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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. Using strategy rm

3. Applied distribute-rgt-in13.9

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

4. Applied associate-+r+13.9

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

6. Applied add-exp-log13.9

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

8. Applied add-sqr-sqrt13.9

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

9. Applied log-prod13.9

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

10. Applied exp-sum13.9

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

12. Applied add-cube-cbrt13.9

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

13. Final simplification13.9

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018354 +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)))))))