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. Taylor expanded around 0 1.3

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

3. Applied simplify1.2

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

5. Applied inv-pow1.2

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

6. Applied pow-pow0.7

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

7. Applied simplify0.7

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

Runtime

Time bar (total: 5.8m)Debug logProfile

herbie shell --seed '#(1072967564 1937075727 894099792 790700740 1036514779 1027793188)' +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)))))))