Jmat.Real.erf

Average Error: 14.0 → 13.2

Time: 9.8s

Precision: binary64

Math

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

↓

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

Error

Bits error versus x

Derivation

1. Initial program 14.0

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

2. Taylor expanded around 0 14.0

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

4. Applied add-exp-log 14.0

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

6. Applied add-cube-cbrt 14.0

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

7. Applied exp-prod 14.0

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

9. Applied add-log-exp 14.0

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

10. Applied add-log-exp 14.0

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

11. Applied diff-log 14.3

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

12. Simplified 13.2

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

13. Final simplification 13.2

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

Reproduce

herbie shell --seed 2020152 
(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)))))))