\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
Test:
Jmat.Real.erfi, branch x less than or equal to 0.5
Bits:
128 bits
Bits error versus x
Time: 21.8 s
Input Error: 0.3
Output Error: 0.3
Log: ⚲
Profile: 🕒
\(\left|\frac{\left(\left(\left|x\right| \cdot \frac{1}{21}\right) \cdot \left({\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^3\right) + {\left(\left|x\right|\right)}^3 \cdot \left(\left(\left|x\right| \cdot \frac{1}{5}\right) \cdot \left|x\right|\right)\right) + \left(2 \cdot \left|x\right| + \frac{2}{3} \cdot {\left(\left|x\right|\right)}^3\right)}{\sqrt{\pi}}\right|\)
  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-sqr-sqrt to get
    \[\left|\color{red}{\frac{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}{\sqrt{\pi}}}\right| \leadsto \left|\color{blue}{{\left(\sqrt{\frac{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}{\sqrt{\pi}}}\right)}^2}\right|\]
    0.6
  5. Using strategy rm
    0.6
  6. Applied div-inv to get
    \[\left|{\left(\sqrt{\color{red}{\frac{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}{\sqrt{\pi}}}}\right)}^2\right| \leadsto \left|{\left(\sqrt{\color{blue}{\left(\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)\right) \cdot \frac{1}{\sqrt{\pi}}}}\right)}^2\right|\]
    0.6
  7. Applied sqrt-prod to get
    \[\left|{\color{red}{\left(\sqrt{\left(\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)\right) \cdot \frac{1}{\sqrt{\pi}}}\right)}}^2\right| \leadsto \left|{\color{blue}{\left(\sqrt{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)} \cdot \sqrt{\frac{1}{\sqrt{\pi}}}\right)}}^2\right|\]
    0.9
  8. Applied square-prod to get
    \[\left|\color{red}{{\left(\sqrt{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)} \cdot \sqrt{\frac{1}{\sqrt{\pi}}}\right)}^2}\right| \leadsto \left|\color{blue}{{\left(\sqrt{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}\right)}^2 \cdot {\left(\sqrt{\frac{1}{\sqrt{\pi}}}\right)}^2}\right|\]
    0.9
  9. Applied simplify to get
    \[\left|{\left(\sqrt{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}\right)}^2 \cdot \color{red}{{\left(\sqrt{\frac{1}{\sqrt{\pi}}}\right)}^2}\right| \leadsto \left|{\left(\sqrt{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}\right)}^2 \cdot \color{blue}{\frac{1}{\sqrt{\pi}}}\right|\]
    0.6
  10. Applied taylor to get
    \[\left|{\left(\sqrt{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}\right)}^2 \cdot \frac{1}{\sqrt{\pi}}\right| \leadsto \left|\left(\frac{1}{21} \cdot \left(\left|x\right| \cdot {\left({\left(\left|x\right|\right)}^3\right)}^2\right) + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + \frac{1}{5} \cdot \left({\left(\left|x\right|\right)}^2 \cdot {\left(\left|x\right|\right)}^3\right)\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]
    0.3
  11. Taylor expanded around 0 to get
    \[\left|\color{red}{\left(\frac{1}{21} \cdot \left(\left|x\right| \cdot {\left({\left(\left|x\right|\right)}^3\right)}^2\right) + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + \frac{1}{5} \cdot \left({\left(\left|x\right|\right)}^2 \cdot {\left(\left|x\right|\right)}^3\right)\right)\right)\right)} \cdot \frac{1}{\sqrt{\pi}}\right| \leadsto \left|\color{blue}{\left(\frac{1}{21} \cdot \left(\left|x\right| \cdot {\left({\left(\left|x\right|\right)}^3\right)}^2\right) + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + \frac{1}{5} \cdot \left({\left(\left|x\right|\right)}^2 \cdot {\left(\left|x\right|\right)}^3\right)\right)\right)\right)} \cdot \frac{1}{\sqrt{\pi}}\right|\]
    0.3
  12. Applied simplify to get
    \[\left|\left(\frac{1}{21} \cdot \left(\left|x\right| \cdot {\left({\left(\left|x\right|\right)}^3\right)}^2\right) + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + \frac{1}{5} \cdot \left({\left(\left|x\right|\right)}^2 \cdot {\left(\left|x\right|\right)}^3\right)\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right| \leadsto \left|\frac{\left(\left(\left|x\right| \cdot \frac{1}{21}\right) \cdot \left({\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^3\right) + {\left(\left|x\right|\right)}^3 \cdot \left(\left(\left|x\right| \cdot \frac{1}{5}\right) \cdot \left|x\right|\right)\right) + \left(2 \cdot \left|x\right| + \frac{2}{3} \cdot {\left(\left|x\right|\right)}^3\right)}{\sqrt{\pi}}\right|\]
    0.3
  13. Applied final simplification
  14. Removed slow pow expressions

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)))))))