\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
Test:
Jmat.Real.erfi, branch x less than or equal to 0.5
Bits:
128 bits
Bits error versus x
Time: 25.0 s
Input Error: 0.2
Output Error: 0.2
Log: ⚲
Profile: 🕒
\(\left|\left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right) \cdot \left(\left(\left|x\right| \cdot \frac{1}{5}\right) \cdot {\left(\left|x\right|\right)}^3 + 2\right) + \left({\left(\left|x\right|\right)}^3 \cdot \left(\frac{1}{21} \cdot \left|x\right|\right) + \frac{2}{3}\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {\left(\left|x\right|\right)}^3\right)\right|\)
  1. Started with
    \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
    0.2
  2. Applied simplify to get
    \[\color{red}{\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|} \leadsto \color{blue}{\left|\frac{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}{\sqrt{\pi}}\right|}\]
    0.6
  3. Using strategy rm
    0.6
  4. Applied add-sqr-sqrt to get
    \[\left|\frac{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}{\color{red}{\sqrt{\pi}}}\right| \leadsto \left|\frac{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}{\color{blue}{{\left(\sqrt{\sqrt{\pi}}\right)}^2}}\right|\]
    0.4
  5. Applied taylor to get
    \[\left|\frac{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}{{\left(\sqrt{\sqrt{\pi}}\right)}^2}\right| \leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{21} \cdot \left(\left|x\right| \cdot {\left({\left(\left|x\right|\right)}^3\right)}^2\right) + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + \frac{1}{5} \cdot \left({\left(\left|x\right|\right)}^2 \cdot {\left(\left|x\right|\right)}^3\right)\right)\right)\right)\right|\]
    0.2
  6. Taylor expanded around 0 to get
    \[\left|\color{red}{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{21} \cdot \left(\left|x\right| \cdot {\left({\left(\left|x\right|\right)}^3\right)}^2\right) + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + \frac{1}{5} \cdot \left({\left(\left|x\right|\right)}^2 \cdot {\left(\left|x\right|\right)}^3\right)\right)\right)\right)}\right| \leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{21} \cdot \left(\left|x\right| \cdot {\left({\left(\left|x\right|\right)}^3\right)}^2\right) + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + \frac{1}{5} \cdot \left({\left(\left|x\right|\right)}^2 \cdot {\left(\left|x\right|\right)}^3\right)\right)\right)\right)}\right|\]
    0.2
  7. Applied simplify to get
    \[\left|\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{21} \cdot \left(\left|x\right| \cdot {\left({\left(\left|x\right|\right)}^3\right)}^2\right) + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + \left(2 \cdot \left|x\right| + \frac{1}{5} \cdot \left({\left(\left|x\right|\right)}^2 \cdot {\left(\left|x\right|\right)}^3\right)\right)\right)\right)\right| \leadsto \left|\left(\left(\left(\left|x\right| \cdot \left(\frac{1}{5} \cdot \left|x\right|\right)\right) \cdot {\left(\left|x\right|\right)}^3 + 2 \cdot \left|x\right|\right) + \left(\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot {\left({\left(\left|x\right|\right)}^3\right)}^2 + \left(\frac{2}{3} \cdot \left|x\right|\right) \cdot {\left(\left|x\right|\right)}^2\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right|\]
    0.2
  8. Applied final simplification
  9. Applied simplify to get
    \[\color{red}{\left|\left(\left(\left(\left|x\right| \cdot \left(\frac{1}{5} \cdot \left|x\right|\right)\right) \cdot {\left(\left|x\right|\right)}^3 + 2 \cdot \left|x\right|\right) + \left(\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot {\left({\left(\left|x\right|\right)}^3\right)}^2 + \left(\frac{2}{3} \cdot \left|x\right|\right) \cdot {\left(\left|x\right|\right)}^2\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right|} \leadsto \color{blue}{\left|\left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right) \cdot \left(\left(\left|x\right| \cdot \frac{1}{5}\right) \cdot {\left(\left|x\right|\right)}^3 + 2\right) + \left({\left(\left|x\right|\right)}^3 \cdot \left(\frac{1}{21} \cdot \left|x\right|\right) + \frac{2}{3}\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot {\left(\left|x\right|\right)}^3\right)\right|}\]
    0.2
  10. 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)))))))