Average Error: 13.7 → 13.7
Time: 3.8m
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(\sqrt[3]{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{{0.254829592}^{3} + {\left(\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)}^{3}}{(\left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(-1.453152027 + \frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) + 1.421413741)_*\right) + -0.284496736)_*\right) \cdot \left((\left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(-1.453152027 + \frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) + 1.421413741)_*\right) + -0.284496736)_*\right) \cdot \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) + \left(\frac{-0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + \left(0.254829592 \cdot 0.254829592\right))_*}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt[3]{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{{0.254829592}^{3} + {\left(\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)}^{3}}{(\left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(-1.453152027 + \frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) + 1.421413741)_*\right) + -0.284496736)_*\right) \cdot \left((\left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(-1.453152027 + \frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) + 1.421413741)_*\right) + -0.284496736)_*\right) \cdot \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) + \left(\frac{-0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + \left(0.254829592 \cdot 0.254829592\right))_*}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right) \cdot \sqrt[3]{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \frac{{0.254829592}^{3} + {\left(\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)}^{3}}{(\left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(-1.453152027 + \frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) + 1.421413741)_*\right) + -0.284496736)_*\right) \cdot \left((\left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(-1.453152027 + \frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) + 1.421413741)_*\right) + -0.284496736)_*\right) \cdot \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) + \left(\frac{-0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + \left(0.254829592 \cdot 0.254829592\right))_*}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\]

Error

Bits error versus x

Derivation

  1. Initial program 13.7

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

  3. Applied flip3-+14.8

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

  4. Applied simplify13.7

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

  6. Applied add-cube-cbrt13.7

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

Runtime

Time bar (total: 3.8m)Debug logProfile

herbie shell --seed 2018178 +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)))))))