Average Error: 13.9 → 13.9
Time: 5.0m
Precision: 64
Internal Precision: 576
\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[\left(\sqrt[3]{\log \left(e^{1 - \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{\left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right) + \frac{1.061405429}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)} + \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot -0.284496736 + 0.254829592\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{\left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right) + \frac{1.061405429}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)} + \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot -0.284496736 + 0.254829592\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{\left(\left(\frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{2}} \cdot 0.284496736 + 1\right) + \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{4}} \cdot 1.453152027\right) - \left(\left(\frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{3}} \cdot 1.421413741 + \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{1 + \left|x\right| \cdot 0.3275911} \cdot 0.254829592\right) + \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{5}} \cdot 1.061405429\right)}\right)}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.9

    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
  2. Using strategy rm

  3. Applied distribute-rgt-in13.9

    \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \color{blue}{\left(-0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right) \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

  4. Applied associate-+r+13.9

    \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(\left(0.254829592 + -0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) + \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right) \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right)}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

  5. Simplified13.9

    \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + -0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) + \color{blue}{\frac{\frac{1.061405429}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)} + \left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right)}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
  6. Using strategy rm

  7. Applied add-log-exp13.9

    \[\leadsto \color{blue}{\log \left(e^{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + -0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) + \frac{\frac{1.061405429}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)} + \left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right)}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt13.9

    \[\leadsto \color{blue}{\left(\sqrt[3]{\log \left(e^{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + -0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) + \frac{\frac{1.061405429}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)} + \left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right)}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + -0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) + \frac{\frac{1.061405429}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)} + \left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right)}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + -0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) + \frac{\frac{1.061405429}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)} + \left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right)}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}}\]

  10. Taylor expanded around inf 13.9

    \[\leadsto \left(\sqrt[3]{\log \left(e^{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + -0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) + \frac{\frac{1.061405429}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)} + \left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right)}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + -0.284496736 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) + \frac{\frac{1.061405429}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)} + \left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right)}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{\color{blue}{\left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{4}} + \left(0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}} + 1\right)\right) - \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{5}} + \left(1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{3}} + 0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{0.3275911 \cdot \left|x\right| + 1}\right)\right)}}\right)}\]

  11. Final simplification13.9

    \[\leadsto \left(\sqrt[3]{\log \left(e^{1 - \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{\left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right) + \frac{1.061405429}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)} + \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot -0.284496736 + 0.254829592\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{\left(\frac{-1.453152027}{1 + \left|x\right| \cdot 0.3275911} + 1.421413741\right) + \frac{1.061405429}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}}{\left(1 + \left|x\right| \cdot 0.3275911\right) \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)} + \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot -0.284496736 + 0.254829592\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{\left(\left(\frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{2}} \cdot 0.284496736 + 1\right) + \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{4}} \cdot 1.453152027\right) - \left(\left(\frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{3}} \cdot 1.421413741 + \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{1 + \left|x\right| \cdot 0.3275911} \cdot 0.254829592\right) + \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{5}} \cdot 1.061405429\right)}\right)}\]

Runtime

Time bar (total: 5.0m)Debug logProfile

herbie shell --seed 2018274 
(FPCore (x)
  :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)))))))