Error

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-sqr-sqrt13.8

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

5. Applied add-exp-log13.8

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

6. Applied simplify13.8

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

8. Applied add-cbrt-cube13.8

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

9. Applied simplify13.8

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

11. Applied add-cube-cbrt13.8

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

Runtime

Time bar (total: 3.9m)Debug logProfile

herbie shell --seed 2018199 +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)))))))