\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
Test:
Jmat.Real.erf
Bits:
128 bits
Bits error versus x
Time: 30.7 s
Input Error: 14.1
Output Error: 10.8
Log: ⚲
Profile: 🕒
\(\frac{{1}^2 - \sqrt[3]{{\left({\left(\frac{(\left((\left(\frac{1.061405429}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) * \left(\frac{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(\frac{1.421413741}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} + -0.284496736\right))_*\right) * \left(\frac{{\left(\sqrt[3]{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}\right)}^3}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(\frac{0.254829592}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right))_*}{e^{\left|x\right| \cdot \left|x\right|}}\right)}^2\right)}^3}}{1 + \frac{(\left((\left(-1.453152027 + \frac{1.061405429}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) * \left(\frac{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(-0.284496736 + \frac{1.421413741}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right))_*\right) * \left(\frac{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(\frac{0.254829592}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right))_*}{e^{\left|x\right| \cdot \left|x\right|}}}\)
  1. Started with
    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
    14.1
  2. Applied simplify to get
    \[\color{red}{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \leadsto \color{blue}{1 - \frac{(\left((\left(\frac{1.061405429}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) * \left(\frac{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(\frac{1.421413741}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} + -0.284496736\right))_*\right) * \left(\frac{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(\frac{0.254829592}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right))_*}{e^{\left|x\right| \cdot \left|x\right|}}}\]
    14.1
  3. Using strategy rm
    14.1
  4. Applied add-cube-cbrt to get
    \[1 - \frac{(\left((\left(\frac{1.061405429}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) * \left(\frac{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(\frac{1.421413741}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} + -0.284496736\right))_*\right) * \left(\frac{\color{red}{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(\frac{0.254829592}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right))_*}{e^{\left|x\right| \cdot \left|x\right|}} \leadsto 1 - \frac{(\left((\left(\frac{1.061405429}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) * \left(\frac{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(\frac{1.421413741}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} + -0.284496736\right))_*\right) * \left(\frac{\color{blue}{{\left(\sqrt[3]{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}\right)}^3}}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(\frac{0.254829592}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right))_*}{e^{\left|x\right| \cdot \left|x\right|}}\]
    14.1
  5. Using strategy rm
    14.1
  6. Applied flip-- to get
    \[\color{red}{1 - \frac{(\left((\left(\frac{1.061405429}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) * \left(\frac{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(\frac{1.421413741}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} + -0.284496736\right))_*\right) * \left(\frac{{\left(\sqrt[3]{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}\right)}^3}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(\frac{0.254829592}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right))_*}{e^{\left|x\right| \cdot \left|x\right|}}} \leadsto \color{blue}{\frac{{1}^2 - {\left(\frac{(\left((\left(\frac{1.061405429}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) * \left(\frac{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(\frac{1.421413741}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} + -0.284496736\right))_*\right) * \left(\frac{{\left(\sqrt[3]{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}\right)}^3}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(\frac{0.254829592}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right))_*}{e^{\left|x\right| \cdot \left|x\right|}}\right)}^2}{1 + \frac{(\left((\left(\frac{1.061405429}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) * \left(\frac{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(\frac{1.421413741}{(0.3275911 * \left(\left|x\right|\right) + 1)_*} + -0.284496736\right))_*\right) * \left(\frac{{\left(\sqrt[3]{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}\right)}^3}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(\frac{0.254829592}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right))_*}{e^{\left|x\right| \cdot \left|x\right|}}}}\]
    14.1
  7. Applied simplify to get
    \[\frac{{1}^2 - {\left(\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{{\left(\sqrt[3]{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}\right)}^3}{(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|}}\right)}^2}{\color{red}{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{{\left(\sqrt[3]{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}\right)}^3}{(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 \frac{{1}^2 - {\left(\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{{\left(\sqrt[3]{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}\right)}^3}{(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|}}\right)}^2}{\color{blue}{1 + \frac{(\left((\left(-1.453152027 + \frac{1.061405429}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) * \left(\frac{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(-0.284496736 + \frac{1.421413741}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\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|}}}}\]
    14.1
  8. Using strategy rm
    14.1
  9. Applied add-cbrt-cube to get
    \[\frac{{1}^2 - \color{red}{{\left(\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{{\left(\sqrt[3]{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}\right)}^3}{(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|}}\right)}^2}}{1 + \frac{(\left((\left(-1.453152027 + \frac{1.061405429}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) * \left(\frac{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(-0.284496736 + \frac{1.421413741}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\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 \frac{{1}^2 - \color{blue}{\sqrt[3]{{\left({\left(\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{{\left(\sqrt[3]{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}\right)}^3}{(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|}}\right)}^2\right)}^3}}}{1 + \frac{(\left((\left(-1.453152027 + \frac{1.061405429}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) * \left(\frac{\frac{1}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\right) + \left(-0.284496736 + \frac{1.421413741}{(0.3275911 * \left(\left|x\right|\right) + 1)_*}\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|}}}\]
    10.8

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)))))))