Average Error: 14.0 → 13.9
Time: 5.4m
Precision: 64
Internal Precision: 576
\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[\sqrt[3]{e^{\log \left(\log \left(e^{1 - \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \left(1.061405429 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911}\right)\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot -0.284496736 + 0.254829592\right)\right)\right) \cdot e^{\left|x\right| \cdot \left(-\left|x\right|\right)}}\right)\right)}} \cdot \left(\sqrt[3]{e^{\log \left(\log \left(e^{1 - \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \left(1.061405429 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911}\right)\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot -0.284496736 + 0.254829592\right)\right)\right) \cdot e^{\left|x\right| \cdot \left(-\left|x\right|\right)}}\right)\right)}} \cdot \sqrt[3]{e^{\log \left(\log \left(e^{1 - \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \left(1.061405429 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911} + -1.453152027\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911}\right)\right) \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911} + \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot -0.284496736 + 0.254829592\right)\right)\right) \cdot e^{\left|x\right| \cdot \left(-\left|x\right|\right)}}\right)\right)}}\right)\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.0

    \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
  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 distribute-lft-in13.9

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

  6. Applied associate-+r+13.9

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

  8. Applied add-exp-log13.9

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

  10. Applied add-cube-cbrt13.9

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

  11. Final simplification13.9

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

Runtime

Time bar (total: 5.4m)Debug logProfile

herbie shell --seed 2018242 +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)))))))