Average Error: 13.6 → 2.5
Time: 3.4m
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]{e^{\log \left((\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(-1.453152027 + \frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + 1)_*\right)} \cdot \left((\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(-1.453152027 + \frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + 1)_* \cdot \left(\left(\left(e^{\left(-\left|x\right|\right) \cdot \left|x\right|} \cdot \left(\frac{1.453152027}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{4}} - \frac{\frac{1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_* \cdot (0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) - \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + \frac{1.061405429}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{5}}\right) \cdot e^{\left(-\left|x\right|\right) \cdot \left|x\right|}\right) + 1\right) + \frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_* \cdot (0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right)\right)}\]

Error

Bits error versus x

Derivation

  1. Initial program 13.6

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

  2. Initial simplification13.6

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

  4. Applied add-cbrt-cube13.6

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

  5. Taylor expanded around -inf 12.8

    \[\leadsto \sqrt[3]{\left(\color{blue}{\left(\left(1.453152027 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911 \cdot \left|x\right| + 1\right)}^{4}} + \left(0.284496736 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}} + 1\right)\right) - \left(1.421413741 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911 \cdot \left|x\right| + 1\right)}^{3}} + \left(1.061405429 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911 \cdot \left|x\right| + 1\right)}^{5}} + 0.254829592 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(0.3275911 \cdot \left|x\right| + 1\right)}\right)\right)\right)} \cdot (\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + 1)_*\right) \cdot (\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + 1)_*}\]

  6. Simplified12.8

    \[\leadsto \sqrt[3]{\left(\color{blue}{\left(\left(\frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_* \cdot (0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1\right) + \left(e^{\left(-\left|x\right|\right) \cdot \left|x\right|} \cdot \left(\frac{1.453152027}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{4}} - \frac{\frac{1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_* \cdot (0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) - e^{\left(-\left|x\right|\right) \cdot \left|x\right|} \cdot \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + \frac{1.061405429}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{5}}\right)\right)\right)} \cdot (\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + 1)_*\right) \cdot (\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + 1)_*}\]
  7. Using strategy rm

  8. Applied associate-+l+2.6

    \[\leadsto \sqrt[3]{\left(\color{blue}{\left(\frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_* \cdot (0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + \left(1 + \left(e^{\left(-\left|x\right|\right) \cdot \left|x\right|} \cdot \left(\frac{1.453152027}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{4}} - \frac{\frac{1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_* \cdot (0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) - e^{\left(-\left|x\right|\right) \cdot \left|x\right|} \cdot \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + \frac{1.061405429}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{5}}\right)\right)\right)\right)} \cdot (\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + 1)_*\right) \cdot (\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + 1)_*}\]
  9. Using strategy rm

  10. Applied add-exp-log2.5

    \[\leadsto \sqrt[3]{\left(\left(\frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_* \cdot (0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + \left(1 + \left(e^{\left(-\left|x\right|\right) \cdot \left|x\right|} \cdot \left(\frac{1.453152027}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{4}} - \frac{\frac{1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_* \cdot (0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) - e^{\left(-\left|x\right|\right) \cdot \left|x\right|} \cdot \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + \frac{1.061405429}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{5}}\right)\right)\right)\right) \cdot (\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + 1)_*\right) \cdot \color{blue}{e^{\log \left((\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + 1)_*\right)}}}\]

  11. Final simplification2.5

    \[\leadsto \sqrt[3]{e^{\log \left((\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(-1.453152027 + \frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + 1)_*\right)} \cdot \left((\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(-1.453152027 + \frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + 1)_* \cdot \left(\left(\left(e^{\left(-\left|x\right|\right) \cdot \left|x\right|} \cdot \left(\frac{1.453152027}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{4}} - \frac{\frac{1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_* \cdot (0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) - \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + \frac{1.061405429}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{5}}\right) \cdot e^{\left(-\left|x\right|\right) \cdot \left|x\right|}\right) + 1\right) + \frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_* \cdot (0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right)\right)}\]

Runtime

Time bar (total: 3.4m)Debug logProfile

BaselineHerbieOracleSpan%
Regimes2.52.52.50.00%
herbie shell --seed 2018353 +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)))))))