\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
Test:
Jmat.Real.erfi, branch x less than or equal to 0.5
Bits:
128 bits
Bits error versus x
Time: 26.3 s
Input Error: 0.2
Output Error: 0.2
Log: ⚲
Profile: 🕒
\(\left|\left(\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot 2\right) + \left(\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left({\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^3\right)\right) + \sqrt{\frac{1}{\pi}} \cdot \left({\left(\left|x\right|\right)}^3 \cdot \frac{2}{3} + {\left(\left|x\right|\right)}^{5} \cdot \frac{1}{5}\right)\right|\)
  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. Applied taylor to get
    \[\left|\frac{\left(2 \cdot \left|x\right| + \left(\left|x\right| \cdot \frac{2}{3}\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}{\sqrt{\pi}}\right| \leadsto \left|\frac{\left(2 \cdot \left|x\right| + \left(\frac{2}{3} \cdot \left|x\right|\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}{\sqrt{\pi}}\right|\]
    0.6
  4. Taylor expanded around 0 to get
    \[\left|\frac{\left(2 \cdot \left|x\right| + \color{red}{\left(\frac{2}{3} \cdot \left|x\right|\right)} \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}{\sqrt{\pi}}\right| \leadsto \left|\frac{\left(2 \cdot \left|x\right| + \color{blue}{\left(\frac{2}{3} \cdot \left|x\right|\right)} \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}{\sqrt{\pi}}\right|\]
    0.6
  5. Applied simplify to get
    \[\color{red}{\left|\frac{\left(2 \cdot \left|x\right| + \left(\frac{2}{3} \cdot \left|x\right|\right) \cdot {\left(\left|x\right|\right)}^2\right) + \left(\frac{{\left({\left(\left|x\right|\right)}^3\right)}^2}{\frac{21}{\left|x\right|}} + \frac{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^2}{5}\right)}{\sqrt{\pi}}\right|} \leadsto \color{blue}{\left|\frac{\left(\left|x\right| \cdot 2 + \frac{{\left(\left|x\right| \cdot \left|x\right|\right)}^2}{\frac{5}{\left|x\right|}}\right) + {\left(\left|x\right|\right)}^3 \cdot \left(\frac{2}{3} + \frac{\left|x\right|}{21} \cdot {\left(\left|x\right|\right)}^3\right)}{\sqrt{\pi}}\right|}\]
    0.6
  6. Applied taylor to get
    \[\left|\frac{\left(\left|x\right| \cdot 2 + \frac{{\left(\left|x\right| \cdot \left|x\right|\right)}^2}{\frac{5}{\left|x\right|}}\right) + {\left(\left|x\right|\right)}^3 \cdot \left(\frac{2}{3} + \frac{\left|x\right|}{21} \cdot {\left(\left|x\right|\right)}^3\right)}{\sqrt{\pi}}\right| \leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{21} \cdot \left(\left|x\right| \cdot {\left({\left(\left|x\right|\right)}^3\right)}^2\right) + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + \frac{2}{3} \cdot {\left(\left|x\right|\right)}^3\right)\right)\right)\right|\]
    0.2
  7. Taylor expanded around 0 to get
    \[\left|\color{red}{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{21} \cdot \left(\left|x\right| \cdot {\left({\left(\left|x\right|\right)}^3\right)}^2\right) + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + \frac{2}{3} \cdot {\left(\left|x\right|\right)}^3\right)\right)\right)}\right| \leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{21} \cdot \left(\left|x\right| \cdot {\left({\left(\left|x\right|\right)}^3\right)}^2\right) + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + \frac{2}{3} \cdot {\left(\left|x\right|\right)}^3\right)\right)\right)}\right|\]
    0.2
  8. Applied simplify to get
    \[\left|\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{21} \cdot \left(\left|x\right| \cdot {\left({\left(\left|x\right|\right)}^3\right)}^2\right) + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + \frac{2}{3} \cdot {\left(\left|x\right|\right)}^3\right)\right)\right)\right| \leadsto \left|\left(2 \cdot \left|x\right| + \left(\frac{2}{3} \cdot \left|x\right|\right) \cdot {\left(\left|x\right|\right)}^2\right) \cdot \sqrt{\frac{1}{\pi}} + \left(\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right) \cdot \left({\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^3\right) + {\left(\left|x\right|\right)}^{5} \cdot \left(\frac{1}{5} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right|\]
    0.2
  9. Applied final simplification
  10. Applied simplify to get
    \[\color{red}{\left|\left(2 \cdot \left|x\right| + \left(\frac{2}{3} \cdot \left|x\right|\right) \cdot {\left(\left|x\right|\right)}^2\right) \cdot \sqrt{\frac{1}{\pi}} + \left(\left(\sqrt{\frac{1}{\pi}} \cdot \left(\frac{1}{21} \cdot \left|x\right|\right)\right) \cdot \left({\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^3\right) + {\left(\left|x\right|\right)}^{5} \cdot \left(\frac{1}{5} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right|} \leadsto \color{blue}{\left|\left(\sqrt{\frac{1}{\pi}} \cdot \left(\left|x\right| \cdot 2\right) + \left(\left(\frac{1}{21} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \left({\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^3\right)\right) + \sqrt{\frac{1}{\pi}} \cdot \left({\left(\left|x\right|\right)}^3 \cdot \frac{2}{3} + {\left(\left|x\right|\right)}^{5} \cdot \frac{1}{5}\right)\right|}\]
    0.2
  11. 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)))))))