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 simplification1.1

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

4. Applied add-cube-cbrt1.1

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

6. Applied add-sqr-sqrt1.1

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

7. Final simplification1.1

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

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018251 +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)))))))