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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. Using strategy rm

3. Applied *-un-lft-identity0.2

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

4. Applied associate-*r*0.2

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

5. Simplified0.2

   \[\leadsto \left|\left(\frac{1}{\sqrt{\pi}} \cdot 1\right) \cdot \color{blue}{\left(\left(\left|x\right| \cdot {\left(\left|x\right|\right)}^{4}\right) \cdot \left(\frac{1}{5} + \left|x\right| \cdot \left(\left|x\right| \cdot \frac{1}{21}\right)\right) + \left(\left(\left|x\right| \cdot \frac{2}{3}\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left|x\right| \cdot 2\right)\right)}\right|\]
6. Using strategy rm

7. Applied add-sqr-sqrt0.2

   \[\leadsto \left|\left(\frac{1}{\sqrt{\pi}} \cdot 1\right) \cdot \left(\left(\left|x\right| \cdot {\left(\left|x\right|\right)}^{4}\right) \cdot \left(\frac{1}{5} + \left|x\right| \cdot \color{blue}{\left(\sqrt{\left|x\right| \cdot \frac{1}{21}} \cdot \sqrt{\left|x\right| \cdot \frac{1}{21}}\right)}\right) + \left(\left(\left|x\right| \cdot \frac{2}{3}\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left|x\right| \cdot 2\right)\right)\right|\]

8. Final simplification0.2

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left|x\right| \cdot {\left(\left|x\right|\right)}^{4}\right) \cdot \left(\left(\sqrt{\frac{1}{21} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{21} \cdot \left|x\right|}\right) \cdot \left|x\right| + \frac{1}{5}\right) + \left(\left(\frac{2}{3} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right) + \left|x\right| \cdot 2\right)\right)\right|\]

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018353 
(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)))))))