\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
Test:
Jmat.Real.erfi, branch x greater than or equal to 5
Bits:
128 bits
Bits error versus x
Time: 1.4 m
Input Error: 1.5
Output Error: 0.9
Log: ⚲
Profile: 🕒
\((\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left(\frac{\frac{\frac{3}{4}}{\left|x\right|}}{{\left(\left|x\right|\right)}^3}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right))_*\right) + \left(\frac{\frac{15}{8}}{{\left({\left(\left|x\right|\right)}^3\right)}^2} \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi} \cdot \left|x\right|}\right))_*\)
  1. Started with
    \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
    1.5
  2. Applied simplify to get
    \[\color{red}{\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)} \leadsto \color{blue}{(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{{\left(\frac{1}{\left|x\right|}\right)}^3}{2}\right))_*\right) + \left(\frac{\frac{15}{8} \cdot e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^3 \cdot {\left(\frac{1}{\left|x\right|}\right)}^3}{\left|x\right|}\right))_*}\]
    1.4
  3. Using strategy rm
    1.4
  4. Applied cube-div to get
    \[(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{{\left(\frac{1}{\left|x\right|}\right)}^3}{2}\right))_*\right) + \left(\frac{\frac{15}{8} \cdot e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \color{red}{{\left(\frac{1}{\left|x\right|}\right)}^3}}{\left|x\right|}\right))_* \leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{{\left(\frac{1}{\left|x\right|}\right)}^3}{2}\right))_*\right) + \left(\frac{\frac{15}{8} \cdot e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \color{blue}{\frac{{1}^3}{{\left(\left|x\right|\right)}^3}}}{\left|x\right|}\right))_*\]
    1.1
  5. Applied taylor to get
    \[(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{{\left(\frac{1}{\left|x\right|}\right)}^3}{2}\right))_*\right) + \left(\frac{\frac{15}{8} \cdot e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \frac{{1}^3}{{\left(\left|x\right|\right)}^3}}{\left|x\right|}\right))_* \leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{{\left(\frac{1}{\left|\frac{1}{x}\right|}\right)}^3}{2}\right))_*\right) + \left(\frac{\frac{15}{8} \cdot e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \frac{{1}^3}{{\left(\left|x\right|\right)}^3}}{\left|x\right|}\right))_*\]
    5.0
  6. Taylor expanded around inf to get
    \[(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\color{red}{{\left(\frac{1}{\left|\frac{1}{x}\right|}\right)}^3}}{2}\right))_*\right) + \left(\frac{\frac{15}{8} \cdot e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \frac{{1}^3}{{\left(\left|x\right|\right)}^3}}{\left|x\right|}\right))_* \leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\color{blue}{{\left(\frac{1}{\left|\frac{1}{x}\right|}\right)}^3}}{2}\right))_*\right) + \left(\frac{\frac{15}{8} \cdot e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \frac{{1}^3}{{\left(\left|x\right|\right)}^3}}{\left|x\right|}\right))_*\]
    5.0
  7. Applied simplify to get
    \[\color{red}{(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{{\left(\frac{1}{\left|\frac{1}{x}\right|}\right)}^3}{2}\right))_*\right) + \left(\frac{\frac{15}{8} \cdot e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \frac{{1}^3}{{\left(\left|x\right|\right)}^3}}{\left|x\right|}\right))_*} \leadsto \color{blue}{(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left(\frac{\frac{1}{\left|x\right|}}{{\left(\left|x\right|\right)}^3} \cdot \frac{3}{4}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{1}{2}}{{\left(\left|\frac{1}{x}\right|\right)}^3}\right))_*\right) + \left(\frac{\frac{1}{{\left(\left|x\right|\right)}^3}}{{\left(\left|x\right|\right)}^3 \cdot \frac{\left|x\right|}{1}} \cdot \left(\frac{\frac{15}{8}}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right)\right))_*}\]
    5.0
  8. Applied taylor to get
