Average Error: 13.9 → 13.9
Time: 12.1s
Precision: binary64
\[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[\sqrt[3]{{\left(\sqrt[3]{{\left(\sqrt[3]{{\left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\left(0.25482959199999999 + 1 \cdot \frac{\left(1.42141374100000006 \cdot \frac{1}{0.32759110000000002 \cdot \left|x\right| + 1} + 1.0614054289999999 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}}\right) - \left(0.284496735999999972 + 1.45315202700000001 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}}\right)}{0.32759110000000002 \cdot \left|x\right| + 1}\right) \cdot \frac{-1}{e^{{\left(\left|x\right|\right)}^{2}}}\right) + 1\right)}^{3}}\right)}^{3}}\right)}^{3}}\]

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. Taylor expanded around 0 13.9

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

  4. Applied add-cbrt-cube13.9

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

  5. Simplified13.9

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

  7. Applied add-cbrt-cube13.9

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

  8. Simplified13.9

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

  10. Applied add-cbrt-cube13.9

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

  11. Simplified13.9

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

  12. Final simplification13.9

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

Reproduce

herbie shell --seed 2020148 
(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 (- (* (fabs x) (fabs x)))))))