Error

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Derivation

1. Initial program 14.1

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

3. Applied add-log-exp14.1

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

5. Applied flip--14.1

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

7. Applied flip3--14.1

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

8. Final simplification14.1

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

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed 2018216 +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)))))))