Error

Try it out

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 add-log-exp13.9

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

5. Applied add-exp-log13.9

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

7. Applied distribute-rgt-in13.9

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

8. Applied associate-+r+13.9

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

9. Simplified13.9

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

10. Final simplification13.9

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

Reproduce

herbie shell --seed 2019002 
(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

Time bar (total: 47.9s)Debug log