\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
Test:
Jmat.Real.erfi, branch x greater than or equal to 5
Bits:
128 bits
Bits error versus x
Time: 1.3 m
Input Error: 1.5
Output Error: 0.8
Log: ⚲
Profile: 🕒
\(\frac{\left(\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^3} + \frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^{\left(3 + 1\right)}}\right) + \frac{1 + \frac{\frac{\frac{3}{4}}{\left|x\right|}}{{\left(\left|x\right|\right)}^3}}{\left|x\right|}}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}}\)
  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
    \[\color{red}{\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)} \leadsto \color{blue}{\frac{\left(\left(\frac{{\left(\frac{1}{\left|x\right|}\right)}^3}{2} + \frac{1}{\left|x\right|}\right) + \left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \frac{{\left({\left(\frac{1}{\left|x\right|}\right)}^3\right)}^2}{\left|x\right|}}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}}}\]
    1.5
  3. Using strategy rm
    1.5
  4. Applied cube-div to get
    \[\frac{\left(\left(\frac{{\left(\frac{1}{\left|x\right|}\right)}^3}{2} + \frac{1}{\left|x\right|}\right) + \left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \frac{{\color{red}{\left({\left(\frac{1}{\left|x\right|}\right)}^3\right)}}^2}{\left|x\right|}}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}} \leadsto \frac{\left(\left(\frac{{\left(\frac{1}{\left|x\right|}\right)}^3}{2} + \frac{1}{\left|x\right|}\right) + \left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \frac{{\color{blue}{\left(\frac{{1}^3}{{\left(\left|x\right|\right)}^3}\right)}}^2}{\left|x\right|}}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}}\]
    1.0
  5. Applied taylor to get
    \[\frac{\left(\left(\frac{{\left(\frac{1}{\left|x\right|}\right)}^3}{2} + \frac{1}{\left|x\right|}\right) + \left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \frac{{\left(\frac{{1}^3}{{\left(\left|x\right|\right)}^3}\right)}^2}{\left|x\right|}}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}} \leadsto \frac{\left(\left(\frac{{\left(\frac{1}{\left|x\right|}\right)}^3}{2} + \frac{1}{\left|x\right|}\right) + \left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \frac{{\left(\frac{{1}^3}{{\left(\left|x\right|\right)}^3}\right)}^2}{\left|x\right|}}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}}\]
    1.0
  6. Taylor expanded around 0 to get
    \[\frac{\left(\left(\frac{{\left(\frac{1}{\left|x\right|}\right)}^3}{2} + \frac{1}{\left|x\right|}\right) + \left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \frac{{\color{red}{\left(\frac{{1}^3}{{\left(\left|x\right|\right)}^3}\right)}}^2}{\left|x\right|}}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}} \leadsto \frac{\left(\left(\frac{{\left(\frac{1}{\left|x\right|}\right)}^3}{2} + \frac{1}{\left|x\right|}\right) + \left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \frac{{\color{blue}{\left(\frac{{1}^3}{{\left(\left|x\right|\right)}^3}\right)}}^2}{\left|x\right|}}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}}\]
    1.0
  7. Applied simplify to get
    \[\color{red}{\frac{\left(\left(\frac{{\left(\frac{1}{\left|x\right|}\right)}^3}{2} + \frac{1}{\left|x\right|}\right) + \left({\left(\frac{1}{\left|x\right|}\right)}^3 \cdot \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right)\right) \cdot \frac{1}{\left|x\right|}\right) + \frac{15}{8} \cdot \frac{{\left(\frac{{1}^3}{{\left(\left|x\right|\right)}^3}\right)}^2}{\left|x\right|}}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}}} \leadsto \color{blue}{\frac{\frac{\frac{\frac{1}{\left|x\right|}}{{\left(\left|x\right|\right)}^3} \cdot \frac{3}{4} + 1}{\left|x\right|} + \left(\frac{\frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^3}}{\left|x\right|} + \frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right)}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}}}\]
    1.0
  8. Applied taylor to get
    \[\frac{\frac{\frac{\frac{1}{\left|x\right|}}{{\left(\left|x\right|\right)}^3} \cdot \frac{3}{4} + 1}{\left|x\right|} + \left(\frac{\frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^3}}{\left|x\right|} + \frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right)}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}} \leadsto \frac{\frac{\frac{\frac{1}{\left|x\right|}}{{\left(\left|x\right|\right)}^3} \cdot \frac{3}{4} + 1}{\left|x\right|} + \left(\frac{\frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^3}}{\left|x\right|} + \frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^{3}}\right)}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}}\]
    1.0
  9. Taylor expanded around 0 to get
    \[\frac{\frac{\frac{\frac{1}{\left|x\right|}}{{\left(\left|x\right|\right)}^3} \cdot \frac{3}{4} + 1}{\left|x\right|} + \left(\frac{\frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^3}}{\left|x\right|} + \color{red}{\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^{3}}}\right)}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}} \leadsto \frac{\frac{\frac{\frac{1}{\left|x\right|}}{{\left(\left|x\right|\right)}^3} \cdot \frac{3}{4} + 1}{\left|x\right|} + \left(\frac{\frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^3}}{\left|x\right|} + \color{blue}{\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^{3}}}\right)}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}}\]
    1.0
  10. Applied simplify to get
    \[\frac{\frac{\frac{\frac{1}{\left|x\right|}}{{\left(\left|x\right|\right)}^3} \cdot \frac{3}{4} + 1}{\left|x\right|} + \left(\frac{\frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^3}}{\left|x\right|} + \frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^{3}}\right)}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}} \leadsto \frac{\left(\frac{\frac{1}{2}}{{\left(\left|x\right|\right)}^3} + \frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^3 \cdot {\left(\left|x\right|\right)}^{\left(3 + 1\right)}}\right) + \frac{1 + \frac{\frac{\frac{3}{4}}{\left|x\right|}}{{\left(\left|x\right|\right)}^3}}{\left|x\right|}}{\frac{\sqrt{\pi}}{e^{\left|x\right| \cdot \left|x\right|}}}\]
    0.8
  11. Applied final simplification
  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)))))))