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. Simplified1.2

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

4. Applied pow11.2

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

5. Applied inv-pow1.2

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

6. Applied pow-div1.1

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

7. Applied pow11.1

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

8. Applied inv-pow1.1

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

9. Applied pow-div1.0

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

10. Applied pow-sqr0.9

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

11. Applied pow10.9

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

12. Applied pow10.9

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

13. Applied pow-flip0.9

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

14. Applied pow-div0.8

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

15. Applied pow-prod-up0.7

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

16. Simplified0.7

    \[\leadsto \frac{(\left(\frac{1}{\left|x\right|}\right) \cdot \left((\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|} \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right) \cdot \frac{3}{4} + \left(\frac{15}{8} \cdot {\left(\left|x\right|\right)}^{\color{blue}{-6}}\right))_*\right) + \left((\frac{1}{2} \cdot \left(\frac{\frac{\frac{1}{\left|x\right|}}{\left|x\right|}}{\left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right))_*}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}}\]

17. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{\left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}}\]

18. Simplified0.5

    \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot (\left(e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{\frac{3}{4}}{{\left(\left|x\right|\right)}^{5}} + \frac{\frac{1}{2}}{\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|}\right) + \frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^{7}}\right) + \left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\left|x\right|}\right))_*}\]

19. Final simplification0.5

    \[\leadsto \sqrt{\frac{1}{\pi}} \cdot (\left(e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{\frac{3}{4}}{{\left(\left|x\right|\right)}^{5}} + \frac{\frac{1}{2}}{\left|x\right| \cdot \left(\left|x\right| \cdot \left|x\right|\right)}\right)\right) + \left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\left|x\right|}\right))_*\]

Reproduce

herbie shell --seed 2019090 +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)))))))