\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
Test:
Jmat.Real.erfi, branch x less than or equal to 0.5
Bits:
128 bits
Bits error versus x
Time: 21.2 s
Input Error: 0.3
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(\left|\frac{\left(\left(\left|x\right| \cdot \frac{1}{21}\right) \cdot \left({\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^3\right) + \left|x\right| \cdot 2\right) + {\left(\left|x\right|\right)}^3 \cdot \left(\frac{2}{3} + \left|x\right| \cdot \left(\frac{1}{5} \cdot \left|x\right|\right)\right)}{\sqrt{\pi}}\right|\)
  1. Started with
    \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
    0.3
  2. Applied simplify to get
    \[\color{red}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|} \leadsto \color{blue}{\left|\frac{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}{\sqrt{\pi}}\right|}\]
    0.4
  3. Using strategy rm
    0.4
  4. Applied add-cube-cbrt to get
    \[\left|\frac{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}{\color{red}{\sqrt{\pi}}}\right| \leadsto \left|\frac{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}{\color{blue}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^3}}\right|\]
    1.2
  5. Applied add-cube-cbrt to get
    \[\left|\frac{\color{red}{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^3}\right| \leadsto \left|\frac{\color{blue}{{\left(\sqrt[3]{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}\right)}^3}}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^3}\right|\]
    1.2
  6. Applied cube-undiv to get
    \[\left|\color{red}{\frac{{\left(\sqrt[3]{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}\right)}^3}{{\left(\sqrt[3]{\sqrt{\pi}}\right)}^3}}\right| \leadsto \left|\color{blue}{{\left(\frac{\sqrt[3]{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}}{\sqrt[3]{\sqrt{\pi}}}\right)}^3}\right|\]
    1.2
  7. Applied simplify to get
    \[\left|{\color{red}{\left(\frac{\sqrt[3]{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}}{\sqrt[3]{\sqrt{\pi}}}\right)}}^3\right| \leadsto \left|{\color{blue}{\left(\frac{\sqrt[3]{\left|x\right| \cdot \left(2 + \frac{{\left(\left|x\right|\right)}^3}{\frac{5}{\left|x\right|}}\right) + {\left(\left|x\right|\right)}^3 \cdot \left(\frac{\left|x\right|}{21} \cdot {\left(\left|x\right|\right)}^3 + \frac{2}{3}\right)}}{\sqrt[3]{\sqrt{\pi}}}\right)}}^3\right|\]
    1.2
  8. Applied taylor to get
    \[\left|{\left(\frac{\sqrt[3]{\left|x\right| \cdot \left(2 + \frac{{\left(\left|x\right|\right)}^3}{\frac{5}{\left|x\right|}}\right) + {\left(\left|x\right|\right)}^3 \cdot \left(\frac{\left|x\right|}{21} \cdot {\left(\left|x\right|\right)}^3 + \frac{2}{3}\right)}}{\sqrt[3]{\sqrt{\pi}}}\right)}^3\right| \leadsto \left|{\left(\frac{\sqrt[3]{\frac{1}{21} \cdot \left(\left|x\right| \cdot {\left({\left(\left|x\right|\right)}^3\right)}^2\right) + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^3 + \left(\frac{1}{5} \cdot \left({\left(\left|x\right|\right)}^2 \cdot {\left(\left|x\right|\right)}^3\right) + 2 \cdot \left|x\right|\right)\right)}}{\sqrt[3]{\sqrt{\pi}}}\right)}^3\right|\]
    1.2
  9. Taylor expanded around 0 to get
    \[\left|\color{red}{{\left(\frac{\sqrt[3]{\frac{1}{21} \cdot \left(\left|x\right| \cdot {\left({\left(\left|x\right|\right)}^3\right)}^2\right) + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^3 + \left(\frac{1}{5} \cdot \left({\left(\left|x\right|\right)}^2 \cdot {\left(\left|x\right|\right)}^3\right) + 2 \cdot \left|x\right|\right)\right)}}{\sqrt[3]{\sqrt{\pi}}}\right)}^3}\right| \leadsto \left|\color{blue}{{\left(\frac{\sqrt[3]{\frac{1}{21} \cdot \left(\left|x\right| \cdot {\left({\left(\left|x\right|\right)}^3\right)}^2\right) + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^3 + \left(\frac{1}{5} \cdot \left({\left(\left|x\right|\right)}^2 \cdot {\left(\left|x\right|\right)}^3\right) + 2 \cdot \left|x\right|\right)\right)}}{\sqrt[3]{\sqrt{\pi}}}\right)}^3}\right|\]
    1.2
  10. Applied simplify to get
    \[\left|{\left(\frac{\sqrt[3]{\frac{1}{21} \cdot \left(\left|x\right| \cdot {\left({\left(\left|x\right|\right)}^3\right)}^2\right) + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^3 + \left(\frac{1}{5} \cdot \left({\left(\left|x\right|\right)}^2 \cdot {\left(\left|x\right|\right)}^3\right) + 2 \cdot \left|x\right|\right)\right)}}{\sqrt[3]{\sqrt{\pi}}}\right)}^3\right| \leadsto \left|\frac{\left(\left(\left|x\right| \cdot \frac{1}{21}\right) \cdot \left({\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^3\right) + \left|x\right| \cdot 2\right) + {\left(\left|x\right|\right)}^3 \cdot \left(\frac{2}{3} + \left|x\right| \cdot \left(\frac{1}{5} \cdot \left|x\right|\right)\right)}{\sqrt{\pi}}\right|\]
    0.3
  11. Applied final simplification

Original test:


(lambda ((x default))
  #: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)))))))