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 simplify1.1

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

3. Taylor expanded around 0 0.7

   \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\frac{15}{8}\right) \cdot \left(\color{blue}{\frac{1}{{\left(\left|x\right|\right)}^{6}}} \cdot \frac{1}{\left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right) + \left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot (\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right|} \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{3}{4}\right) + \left(\frac{\frac{1}{\left|x\right|}}{2} \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right|}\right))_*\right))_*\]

4. Applied simplify0.7

   \[\leadsto \color{blue}{(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\frac{15}{8}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{{\left(\left|x\right|\right)}^{6}}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right) + \left((\left(\frac{\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{3}{4}\right) + \left(\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right))_* \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right))_*}\]
5. Using strategy rm

6. Applied add-sqr-sqrt0.7

   \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}}\right) \cdot \left((\left(\frac{15}{8}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{{\left(\left|x\right|\right)}^{6}}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right) + \left((\left(\frac{\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{3}{4}\right) + \left(\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right))_* \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right))_*\]
7. Using strategy rm

8. Applied inv-pow0.7

   \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}\right) \cdot \left((\left(\frac{15}{8}\right) \cdot \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{-1}}}{{\left(\left|x\right|\right)}^{6}}\right) + \left(\frac{1}{\left|x\right|}\right))_*\right) + \left((\left(\frac{\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{3}{4}\right) + \left(\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right))_* \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right))_*\]

9. Applied pow-div0.5

   \[\leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}\right) \cdot \left((\left(\frac{15}{8}\right) \cdot \color{blue}{\left({\left(\left|x\right|\right)}^{\left(-1 - 6\right)}\right)} + \left(\frac{1}{\left|x\right|}\right))_*\right) + \left((\left(\frac{\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{3}{4}\right) + \left(\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right))_* \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right))_*\]

Runtime

Time bar (total: 2.8m)Debug logProfile

herbie shell --seed '#(1070100504 930361288 1279167582 284574201 1450237281 2578255382)' +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)))))))