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 simplify0.6

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

4. Applied add-sqr-sqrt0.4

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

5. Taylor expanded around 0 0.4

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

6. Applied simplify0.5

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

8. Applied add-sqr-sqrt0.3

   \[\leadsto \left|\frac{\left(\left(\left|x\right| + \left|x\right|\right) + \left(\frac{2}{3} \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) + \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\frac{\left|x\right| \cdot \left|x\right|}{\frac{5}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^{\left(3 + 2\right)}}{21}\right)}{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}}\right|\]
9. Removed slow pow expressions.

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1063282112 2455465480 4141627379 3773598652 1647277307 776739644)' 
(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)))))))