Average Error: 1.5 → 0.7
Time: 2.1m
Precision: 64
Internal Precision: 576
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}} \cdot \left(\frac{\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}}{2} + \left(\frac{1}{\left|x\right|} \cdot \frac{3}{4}\right) \cdot {\left(\frac{1}{\left|x\right|}\right)}^{\left(3 + 1\right)}\right) + \left(\left(e^{\left|x\right| \cdot \left|x\right|} \cdot \frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) \cdot \left({\left(\left|x\right|\right)}^{\left(-1 - 3\right)} \cdot \frac{\frac{15}{8}}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}\right) + \frac{\frac{1}{\sqrt{\pi}}}{\left|x\right|} \cdot e^{\left|x\right| \cdot \left|x\right|}\right)\]

Error

Bits error versus x

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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. Applied simplify1.5

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

  4. Applied add-cube-cbrt1.5

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

  5. Taylor expanded around 0 1.5

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

  6. Applied simplify1.3

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

  8. Applied inv-pow1.3

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

  9. Applied pow-pow1.0

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

  10. Applied simplify1.0

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

  12. Applied add-sqr-sqrt0.7

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2020178 
(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)))))))