Average Error: 1.5 → 0.6
Time: 4.6m
Precision: 64
Internal Precision: 384
\[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right)\]
↓
\[(\left(\frac{e^{\left|x\right| \cdot \left|x\right|}}{\sqrt{\pi}}\right) \cdot \left((\left(\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\frac{1}{2}\right) + \left(\left(\sqrt[3]{(\left(\frac{\frac{3}{\left|x\right|}}{4 \cdot \left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*} \cdot \sqrt[3]{(\left(\frac{\frac{3}{\left|x\right|}}{4 \cdot \left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*}\right) \cdot \sqrt[3]{(\left(\frac{\frac{3}{\left|x\right|}}{4 \cdot \left|x\right|}\right) \cdot \left(\frac{\frac{1}{\left|x\right|}}{\left|x\right| \cdot \left|x\right|}\right) + \left(\frac{1}{\left|x\right|}\right))_*}\right))_*\right) + \left(\left(\frac{\frac{15}{8}}{{\left(\left|x\right|\right)}^{7}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \sqrt{\frac{1}{\pi}}\right))_*\]

Error

Bits error versus x

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 simplify1.4

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

  4. Applied inv-pow1.4

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

  5. Applied pow-pow0.9

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

  6. Applied simplify0.9

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

  8. Applied inv-pow0.9

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

  9. Applied pow-pow0.6

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

  10. Applied simplify0.6

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

  11. Taylor expanded around 0 0.5

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

  12. Applied simplify0.6

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

  14. Applied add-cube-cbrt0.6

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

Runtime

Time bar (total: 4.6m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' +o rules:numerics
(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)))))))