\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
Test:
Jmat.Real.erf
Bits:
128 bits
Bits error versus x
Time: 15.7 s
Input Error: 13.9
Output Error: 13.9
Log: ⚲
Profile: 🕒
\(1 - \left({\left(e^{\left|x\right|}\right)}^{\left(-\left|x\right|\right)} \cdot \left(\frac{\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \frac{0.254829592}{1 + \left|x\right| \cdot 0.3275911}\right) + \left(\frac{1}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^3} \cdot {\left(e^{\left|x\right|}\right)}^{\left(-\left|x\right|\right)}\right) \cdot \left(\frac{1.061405429}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)} - \left(\frac{1.453152027}{1 + \left|x\right| \cdot 0.3275911} - 1.421413741\right)\right)\right)\)
  1. Started with
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.9
  2. Using strategy rm
    13.9
  3. Applied add-log-exp to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \color{red}{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \color{blue}{\log \left(e^{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right)} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.9
  4. Applied taylor to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \log \left(e^{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \log \left(e^{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot \left(1.421413741 + \frac{1.061405429 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} - 1.453152027}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.9
  5. Taylor expanded around 0 to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \log \left(e^{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot \left(1.421413741 + \color{red}{\frac{1.061405429 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} - 1.453152027}{1 + 0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \log \left(e^{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot \left(1.421413741 + \color{blue}{\frac{1.061405429 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} - 1.453152027}{1 + 0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.9
  6. Applied simplify to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \log \left(e^{\frac{1}{1 + 0.3275911 \cdot \left|x\right|}}\right) \cdot \left(1.421413741 + \frac{1.061405429 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} - 1.453152027}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto 1 - \left(\left(\frac{0.254829592}{\left|x\right| \cdot 0.3275911 + 1} + \frac{1}{\left|x\right| \cdot 0.3275911 + 1} \cdot \frac{-0.284496736}{\left|x\right| \cdot 0.3275911 + 1}\right) + \left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1} \cdot \left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1} \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)\right) \cdot \left(\frac{\frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1} - \left(\frac{1.453152027}{\left|x\right| \cdot 0.3275911 + 1} - 1.421413741\right)\right)\right) \cdot {\left(e^{\left|x\right|}\right)}^{\left(-\left|x\right|\right)}\]
    13.9
  7. Applied final simplification
  8. Applied simplify to get
    \[\color{red}{1 - \left(\left(\frac{0.254829592}{\left|x\right| \cdot 0.3275911 + 1} + \frac{1}{\left|x\right| \cdot 0.3275911 + 1} \cdot \frac{-0.284496736}{\left|x\right| \cdot 0.3275911 + 1}\right) + \left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1} \cdot \left(\frac{1}{\left|x\right| \cdot 0.3275911 + 1} \cdot \frac{1}{\left|x\right| \cdot 0.3275911 + 1}\right)\right) \cdot \left(\frac{\frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1} - \left(\frac{1.453152027}{\left|x\right| \cdot 0.3275911 + 1} - 1.421413741\right)\right)\right) \cdot {\left(e^{\left|x\right|}\right)}^{\left(-\left|x\right|\right)}} \leadsto \color{blue}{1 - \left({\left(e^{\left|x\right|}\right)}^{\left(-\left|x\right|\right)} \cdot \left(\frac{\frac{-0.284496736}{1 + \left|x\right| \cdot 0.3275911}}{1 + \left|x\right| \cdot 0.3275911} + \frac{0.254829592}{1 + \left|x\right| \cdot 0.3275911}\right) + \left(\frac{1}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^3} \cdot {\left(e^{\left|x\right|}\right)}^{\left(-\left|x\right|\right)}\right) \cdot \left(\frac{1.061405429}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)} - \left(\frac{1.453152027}{1 + \left|x\right| \cdot 0.3275911} - 1.421413741\right)\right)\right)}\]
    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)))))))