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-cube-cbrt13.8

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

4. Applied associate-/r*13.8

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

5. Simplified13.8

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

7. Applied log1p-expm1-u13.8

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

9. Applied add-cube-cbrt13.8

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

10. Taylor expanded around inf 13.0

    \[\leadsto \left(\sqrt[3]{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \log_* (1 + (e^{0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{\frac{1}{\sqrt[3]{1 + 0.3275911 \cdot \left|x\right|} \cdot \sqrt[3]{1 + 0.3275911 \cdot \left|x\right|}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot 1.061405429\right)\right)\right)} - 1)^*)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}} \cdot \sqrt[3]{1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \log_* (1 + (e^{0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{\frac{1}{\sqrt[3]{1 + 0.3275911 \cdot \left|x\right|} \cdot \sqrt[3]{1 + 0.3275911 \cdot \left|x\right|}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot 1.061405429\right)\right)\right)} - 1)^*)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}}\right) \cdot \color{blue}{{\left(\left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{4}} + \left(0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}} + 1\right)\right) - \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{5}} + \left(1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{3}} + 0.254829592 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right)}^{\frac{1}{3}}}\]

11. Final simplification13.0

    \[\leadsto \left(\sqrt[3]{1 - e^{\left(-\left|x\right|\right) \cdot \left|x\right|} \cdot \left(\log_* (1 + (e^{0.254829592 + \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{\frac{1}{\sqrt[3]{1 + \left|x\right| \cdot 0.3275911} \cdot \sqrt[3]{1 + \left|x\right| \cdot 0.3275911}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot 1.061405429 + -1.453152027\right)\right)\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911}} - 1)^*) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911}\right)} \cdot \sqrt[3]{1 - e^{\left(-\left|x\right|\right) \cdot \left|x\right|} \cdot \left(\log_* (1 + (e^{0.254829592 + \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{\frac{1}{\sqrt[3]{1 + \left|x\right| \cdot 0.3275911} \cdot \sqrt[3]{1 + \left|x\right| \cdot 0.3275911}}}{\sqrt[3]{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot 1.061405429 + -1.453152027\right)\right)\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911}} - 1)^*) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911}\right)}\right) \cdot {\left(\left(1.453152027 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{4}} + \left(\frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{2}} \cdot 0.284496736 + 1\right)\right) - \left(\frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{5}} \cdot 1.061405429 + \left(\frac{e^{-{\left(\left|x\right|\right)}^{2}}}{1 + \left|x\right| \cdot 0.3275911} \cdot 0.254829592 + \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{3}} \cdot 1.421413741\right)\right)\right)}^{\frac{1}{3}}\]

Reproduce

herbie shell --seed 2019004 +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)))))))

Details

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