Average Error: 1.5 → 0.6
Time: 3.1m
Precision: 64
Internal Precision: 576
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\left(\sqrt{\frac{1}{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{\frac{3}{4}}{{\left(\left|x\right|\right)}^{5}} + \frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^{7}}\right) + \left(\frac{1}{\left|x\right|} + \frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right)\right)\]

Error

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Your Program's Arguments

Results

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Derivation

  1. Initial program 1.5

    \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
  2. Using strategy rm

  3. Applied frac-times1.3

    \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\color{blue}{\frac{1 \cdot 1}{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]

  4. Simplified1.3

    \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\frac{\color{blue}{1}}{\left|x\right| \cdot \left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]

  5. Simplified1.0

    \[\leadsto \color{blue}{\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\frac{\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{\frac{15}{8}}{\left|x\right| \cdot \left|x\right|} + \frac{3}{4}\right) + \left(\frac{1}{\left|x\right|} + \frac{\frac{\frac{1}{2}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right|}\right)\right)}\]
  6. Using strategy rm

  7. Applied pow21.0

    \[\leadsto \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\frac{\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}}{\color{blue}{{\left(\left|x\right|\right)}^{2}}} \cdot \left(\frac{\frac{15}{8}}{\left|x\right| \cdot \left|x\right|} + \frac{3}{4}\right) + \left(\frac{1}{\left|x\right|} + \frac{\frac{\frac{1}{2}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right|}\right)\right)\]

  8. Applied pow11.0

    \[\leadsto \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\frac{\frac{\frac{1}{\left|x\right|}}{\color{blue}{{\left(\left|x\right|\right)}^{1}} \cdot \left|x\right|}}{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{\frac{15}{8}}{\left|x\right| \cdot \left|x\right|} + \frac{3}{4}\right) + \left(\frac{1}{\left|x\right|} + \frac{\frac{\frac{1}{2}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right|}\right)\right)\]

  9. Applied pow-plus1.0

    \[\leadsto \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\frac{\frac{\frac{1}{\left|x\right|}}{\color{blue}{{\left(\left|x\right|\right)}^{\left(1 + 1\right)}}}}{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{\frac{15}{8}}{\left|x\right| \cdot \left|x\right|} + \frac{3}{4}\right) + \left(\frac{1}{\left|x\right|} + \frac{\frac{\frac{1}{2}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right|}\right)\right)\]

  10. Applied pow11.0

    \[\leadsto \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\frac{\frac{\frac{1}{\color{blue}{{\left(\left|x\right|\right)}^{1}}}}{{\left(\left|x\right|\right)}^{\left(1 + 1\right)}}}{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{\frac{15}{8}}{\left|x\right| \cdot \left|x\right|} + \frac{3}{4}\right) + \left(\frac{1}{\left|x\right|} + \frac{\frac{\frac{1}{2}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right|}\right)\right)\]

  11. Applied pow-flip1.0

    \[\leadsto \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\frac{\frac{\color{blue}{{\left(\left|x\right|\right)}^{\left(-1\right)}}}{{\left(\left|x\right|\right)}^{\left(1 + 1\right)}}}{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{\frac{15}{8}}{\left|x\right| \cdot \left|x\right|} + \frac{3}{4}\right) + \left(\frac{1}{\left|x\right|} + \frac{\frac{\frac{1}{2}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right|}\right)\right)\]

  12. Applied pow-div0.8

    \[\leadsto \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\frac{\color{blue}{{\left(\left|x\right|\right)}^{\left(\left(-1\right) - \left(1 + 1\right)\right)}}}{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{\frac{15}{8}}{\left|x\right| \cdot \left|x\right|} + \frac{3}{4}\right) + \left(\frac{1}{\left|x\right|} + \frac{\frac{\frac{1}{2}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right|}\right)\right)\]

  13. Applied pow-div0.7

    \[\leadsto \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\color{blue}{{\left(\left|x\right|\right)}^{\left(\left(\left(-1\right) - \left(1 + 1\right)\right) - 2\right)}} \cdot \left(\frac{\frac{15}{8}}{\left|x\right| \cdot \left|x\right|} + \frac{3}{4}\right) + \left(\frac{1}{\left|x\right|} + \frac{\frac{\frac{1}{2}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right|}\right)\right)\]

  14. Simplified0.7

    \[\leadsto \frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left({\left(\left|x\right|\right)}^{\color{blue}{-5}} \cdot \left(\frac{\frac{15}{8}}{\left|x\right| \cdot \left|x\right|} + \frac{3}{4}\right) + \left(\frac{1}{\left|x\right|} + \frac{\frac{\frac{1}{2}}{\left|x\right| \cdot \left|x\right|}}{\left|x\right|}\right)\right)\]

  15. Taylor expanded around 0 0.6

    \[\leadsto \color{blue}{\left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{1}{\left|x\right|} + \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right)\right)\right) \cdot \sqrt{\frac{1}{\pi}}}\]

  16. Simplified0.6

    \[\leadsto \color{blue}{\left(\left(\frac{1}{\left|x\right|} + \frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) + \left(\frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^{7}} + \frac{\frac{3}{4}}{{\left(\left|x\right|\right)}^{5}}\right)\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right)}\]

  17. Final simplification0.6

    \[\leadsto \left(\sqrt{\frac{1}{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\frac{\frac{3}{4}}{{\left(\left|x\right|\right)}^{5}} + \frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^{7}}\right) + \left(\frac{1}{\left|x\right|} + \frac{\frac{\frac{1}{2}}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right)\right)\]

Runtime

Time bar (total: 3.1m)Debug logProfile

herbie shell --seed 2018248 
(FPCore (x)
  :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)))))))