Jmat.Real.erf

Average Error: 13.9 → 13.2

Time: 10.0s

Precision: binary64

Math

\[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

↓

\[\left(\sqrt[3]{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt[3]{\log \left(e^{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\]

Error

Bits error versus x

Derivation

1. Initial program 13.9

   \[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
2. Using strategy rm

3. Applied distribute-lft-in 13.9

   \[\leadsto 1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \color{blue}{\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

4. Simplified 13.9

   \[\leadsto 1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + \color{blue}{1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
5. Using strategy rm

6. Applied add-cube-cbrt 13.9

   \[\leadsto \color{blue}{\left(\sqrt[3]{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt[3]{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right) \cdot \sqrt[3]{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}}\]
7. Using strategy rm

8. Applied add-log-exp 13.9

   \[\leadsto \left(\sqrt[3]{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt[3]{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right) \cdot \sqrt[3]{1 - \color{blue}{\log \left(e^{\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}}\]

9. Applied add-log-exp 13.9

   \[\leadsto \left(\sqrt[3]{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt[3]{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\color{blue}{\log \left(e^{1}\right)} - \log \left(e^{\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\]

10. Applied diff-log 14.3

    \[\leadsto \left(\sqrt[3]{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt[3]{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\color{blue}{\log \left(\frac{e^{1}}{e^{\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}}\right)}}\]

11. Simplified 13.2

    \[\leadsto \left(\sqrt[3]{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt[3]{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right) \cdot \sqrt[3]{\log \color{blue}{\left(e^{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}}\]
12. Using strategy rm

13. Applied add-log-exp 13.2

    \[\leadsto \left(\sqrt[3]{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt[3]{1 - \color{blue}{\log \left(e^{\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}}\right) \cdot \sqrt[3]{\log \left(e^{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\]

14. Applied add-log-exp 13.2

    \[\leadsto \left(\sqrt[3]{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt[3]{\color{blue}{\log \left(e^{1}\right)} - \log \left(e^{\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\]

15. Applied diff-log 13.9

    \[\leadsto \left(\sqrt[3]{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt[3]{\color{blue}{\log \left(\frac{e^{1}}{e^{\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}}\right)}}\right) \cdot \sqrt[3]{\log \left(e^{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\]

16. Simplified 13.2

    \[\leadsto \left(\sqrt[3]{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt[3]{\log \color{blue}{\left(e^{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}}\right) \cdot \sqrt[3]{\log \left(e^{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\]

17. Final simplification 13.2

    \[\leadsto \left(\sqrt[3]{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt[3]{\log \left(e^{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot -1.45315202700000001 + 1.0614054289999999 \cdot \frac{1}{\frac{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}{1}}\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right)}\]

Reproduce

herbie shell --seed 2020162 
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 1.0 (* (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (neg (* (fabs x) (fabs x)))))))