\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
Test:
Jmat.Real.erfi, branch x greater than or equal to 5
Bits:
128 bits
Bits error versus x
Time: 1.8 m
Input Error: 1.5
Output Error: 0.8
Log: ⚲
Profile: 🕒
\((\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left((\left(\frac{\frac{\frac{3}{\left|x\right|}}{{\left(\left|x\right|\right)}^3}}{4}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^3}\right))_* + \frac{1}{\left|x\right|}\right) + \left(\frac{\frac{\sqrt{\frac{1}{\pi}}}{{\left(\left|x\right|\right)}^3} \cdot \frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^3}}{\frac{\left|x\right|}{e^{\left|x\right| \cdot \left|x\right|}}}\right))_*\)
  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. Applied taylor to get
    \[(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{{\left(\frac{1}{\left|x\right|}\right)}^3}{2}\right))_*\right) + \left(\frac{\frac{15}{8} \cdot e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \frac{{\left(\frac{1}{\left|x\right|}\right)}^3 \cdot {\left(\frac{1}{\left|x\right|}\right)}^3}{\left|x\right|}\right))_* \leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{{\left(\frac{1}{\left|x\right|}\right)}^3}{2}\right))_*\right) + \left(\frac{15}{8} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{{\left({\left(\frac{1}{\left|x\right|}\right)}^3\right)}^2 \cdot e^{{\left(\left|x\right|\right)}^2}}{\left|x\right|}\right)\right))_*\]
    1.4
  4. Taylor expanded around 0 to get
    \[(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{{\left(\frac{1}{\left|x\right|}\right)}^3}{2}\right))_*\right) + \color{red}{\left(\frac{15}{8} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{{\left({\left(\frac{1}{\left|x\right|}\right)}^3\right)}^2 \cdot e^{{\left(\left|x\right|\right)}^2}}{\left|x\right|}\right)\right)})_* \leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{{\left(\frac{1}{\left|x\right|}\right)}^3}{2}\right))_*\right) + \color{blue}{\left(\frac{15}{8} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{{\left({\left(\frac{1}{\left|x\right|}\right)}^3\right)}^2 \cdot e^{{\left(\left|x\right|\right)}^2}}{\left|x\right|}\right)\right)})_*\]
    1.4
  5. Applied simplify to get
    \[\color{red}{(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left(\frac{1}{\left|x\right|} + (\left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{{\left(\frac{1}{\left|x\right|}\right)}^3}{2}\right))_*\right) + \left(\frac{15}{8} \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \frac{{\left({\left(\frac{1}{\left|x\right|}\right)}^3\right)}^2 \cdot e^{{\left(\left|x\right|\right)}^2}}{\left|x\right|}\right)\right))_*} \leadsto \color{blue}{(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left((\left(\frac{\frac{\frac{3}{\left|x\right|}}{{\left(\left|x\right|\right)}^3}}{4}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^3}\right))_* + \frac{1}{\left|x\right|}\right) + \left(\frac{\frac{\frac{15}{8} \cdot \sqrt{\frac{1}{\pi}}}{{\left({\left(\left|x\right|\right)}^3\right)}^2}}{\frac{\left|x\right|}{e^{\left|x\right| \cdot \left|x\right|}}}\right))_*}\]
    0.8
  6. Using strategy rm
    0.8
  7. Applied *-un-lft-identity to get
    \[(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left((\left(\frac{\frac{\frac{3}{\left|x\right|}}{{\left(\left|x\right|\right)}^3}}{4}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^3}\right))_* + \frac{1}{\left|x\right|}\right) + \left(\frac{\frac{\frac{15}{8} \cdot \sqrt{\frac{1}{\pi}}}{{\left({\left(\left|x\right|\right)}^3\right)}^2}}{\frac{\left|x\right|}{\color{red}{e^{\left|x\right| \cdot \left|x\right|}}}}\right))_* \leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left((\left(\frac{\frac{\frac{3}{\left|x\right|}}{{\left(\left|x\right|\right)}^3}}{4}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^3}\right))_* + \frac{1}{\left|x\right|}\right) + \left(\frac{\frac{\frac{15}{8} \cdot \sqrt{\frac{1}{\pi}}}{{\left({\left(\left|x\right|\right)}^3\right)}^2}}{\frac{\left|x\right|}{\color{blue}{1 \cdot e^{\left|x\right| \cdot \left|x\right|}}}}\right))_*\]
