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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. Using strategy rm

3. Applied frac-times1.3

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

4. Simplified1.3

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

5. Simplified1.0

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

6. Taylor expanded around inf 0.7

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

7. Simplified0.7

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

8. Taylor expanded around inf 0.6

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

9. Simplified0.6

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

10. Final simplification0.6

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

Runtime

Time bar (total: 3.2m)Debug logProfile

herbie shell --seed 2018273 
(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)))))))