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. Initial simplification1.4

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

7. Applied inv-pow0.9

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

8. Applied pow-pow0.6

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

10. Applied add-cube-cbrt0.6

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

12. Applied add-sqr-sqrt0.6

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

13. Applied cbrt-prod0.6

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

14. Final simplification0.6

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

Runtime

Time bar (total: 2.3m)Debug logProfile

herbie shell --seed 2018215 +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)))))))