    0.8
  8. Applied *-un-lft-identity to get
    \[(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left((\left(\frac{\frac{\frac{3}{\left|x\right|}}{{\left(\left|x\right|\right)}^3}}{4}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^3}\right))_* + \frac{1}{\left|x\right|}\right) + \left(\frac{\frac{\frac{15}{8} \cdot \sqrt{\frac{1}{\pi}}}{{\left({\left(\left|x\right|\right)}^3\right)}^2}}{\frac{\color{red}{\left|x\right|}}{1 \cdot e^{\left|x\right| \cdot \left|x\right|}}}\right))_* \leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left((\left(\frac{\frac{\frac{3}{\left|x\right|}}{{\left(\left|x\right|\right)}^3}}{4}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^3}\right))_* + \frac{1}{\left|x\right|}\right) + \left(\frac{\frac{\frac{15}{8} \cdot \sqrt{\frac{1}{\pi}}}{{\left({\left(\left|x\right|\right)}^3\right)}^2}}{\frac{\color{blue}{1 \cdot \left|x\right|}}{1 \cdot e^{\left|x\right| \cdot \left|x\right|}}}\right))_*\]
    0.8
  9. Applied times-frac to get
    \[(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left((\left(\frac{\frac{\frac{3}{\left|x\right|}}{{\left(\left|x\right|\right)}^3}}{4}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^3}\right))_* + \frac{1}{\left|x\right|}\right) + \left(\frac{\frac{\frac{15}{8} \cdot \sqrt{\frac{1}{\pi}}}{{\left({\left(\left|x\right|\right)}^3\right)}^2}}{\color{red}{\frac{1 \cdot \left|x\right|}{1 \cdot e^{\left|x\right| \cdot \left|x\right|}}}}\right))_* \leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left((\left(\frac{\frac{\frac{3}{\left|x\right|}}{{\left(\left|x\right|\right)}^3}}{4}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^3}\right))_* + \frac{1}{\left|x\right|}\right) + \left(\frac{\frac{\frac{15}{8} \cdot \sqrt{\frac{1}{\pi}}}{{\left({\left(\left|x\right|\right)}^3\right)}^2}}{\color{blue}{\frac{1}{1} \cdot \frac{\left|x\right|}{e^{\left|x\right| \cdot \left|x\right|}}}}\right))_*\]
    0.8
  10. Applied associate-/r* to get
    \[(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left((\left(\frac{\frac{\frac{3}{\left|x\right|}}{{\left(\left|x\right|\right)}^3}}{4}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^3}\right))_* + \frac{1}{\left|x\right|}\right) + \color{red}{\left(\frac{\frac{\frac{15}{8} \cdot \sqrt{\frac{1}{\pi}}}{{\left({\left(\left|x\right|\right)}^3\right)}^2}}{\frac{1}{1} \cdot \frac{\left|x\right|}{e^{\left|x\right| \cdot \left|x\right|}}}\right)})_* \leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left((\left(\frac{\frac{\frac{3}{\left|x\right|}}{{\left(\left|x\right|\right)}^3}}{4}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^3}\right))_* + \frac{1}{\left|x\right|}\right) + \color{blue}{\left(\frac{\frac{\frac{\frac{15}{8} \cdot \sqrt{\frac{1}{\pi}}}{{\left({\left(\left|x\right|\right)}^3\right)}^2}}{\frac{1}{1}}}{\frac{\left|x\right|}{e^{\left|x\right| \cdot \left|x\right|}}}\right)})_*\]
    0.8
  11. Applied simplify to get
    \[(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left((\left(\frac{\frac{\frac{3}{\left|x\right|}}{{\left(\left|x\right|\right)}^3}}{4}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^3}\right))_* + \frac{1}{\left|x\right|}\right) + \left(\frac{\color{red}{\frac{\frac{\frac{15}{8} \cdot \sqrt{\frac{1}{\pi}}}{{\left({\left(\left|x\right|\right)}^3\right)}^2}}{\frac{1}{1}}}}{\frac{\left|x\right|}{e^{\left|x\right| \cdot \left|x\right|}}}\right))_* \leadsto (\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) * \left((\left(\frac{\frac{\frac{3}{\left|x\right|}}{{\left(\left|x\right|\right)}^3}}{4}\right) * \left(\frac{1}{\left|x\right|}\right) + \left(\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^3}\right))_* + \frac{1}{\left|x\right|}\right) + \left(\frac{\color{blue}{\frac{\sqrt{\frac{1}{\pi}}}{{\left(\left|x\right|\right)}^3} \cdot \frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^3}}}{\frac{\left|x\right|}{e^{\left|x\right| \cdot \left|x\right|}}}\right))_*\]
    0.8
  12. Removed slow pow expressions

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)))))))