    \[(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left(\frac{\frac{1}{\left|x\right|}}{{\left(\left|x\right|\right)}^3} \cdot \frac{3}{4}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{1}{2}}{{\left(\left|\frac{1}{x}\right|\right)}^3}\right))_*\right) + \left(\frac{\frac{1}{{\left(\left|x\right|\right)}^3}}{{\left(\left|x\right|\right)}^3 \cdot \frac{\left|x\right|}{1}} \cdot \left(\frac{\frac{15}{8}}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right)\right))_* \leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left(\frac{\frac{1}{\left|x\right|}}{{\left(\left|x\right|\right)}^3} \cdot \frac{3}{4}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^3}\right))_*\right) + \left(\frac{\frac{1}{{\left(\left|x\right|\right)}^3}}{{\left(\left|x\right|\right)}^3 \cdot \frac{\left|x\right|}{1}} \cdot \left(\frac{\frac{15}{8}}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right)\right))_*\]
    1.1
  9. Taylor expanded around inf to get
    \[(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left(\frac{\frac{1}{\left|x\right|}}{{\left(\left|x\right|\right)}^3} \cdot \frac{3}{4}\right) * \left(\frac{1}{\left|x\right|}\right) + \color{red}{\left(\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^3}\right)})_*\right) + \left(\frac{\frac{1}{{\left(\left|x\right|\right)}^3}}{{\left(\left|x\right|\right)}^3 \cdot \frac{\left|x\right|}{1}} \cdot \left(\frac{\frac{15}{8}}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right)\right))_* \leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left(\frac{\frac{1}{\left|x\right|}}{{\left(\left|x\right|\right)}^3} \cdot \frac{3}{4}\right) * \left(\frac{1}{\left|x\right|}\right) + \color{blue}{\left(\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^3}\right)})_*\right) + \left(\frac{\frac{1}{{\left(\left|x\right|\right)}^3}}{{\left(\left|x\right|\right)}^3 \cdot \frac{\left|x\right|}{1}} \cdot \left(\frac{\frac{15}{8}}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right)\right))_*\]
    1.1
  10. Applied simplify to get
    \[(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left(\frac{\frac{1}{\left|x\right|}}{{\left(\left|x\right|\right)}^3} \cdot \frac{3}{4}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^3}\right))_*\right) + \left(\frac{\frac{1}{{\left(\left|x\right|\right)}^3}}{{\left(\left|x\right|\right)}^3 \cdot \frac{\left|x\right|}{1}} \cdot \left(\frac{\frac{15}{8}}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right)\right))_* \leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left((\left(\frac{\frac{\frac{3}{4}}{\left|x\right|}}{{\left(\left|x\right|\right)}^3}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right))_* + \frac{1}{\left|x\right|}\right) + \left(\frac{\frac{e^{\left|x\right| \cdot \left|x\right|}}{\frac{\sqrt{\pi}}{\frac{15}{8}}}}{\left({\left(\left|x\right|\right)}^3 \cdot \frac{\left|x\right|}{1}\right) \cdot {\left(\left|x\right|\right)}^3}\right))_*\]
    1.0
  11. Applied final simplification
  12. Applied simplify to get
    \[\color{red}{(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left((\left(\frac{\frac{\frac{3}{4}}{\left|x\right|}}{{\left(\left|x\right|\right)}^3}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right))_* + \frac{1}{\left|x\right|}\right) + \left(\frac{\frac{e^{\left|x\right| \cdot \left|x\right|}}{\frac{\sqrt{\pi}}{\frac{15}{8}}}}{\left({\left(\left|x\right|\right)}^3 \cdot \frac{\left|x\right|}{1}\right) \cdot {\left(\left|x\right|\right)}^3}\right))_*} \leadsto \color{blue}{(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left(\frac{\frac{\frac{3}{4}}{\left|x\right|}}{{\left(\left|x\right|\right)}^3}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right))_*\right) + \left(\frac{\frac{15}{8}}{{\left({\left(\left|x\right|\right)}^3\right)}^2} \cdot \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi} \cdot \left|x\right|}\right))_*}\]
    0.9

Original test:


(lambda ((x default))
  #:name "Jmat.Real.erfi, branch x greater than or equal to 5"
  (* (* (/ 1 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1 (fabs x)) (* (/ 1 2) (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))))) (* (/ 3 4) (* (* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))))) (* (/ 15 8) (* (* (* (* (* (* (/ 1 (fabs x)) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x))) (/ 1 (fabs x)))))))