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. Applied simplify1.2

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

4. Applied inv-pow1.2

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

5. Applied pow-pow0.7

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

6. Applied simplify0.7

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

7. Taylor expanded around 0 0.5

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

8. Applied simplify0.5

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

Runtime

Time bar (total: 4.5m)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' 
(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)))))))