\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
Test:
Jmat.Real.erf
Bits:
128 bits
Bits error versus x
Time: 10.7 s
Input Error: 13.9
Output Error: 13.9
Log: ⚲
Profile: 🕒
\(e^{\log \left(1 - \frac{\log \left(e^{(\left((\left(1.061405429 \cdot \frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} - 1.453152027\right) * \left(\frac{1}{{\left((0.3275911 * \left(\left|x\right|\right) + 1)_*\right)}^2}\right) + \left(1.421413741 \cdot \frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} - 0.284496736\right))_*\right) * \left(\frac{1}{{\left((0.3275911 * \left(\left|x\right|\right) + 1)_*\right)}^2}\right) + \left(\frac{0.254829592}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right))_*}\right)}{e^{{\left(\left|x\right|\right)}^2}}\right)}\)
  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. Applied simplify to get
    \[\color{red}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \leadsto \color{blue}{1 - \frac{(\left((\left(\frac{1.061405429}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) * \left(\frac{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(\frac{1.421413741}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} + -0.284496736\right))_*\right) * \left(\frac{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(\frac{0.254829592}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right))_*}{e^{\left|x\right| \cdot \left|x\right|}}}\]
    13.9
  3. Applied taylor to get
    \[1 - \frac{(\left((\left(\frac{1.061405429}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) * \left(\frac{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(\frac{1.421413741}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} + -0.284496736\right))_*\right) * \left(\frac{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(\frac{0.254829592}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right))_*}{e^{\left|x\right| \cdot \left|x\right|}} \leadsto 1 - \frac{(\left((\left(1.061405429 \cdot \frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} - 1.453152027\right) * \left(\frac{1}{{\left((0.3275911 * \left(\left|x\right|\right) + 1)_*\right)}^2}\right) + \left(1.421413741 \cdot \frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} - 0.284496736\right))_*\right) * \left(\frac{1}{{\left((0.3275911 * \left(\left|x\right|\right) + 1)_*\right)}^2}\right) + \left(\frac{0.254829592}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right))_*}{e^{{\left(\left|x\right|\right)}^2}}\]
    13.9
  4. Taylor expanded around 0 to get
    \[\color{red}{1 - \frac{(\left((\left(1.061405429 \cdot \frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} - 1.453152027\right) * \left(\frac{1}{{\left((0.3275911 * \left(\left|x\right|\right) + 1)_*\right)}^2}\right) + \left(1.421413741 \cdot \frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} - 0.284496736\right))_*\right) * \left(\frac{1}{{\left((0.3275911 * \left(\left|x\right|\right) + 1)_*\right)}^2}\right) + \left(\frac{0.254829592}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right))_*}{e^{{\left(\left|x\right|\right)}^2}}} \leadsto \color{blue}{1 - \frac{(\left((\left(1.061405429 \cdot \frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} - 1.453152027\right) * \left(\frac{1}{{\left((0.3275911 * \left(\left|x\right|\right) + 1)_*\right)}^2}\right) + \left(1.421413741 \cdot \frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} - 0.284496736\right))_*\right) * \left(\frac{1}{{\left((0.3275911 * \left(\left|x\right|\right) + 1)_*\right)}^2}\right) + \left(\frac{0.254829592}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right))_*}{e^{{\left(\left|x\right|\right)}^2}}}\]
    13.9
  5. Using strategy rm
    13.9
  6. Applied add-log-exp to get
    \[1 - \frac{\color{red}{(\left((\left(1.061405429 \cdot \frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} - 1.453152027\right) * \left(\frac{1}{{\left((0.3275911 * \left(\left|x\right|\right) + 1)_*\right)}^2}\right) + \left(1.421413741 \cdot \frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} - 0.284496736\right))_*\right) * \left(\frac{1}{{\left((0.3275911 * \left(\left|x\right|\right) + 1)_*\right)}^2}\right) + \left(\frac{0.254829592}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right))_*}}{e^{{\left(\left|x\right|\right)}^2}} \leadsto 1 - \frac{\color{blue}{\log \left(e^{(\left((\left(1.061405429 \cdot \frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} - 1.453152027\right) * \left(\frac{1}{{\left((0.3275911 * \left(\left|x\right|\right) + 1)_*\right)}^2}\right) + \left(1.421413741 \cdot \frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} - 0.284496736\right))_*\right) * \left(\frac{1}{{\left((0.3275911 * \left(\left|x\right|\right) + 1)_*\right)}^2}\right) + \left(\frac{0.254829592}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right))_*}\right)}}{e^{{\left(\left|x\right|\right)}^2}}\]
    13.9
  7. Using strategy rm
    13.9
  8. Applied add-exp-log to get
    \[\color{red}{1 - \frac{\log \left(e^{(\left((\left(1.061405429 \cdot \frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} - 1.453152027\right) * \left(\frac{1}{{\left((0.3275911 * \left(\left|x\right|\right) + 1)_*\right)}^2}\right) + \left(1.421413741 \cdot \frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} - 0.284496736\right))_*\right) * \left(\frac{1}{{\left((0.3275911 * \left(\left|x\right|\right) + 1)_*\right)}^2}\right) + \left(\frac{0.254829592}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right))_*}\right)}{e^{{\left(\left|x\right|\right)}^2}}} \leadsto \color{blue}{e^{\log \left(1 - \frac{\log \left(e^{(\left((\left(1.061405429 \cdot \frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} - 1.453152027\right) * \left(\frac{1}{{\left((0.3275911 * \left(\left|x\right|\right) + 1)_*\right)}^2}\right) + \left(1.421413741 \cdot \frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} - 0.284496736\right))_*\right) * \left(\frac{1}{{\left((0.3275911 * \left(\left|x\right|\right) + 1)_*\right)}^2}\right) + \left(\frac{0.254829592}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right))_*}\right)}{e^{{\left(\left|x\right|\right)}^2}}\right)}}\]
    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)))))))