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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. Initial simplification1.4

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

3. Simplified1.3

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

5. Applied inv-pow1.3

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

6. Applied pow-pow1.0

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

8. Applied add-sqr-sqrt0.7

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

9. Final simplification0.7

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018217 
(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)))))))