Average Error: 13.8 → 13.1
Time: 1.0m
Precision: 64
Internal Precision: 128
\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[\sqrt[3]{1 - \log \left(e^{\frac{\frac{(\left(\frac{\sqrt[3]{\frac{1.421413741 + \frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -0.284496736} \cdot \sqrt[3]{\frac{1.421413741 + \frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -0.284496736}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1.421413741 + \frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -0.284496736}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) + 0.254829592)_*}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)} \cdot \left(\sqrt[3]{\log \left(e^{1 - \frac{\frac{(\left(\frac{\sqrt[3]{\frac{1.421413741 + \frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -0.284496736} \cdot \sqrt[3]{\frac{1.421413741 + \frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -0.284496736}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1.421413741 + \frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -0.284496736}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) + 0.254829592)_*}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{(\left(\frac{\sqrt[3]{\frac{1.421413741 + \frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -0.284496736} \cdot \sqrt[3]{\frac{1.421413741 + \frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -0.284496736}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1.421413741 + \frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -0.284496736}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) + 0.254829592)_*}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}\right)\]

Error

Bits error versus x

Derivation

  1. Initial program 13.8

    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

  2. Simplified13.8

    \[\leadsto \color{blue}{1 - \frac{\frac{\frac{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\]
  3. Using strategy rm

  4. Applied add-log-exp13.8

    \[\leadsto \color{blue}{\log \left(e^{1 - \frac{\frac{\frac{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt13.8

    \[\leadsto \log \left(e^{1 - \frac{\frac{\frac{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{\color{blue}{\left(\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}} + 0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)\]

  7. Applied add-cube-cbrt13.8

    \[\leadsto \log \left(e^{1 - \frac{\frac{\frac{\color{blue}{\left(\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}}{\left(\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} + 0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)\]

  8. Applied times-frac13.8

    \[\leadsto \log \left(e^{1 - \frac{\frac{\color{blue}{\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}} + 0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)\]

  9. Applied fma-def13.8

    \[\leadsto \log \left(e^{1 - \frac{\frac{\color{blue}{(\left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) + 0.254829592)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)\]
  10. Using strategy rm

  11. Applied add-cube-cbrt13.8

    \[\leadsto \color{blue}{\left(\sqrt[3]{\log \left(e^{1 - \frac{\frac{(\left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) + 0.254829592)_*}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{(\left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) + 0.254829592)_*}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}\right) \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{(\left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) + 0.254829592)_*}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}}\]
  12. Using strategy rm

  13. Applied exp-diff13.1

    \[\leadsto \left(\sqrt[3]{\log \left(e^{1 - \frac{\frac{(\left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) + 0.254829592)_*}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{(\left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) + 0.254829592)_*}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}\right) \cdot \sqrt[3]{\log \color{blue}{\left(\frac{e^{1}}{e^{\frac{\frac{(\left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) + 0.254829592)_*}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}}\right)}}\]

  14. Applied log-div13.1

    \[\leadsto \left(\sqrt[3]{\log \left(e^{1 - \frac{\frac{(\left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) + 0.254829592)_*}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{(\left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) + 0.254829592)_*}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}\right) \cdot \sqrt[3]{\color{blue}{\log \left(e^{1}\right) - \log \left(e^{\frac{\frac{(\left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) + 0.254829592)_*}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}}\]

  15. Simplified13.1

    \[\leadsto \left(\sqrt[3]{\log \left(e^{1 - \frac{\frac{(\left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) + 0.254829592)_*}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{(\left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) + 0.254829592)_*}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}\right) \cdot \sqrt[3]{\color{blue}{1} - \log \left(e^{\frac{\frac{(\left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \left(\frac{\sqrt[3]{-0.284496736 + \frac{\frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) + 0.254829592)_*}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}\]

  16. Final simplification13.1

    \[\leadsto \sqrt[3]{1 - \log \left(e^{\frac{\frac{(\left(\frac{\sqrt[3]{\frac{1.421413741 + \frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -0.284496736} \cdot \sqrt[3]{\frac{1.421413741 + \frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -0.284496736}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1.421413741 + \frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -0.284496736}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) + 0.254829592)_*}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)} \cdot \left(\sqrt[3]{\log \left(e^{1 - \frac{\frac{(\left(\frac{\sqrt[3]{\frac{1.421413741 + \frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -0.284496736} \cdot \sqrt[3]{\frac{1.421413741 + \frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -0.284496736}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1.421413741 + \frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -0.284496736}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) + 0.254829592)_*}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)} \cdot \sqrt[3]{\log \left(e^{1 - \frac{\frac{(\left(\frac{\sqrt[3]{\frac{1.421413741 + \frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -0.284496736} \cdot \sqrt[3]{\frac{1.421413741 + \frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -0.284496736}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) \cdot \left(\frac{\sqrt[3]{\frac{1.421413741 + \frac{\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -0.284496736}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right) + 0.254829592)_*}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{e^{\left|x\right| \cdot \left|x\right|}}}\right)}\right)\]

Reproduce

herbie shell --seed 2019053 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erf"
  (- 1 (* (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))