\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
Test:
Jmat.Real.erf
Bits:
128 bits
Bits error versus x
Time: 24.0 s
Input Error: 13.6
Output Error: 13.6
Log: ⚲
Profile: 🕒
\(e^{\log \left(\left(1 + \left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right) - \left(0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + 1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)\right)}\)
  1. Started with
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    13.6
  2. Applied taylor to get
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \leadsto \left(1 + \left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right) - \left(0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + 1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)\]
    13.6
  3. Taylor expanded around 0 to get
    \[\color{red}{\left(1 + \left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right) - \left(0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + 1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)} \leadsto \color{blue}{\left(1 + \left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right) - \left(0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + 1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)}\]
    13.6
  4. Using strategy rm
    13.6
  5. Applied add-exp-log to get
    \[\color{red}{\left(1 + \left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right) - \left(0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + 1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)} \leadsto \color{blue}{e^{\log \left(\left(1 + \left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{4}} + 0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^2}\right)\right) - \left(0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{1 + 0.3275911 \cdot \left|x\right|} + \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{5}} + 1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^2}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}}\right)\right)\right)}}\]
    13.6

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)))))))