Error

Derivation

1. Initial program 0.2

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]

2. Applied simplify0.2

   \[\leadsto \color{blue}{\left|(\left(\frac{1}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{5} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left((\left(\left|x\right| \cdot \frac{2}{3}\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(2 \cdot \left|x\right|\right))_*\right))_*\right) + \left(\left({\left(\left|x\right|\right)}^{3} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \left(\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \frac{1}{\sqrt{\pi}}\right)\right))_*\right|}\]

3. Taylor expanded around 0 0.2

   \[\leadsto \left|(\left(\frac{1}{\sqrt{\pi}}\right) \cdot \left((\left(\color{blue}{\left(\frac{1}{5} \cdot \left|x\right|\right)} \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left((\left(\left|x\right| \cdot \frac{2}{3}\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(2 \cdot \left|x\right|\right))_*\right))_*\right) + \left(\left({\left(\left|x\right|\right)}^{3} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \left(\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \frac{1}{\sqrt{\pi}}\right)\right))_*\right|\]

4. Applied simplify0.2

   \[\leadsto \color{blue}{\left|(\left(\frac{1}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{5} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left((\left(\frac{2}{3} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(2 \cdot \left|x\right|\right))_*\right))_*\right) + \left(\frac{{\left(\left|x\right|\right)}^{3}}{\sqrt{\pi}} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{\frac{21}{\left|x\right|}}\right))_*\right|}\]

5. Taylor expanded around 0 0.2

   \[\leadsto \left|(\left(\frac{1}{\sqrt{\pi}}\right) \cdot \left((\left(\left(\frac{1}{5} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left((\left(\frac{2}{3} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(2 \cdot \left|x\right|\right))_*\right))_*\right) + \left(\frac{{\left(\left|x\right|\right)}^{3}}{\color{blue}{\sqrt{\pi}}} \cdot \frac{{\left(\left|x\right|\right)}^{3}}{\frac{21}{\left|x\right|}}\right))_*\right|\]

6. Applied simplify0.2

   \[\leadsto \color{blue}{\left|(\left((\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left((\left(\frac{1}{5} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(\frac{\left|x\right|}{\frac{3}{2}}\right))_*\right) + \left(2 \cdot \left|x\right|\right))_*\right) \cdot \left(\frac{1}{\sqrt{\pi}}\right) + \left(\frac{{\left(\left|x\right|\right)}^{3} \cdot {\left(\left|x\right|\right)}^{3}}{\frac{\sqrt{\pi} \cdot 21}{\left|x\right|}}\right))_*\right|}\]
7. Using strategy rm

8. Applied pow-prod-up0.1

   \[\leadsto \left|(\left((\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left((\left(\frac{1}{5} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(\frac{\left|x\right|}{\frac{3}{2}}\right))_*\right) + \left(2 \cdot \left|x\right|\right))_*\right) \cdot \left(\frac{1}{\sqrt{\pi}}\right) + \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{\left(3 + 3\right)}}}{\frac{\sqrt{\pi} \cdot 21}{\left|x\right|}}\right))_*\right|\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  (fabs (* (/ 1 (sqrt PI)) (+ (+ (+ (* 2 (fabs x)) (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1 21) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))