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. Simplified0.2

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

3. Simplified0.1

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

4. Simplified0.1

   \[\leadsto \color{blue}{\left|(\left(\frac{1}{\sqrt{\pi}}\right) \cdot \left((\left((\left(\frac{1}{5} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(\frac{2}{3} \cdot \left|x\right|\right))_*\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(\left|x\right| \cdot 2\right))_*\right) + \left(\left({\left(\left|x\right|\right)}^{6} \cdot \left|x\right|\right) \cdot \frac{\frac{1}{21}}{\sqrt{\pi}}\right))_*\right|}\]

5. Taylor expanded around 0 0.1

   \[\leadsto \left|(\left(\frac{1}{\sqrt{\pi}}\right) \cdot \left((\left((\left(\frac{1}{5} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \color{blue}{\left(\frac{2}{3} \cdot \left|x\right|\right)})_*\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(\left|x\right| \cdot 2\right))_*\right) + \left(\left({\left(\left|x\right|\right)}^{6} \cdot \left|x\right|\right) \cdot \frac{\frac{1}{21}}{\sqrt{\pi}}\right))_*\right|\]

6. Final simplification0.1

   \[\leadsto \left|(\left(\frac{1}{\sqrt{\pi}}\right) \cdot \left((\left((\left(\left|x\right| \cdot \frac{1}{5}\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(\frac{2}{3} \cdot \left|x\right|\right))_*\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left(2 \cdot \left|x\right|\right))_*\right) + \left(\left(\left|x\right| \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \frac{\frac{1}{21}}{\sqrt{\pi}}\right))_*\right|\]

Reproduce

herbie shell --seed 2019007 +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)))))))

Details

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