Error

Derivation

1. Initial program 1.5

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]

2. Applied simplify1.4

   \[\leadsto \color{blue}{(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) + \left((\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{2}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\left({\left(\frac{1}{\left|x\right|}\right)}^{\left(3 + 1\right)} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)\right)\right))_*}\]
3. Using strategy rm

4. Applied inv-pow1.4

   \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) + \left((\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{2}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\left({\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{\left(3 + 1\right)} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)\right)\right))_*\]

5. Applied pow-pow0.9

   \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) + \left((\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{2}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\left(\color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot \left(3 + 1\right)\right)}} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)\right)\right))_*\]

6. Applied simplify0.9

   \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) + \left((\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{2}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\left({\left(\left|x\right|\right)}^{\color{blue}{\left(-1 - 3\right)}} \cdot {\left(\frac{1}{\left|x\right|}\right)}^{3}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)\right)\right))_*\]
7. Using strategy rm

8. Applied inv-pow0.9

   \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) + \left((\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{2}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\left({\left(\left|x\right|\right)}^{\left(-1 - 3\right)} \cdot {\color{blue}{\left({\left(\left|x\right|\right)}^{-1}\right)}}^{3}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)\right)\right))_*\]

9. Applied pow-pow0.6

   \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) + \left((\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{2}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\left({\left(\left|x\right|\right)}^{\left(-1 - 3\right)} \cdot \color{blue}{{\left(\left|x\right|\right)}^{\left(-1 \cdot 3\right)}}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)\right)\right))_*\]

10. Applied simplify0.6

    \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) + \left((\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{2}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\left({\left(\left|x\right|\right)}^{\left(-1 - 3\right)} \cdot {\left(\left|x\right|\right)}^{\color{blue}{\left(-3\right)}}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)\right)\right))_*\]
11. Using strategy rm

12. Applied pow10.6

    \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) + \left((\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{2}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\left({\left(\left|x\right|\right)}^{\left(-1 - 3\right)} \cdot {\left(\left|x\right|\right)}^{\left(-3\right)}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \color{blue}{{\left(\frac{15}{8}\right)}^{1}}\right)\right)\right))_*\]

13. Applied pow10.6

    \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) + \left((\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{2}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\left({\left(\left|x\right|\right)}^{\left(-1 - 3\right)} \cdot {\left(\left|x\right|\right)}^{\left(-3\right)}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\color{blue}{{\left(\frac{1}{\sqrt{\pi}}\right)}^{1}} \cdot {\left(\frac{15}{8}\right)}^{1}\right)\right)\right))_*\]

14. Applied pow-prod-down0.6

    \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) + \left((\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{2}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\left({\left(\left|x\right|\right)}^{\left(-1 - 3\right)} \cdot {\left(\left|x\right|\right)}^{\left(-3\right)}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \color{blue}{{\left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)}^{1}}\right)\right))_*\]

15. Applied pow10.6

    \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) + \left((\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{2}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\left({\left(\left|x\right|\right)}^{\left(-1 - 3\right)} \cdot {\left(\left|x\right|\right)}^{\left(-3\right)}\right) \cdot \left(\color{blue}{{\left(e^{\left|x\right| \cdot \left|x\right|}\right)}^{1}} \cdot {\left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)}^{1}\right)\right))_*\]

16. Applied pow-prod-down0.6

    \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) + \left((\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{2}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\left({\left(\left|x\right|\right)}^{\left(-1 - 3\right)} \cdot {\left(\left|x\right|\right)}^{\left(-3\right)}\right) \cdot \color{blue}{{\left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)\right)}^{1}}\right))_*\]

17. Applied pow10.6

    \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) + \left((\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{2}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left(\color{blue}{{\left({\left(\left|x\right|\right)}^{\left(-1 - 3\right)} \cdot {\left(\left|x\right|\right)}^{\left(-3\right)}\right)}^{1}} \cdot {\left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)\right)}^{1}\right))_*\]

18. Applied pow-prod-down0.6

    \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) + \left((\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{2}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \color{blue}{\left({\left(\left({\left(\left|x\right|\right)}^{\left(-1 - 3\right)} \cdot {\left(\left|x\right|\right)}^{\left(-3\right)}\right) \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{\sqrt{\pi}} \cdot \frac{15}{8}\right)\right)\right)}^{1}\right)})_*\]

19. Applied simplify0.7

    \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) + \left((\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{2}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left({\color{blue}{\left(\frac{{\left(\left|x\right|\right)}^{\left(-3\right)}}{\frac{\sqrt{\pi}}{\frac{15}{8}}} \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot {\left(\left|x\right|\right)}^{\left(-1 - 3\right)}\right)\right)}}^{1}\right))_*\]
20. Using strategy rm

21. Applied add-sqr-sqrt0.5

    \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) + \left((\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{2}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*\right) + \left({\left(\frac{{\left(\left|x\right|\right)}^{\left(-3\right)}}{\frac{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}}{\frac{15}{8}}} \cdot \left(e^{\left|x\right| \cdot \left|x\right|} \cdot {\left(\left|x\right|\right)}^{\left(-1 - 3\right)}\right)\right)}^{1}\right))_*\]

Runtime

Time bar (total: 4.6m)Debug logProfile

herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)' +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  (* (* (/ 1 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1 (fabs x)) (* (/ 1 2) (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))))) (* (/ 3 4) (* (* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))))) (* (/ 15 8) (* (* (* (* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x)))))))