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Derivation

1. Initial program 13.6

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

3. Applied distribute-rgt-in13.6

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

4. Applied associate-+r+13.6

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

5. Simplified13.6

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

7. Applied flip-+13.6

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

8. Applied associate-*r/13.6

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

9. Simplified13.6

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

11. Applied add-exp-log13.6

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

13. Applied add-cube-cbrt13.6

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

14. Final simplification13.6

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

Runtime

Time bar (total: 3.3m)Debug logProfile

herbie shell --seed 2018230 
(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)))))))