Average Error: 13.8 → 13.0
Time: 3.3m
Precision: 64
Internal Precision: 576
\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[{\left(e^{\sqrt[3]{\log_* (1 + \left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}\right) \cdot (\left(\sqrt[3]{\frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}} \cdot \sqrt[3]{\frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}}\right) \cdot \left((\left((\left(-1.453152027 + \frac{1.061405429}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}\right) \cdot \left(\frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}\right) + 1.421413741)_*\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}}}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}\right) + \left(-0.284496736 \cdot \sqrt[3]{\frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}}\right))_*\right) + 0.254829592)_*)} \cdot \sqrt[3]{\log \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(1 + 0.284496736 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}\right)\right) - \left(1.061405429 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911 \cdot \left|x\right| + 1\right)}^{5}} + \left(1.421413741 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911 \cdot \left|x\right| + 1\right)}^{3}} + 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)}}\right)}^{\left(\sqrt[3]{\log_* (1 + \left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}\right) \cdot (\left(\sqrt[3]{\frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}} \cdot \sqrt[3]{\frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}}\right) \cdot \left((\left((\left(-1.453152027 + \frac{1.061405429}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}\right) \cdot \left(\frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}\right) + 1.421413741)_*\right) \cdot \left(\frac{\sqrt[3]{\frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}}}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}\right) + \left(-0.284496736 \cdot \sqrt[3]{\frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}}\right))_*\right) + 0.254829592)_*)}\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. Using strategy rm

  3. Applied add-cube-cbrt13.8

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

  4. Applied associate-*l*13.8

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

  5. Applied simplify13.8

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

  7. Applied add-exp-log13.8

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

  8. Applied simplify13.8

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

  10. Applied add-cube-cbrt13.8

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

  11. Applied exp-prod13.8

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

  12. Taylor expanded around 0 13.0

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

Runtime

Time bar (total: 3.3m)Debug logProfile

herbie shell --seed '#(1072107073 2127697367 3936270018 2300570620 2134894798 4023771849)' +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)))))))