Error

Derivation

1. Initial program 13.8

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

2. Applied simplify 13.8

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

3. Applied taylor 13.8

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

4. Taylor expanded around 0 13.8

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

5. Applied simplify 13.8

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

7. Applied add-log-exp 13.8

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

8. Applied add-log-exp 13.8

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

9. Applied sum-log 13.8

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

10. Applied add-log-exp 13.8

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

11. Applied sum-log 13.8

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

12. Applied add-log-exp 13.8

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

13. Applied diff-log 13.8

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

14. Applied simplify 13.8

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

16. Applied add-sqr-sqrt 14.9

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

17. Applied add-sqr-sqrt 13.8

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

18. Applied difference-of-squares 13.8

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

19. Applied exp-prod 13.8

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

20. Applied log-pow 12.6

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

21. Applied simplify 12.6

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

Runtime

Time bar (total: 1.9m) Debug log

Please include this information when filing a bug report:

herbie --seed '#(4034646702 4251231237 1276265448 4242016357 811937878 2266925246)'
(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)))))))