
  5. Applied flip-+ to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + {\left(\sqrt{\frac{1}{\color{red}{1 + 0.3275911 \cdot \left|x\right|}}}\right)}^2 \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + {\left(\sqrt{\frac{1}{\color{blue}{\frac{{1}^2 - {\left(0.3275911 \cdot \left|x\right|\right)}^2}{1 - 0.3275911 \cdot \left|x\right|}}}}\right)}^2 \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.9
  6. Applied associate-/r/ to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + {\left(\sqrt{\color{red}{\frac{1}{\frac{{1}^2 - {\left(0.3275911 \cdot \left|x\right|\right)}^2}{1 - 0.3275911 \cdot \left|x\right|}}}}\right)}^2 \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + {\left(\sqrt{\color{blue}{\frac{1}{{1}^2 - {\left(0.3275911 \cdot \left|x\right|\right)}^2} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right)}}\right)}^2 \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.9
  7. Applied sqrt-prod to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + {\color{red}{\left(\sqrt{\frac{1}{{1}^2 - {\left(0.3275911 \cdot \left|x\right|\right)}^2} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right)}\right)}}^2 \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + {\color{blue}{\left(\sqrt{\frac{1}{{1}^2 - {\left(0.3275911 \cdot \left|x\right|\right)}^2}} \cdot \sqrt{1 - 0.3275911 \cdot \left|x\right|}\right)}}^2 \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    44.7
  8. Applied square-prod to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \color{red}{{\left(\sqrt{\frac{1}{{1}^2 - {\left(0.3275911 \cdot \left|x\right|\right)}^2}} \cdot \sqrt{1 - 0.3275911 \cdot \left|x\right|}\right)}^2} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \color{blue}{\left({\left(\sqrt{\frac{1}{{1}^2 - {\left(0.3275911 \cdot \left|x\right|\right)}^2}}\right)}^2 \cdot {\left(\sqrt{1 - 0.3275911 \cdot \left|x\right|}\right)}^2\right)} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    44.7
  9. Applied simplify to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \left({\left(\sqrt{\frac{1}{{1}^2 - {\left(0.3275911 \cdot \left|x\right|\right)}^2}}\right)}^2 \cdot \color{red}{{\left(\sqrt{1 - 0.3275911 \cdot \left|x\right|}\right)}^2}\right) \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \left({\left(\sqrt{\frac{1}{{1}^2 - {\left(0.3275911 \cdot \left|x\right|\right)}^2}}\right)}^2 \cdot \color{blue}{\left(1 - \left|x\right| \cdot 0.3275911\right)}\right) \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    29.0
  10. Applied taylor to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \left({\left(\sqrt{\frac{1}{{1}^2 - {\left(0.3275911 \cdot \left|x\right|\right)}^2}}\right)}^2 \cdot \left(1 - \left|x\right| \cdot 0.3275911\right)\right) \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \left({\left(\sqrt{\frac{1}{1 - 0.10731592879921002 \cdot {\left(\left|x\right|\right)}^2}}\right)}^2 \cdot \left(1 - \left|x\right| \cdot 0.3275911\right)\right) \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    28.9
  11. Taylor expanded around 0 to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \left({\color{red}{\left(\sqrt{\frac{1}{1 - 0.10731592879921002 \cdot {\left(\left|x\right|\right)}^2}}\right)}}^2 \cdot \left(1 - \left|x\right| \cdot 0.3275911\right)\right) \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \left({\color{blue}{\left(\sqrt{\frac{1}{1 - 0.10731592879921002 \cdot {\left(\left|x\right|\right)}^2}}\right)}}^2 \cdot \left(1 - \left|x\right| \cdot 0.3275911\right)\right) \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    28.9
  12. Applied simplify to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \left({\left(\sqrt{\frac{1}{1 - 0.10731592879921002 \cdot {\left(\left|x\right|\right)}^2}}\right)}^2 \cdot \left(1 - \left|x\right| \cdot 0.3275911\right)\right) \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \frac{\frac{\left(\frac{-0.284496736}{\left|x\right| \cdot 0.3275911 + 1} + 0.254829592\right) + \left(\frac{1 - \left|x\right| \cdot 0.3275911}{1 - \left(\left|x\right| \cdot \left|x\right|\right) \cdot 0.10731592879921002} \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right) \cdot \left(\left(1.421413741 + \frac{-1.453152027}{\left|x\right| \cdot 0.3275911 + 1}\right) + \frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1} \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}{e^{\left|x\right| \cdot \left|x\right|}}}{\left|x\right| \cdot 0.3275911 + 1}\]
    13.9

  13. Applied final simplification
  14. Applied simplify to get
    \[\color{red}{1 - \frac{\frac{\left(\frac{-0.284496736}{\left|x\right| \cdot 0.3275911 + 1} + 0.254829592\right) + \left(\frac{1 - \left|x\right| \cdot 0.3275911}{1 - \left(\left|x\right| \cdot \left|x\right|\right) \cdot 0.10731592879921002} \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right) \cdot \left(\left(1.421413741 + \frac{-1.453152027}{\left|x\right| \cdot 0.3275911 + 1}\right) + \frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1} \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)}{e^{\left|x\right| \cdot \left|x\right|}}}{\left|x\right| \cdot 0.3275911 + 1}} \leadsto \color{blue}{1 - \frac{\left(\left(\frac{-1.453152027}{1 + 0.3275911 \cdot \left|x\right|} + 1.421413741\right) + \frac{\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}\right) \cdot \frac{\frac{1 - 0.3275911 \cdot \left|x\right|}{1 + 0.3275911 \cdot \left|x\right|}}{1 - \left(\left|x\right| \cdot \left|x\right|\right) \cdot 0.10731592879921002} + \left(0.254829592 + \frac{-0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)}{e^{\left|x\right| \cdot \left|x\right|} \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)}}\]
    13.9

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)))))))