Jmat.Real.erf

Average Error: 14.0 → 14.0

Time: 34.9s

Precision: binary64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\frac{\frac{{1}^{6} - {\left(\frac{\frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}\right)}^{3}}{\frac{\frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} \cdot \left(1 \cdot 1 + \frac{\frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}\right) + {1}^{4}}}{1 + \frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{\left(1 + 0.32759110000000002 \cdot \left|x\right|\right) \cdot e^{{\left(\left|x\right|\right)}^{2}}}}\]

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. Simplified 14.0

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

4. Applied flip-+ 14.0

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

5. Applied associate-/r/ 14.0

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

6. Simplified 14.0

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

8. Applied flip-- 14.0

   \[\leadsto \color{blue}{\frac{1 \cdot 1 - \frac{\frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}}}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \frac{\frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}}}{1 + 0.32759110000000002 \cdot \left|x\right|}}{1 + \frac{\frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{\left|x\right| \cdot \left|x\right|}}}{1 + 0.32759110000000002 \cdot \left|x\right|}}}\]

9. Simplified 14.0

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

10. Simplified 14.0

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

12. Applied flip3-- 14.0

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

13. Simplified 14.0

    \[\leadsto \frac{\frac{\color{blue}{{1}^{6} - {\left(\frac{\frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}\right)}^{3}}}{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) + \left(\frac{\frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} \cdot \frac{\frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + \left(1 \cdot 1\right) \cdot \frac{\frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}\right)}}{1 + \frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{\left(1 + 0.32759110000000002 \cdot \left|x\right|\right) \cdot e^{{\left(\left|x\right|\right)}^{2}}}}\]

14. Simplified 14.0

    \[\leadsto \frac{\frac{{1}^{6} - {\left(\frac{\frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}\right)}^{3}}{\color{blue}{\frac{\frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} \cdot \left(1 \cdot 1 + \frac{\frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}\right) + {1}^{4}}}}{1 + \frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{\left(1 + 0.32759110000000002 \cdot \left|x\right|\right) \cdot e^{{\left(\left|x\right|\right)}^{2}}}}\]

15. Final simplification 14.0

    \[\leadsto \frac{\frac{{1}^{6} - {\left(\frac{\frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}\right)}^{3}}{\frac{\frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} \cdot \left(1 \cdot 1 + \frac{\frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}} \cdot \frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{e^{{\left(\left|x\right|\right)}^{2}}}}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}\right) + {1}^{4}}}{1 + \frac{1 \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 + \left(\frac{1}{1 \cdot 1 - \left(0.32759110000000002 \cdot 0.32759110000000002\right) \cdot {\left(\left|x\right|\right)}^{2}} \cdot \left(1 - 0.32759110000000002 \cdot \left|x\right|\right)\right) \cdot 1.0614054289999999\right)\right)\right)\right)}{\left(1 + 0.32759110000000002 \cdot \left|x\right|\right) \cdot e^{{\left(\left|x\right|\right)}^{2}}}}\]

Reproduce

herbie shell --seed 2020182 
(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)))))))