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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

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

3. Taylor expanded around 0 1.4

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

4. Applied simplify1.3

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

6. Applied inv-pow1.3

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

7. Applied pow-pow1.0

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

8. Applied simplify1.0

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

10. Applied add-sqr-sqrt0.7

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

Runtime

Time bar (total: 3.2m)Debug logProfile

herbie shell --seed 2018193 
(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)))))))