Error

Derivation

1. Initial program 13.9

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

2. Applied simplify13.9

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

4. Applied add-log-exp13.9

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

5. Applied simplify13.8

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

7. Applied fma-udef13.9

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

8. Applied exp-sum14.6

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

9. Applied log-prod14.6

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

10. Applied simplify13.9

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

11. Applied simplify13.9

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

13. Applied flip3-+13.8

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

14. Applied simplify13.8

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

16. Applied add-log-exp13.1

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

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +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)))))))