Average Error: 14.0 → 1.7
Time: 46.2s
Precision: 64
Internal Precision: 128
\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[\sqrt{e^{\log \left((\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_* \cdot (\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}\right) \cdot \left((\left(\frac{1.061405429}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*} + -1.453152027\right) \cdot \left(\frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_* \cdot (\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}\right) + \left(\frac{1.421413741}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*} + -0.284496736\right))_*\right) + \left((\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left(\frac{0.254829592}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}\right) + 1)_*\right))_*\right)}} \cdot e^{\log \left(\sqrt{(\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\sqrt[3]{\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \left(\sqrt[3]{\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}\right)\right))_*\right) + 1)_*}\right)}\]

Error

Bits error versus x

Derivation

  1. Initial program 14.0

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

  2. Initial simplification14.0

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

  4. Applied add-cube-cbrt14.0

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

  6. Applied add-exp-log14.0

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

  8. Applied add-sqr-sqrt14.0

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

  9. Applied log-prod14.0

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

  10. Applied exp-sum14.0

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

  11. Simplified2.0

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

  13. Applied add-exp-log1.7

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

  14. Final simplification1.7

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

Runtime

Time bar (total: 46.2s)Debug logProfile

herbie shell --seed 2018278 +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)))))))