Average Error: 14.2 → 14.2
Time: 2.1m
Precision: 64
Internal Precision: 128
\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[\frac{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(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left((e^{\log_* (1 + \frac{1}{1 - \left(0.3275911 \cdot \left|x\right|\right) \cdot \left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right))} - 1)^* \cdot 1.061405429 + -1.453152027\right) + 1.421413741\right) + -0.284496736\right)\right)\right) \cdot e^{\left(-\left|x\right|\right) \cdot \left|x\right|}\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(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left((e^{\log_* (1 + \frac{1}{1 - \left(0.3275911 \cdot \left|x\right|\right) \cdot \left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right))} - 1)^* \cdot 1.061405429 + -1.453152027\right) + 1.421413741\right) + -0.284496736\right)\right)\right) \cdot e^{\left(-\left|x\right|\right) \cdot \left|x\right|}\right) \cdot \left(\left(\left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(-1.453152027 + (e^{\log_* (1 + \sqrt{\frac{1}{1 - \left(0.3275911 \cdot \left|x\right|\right) \cdot \left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right)} \cdot \sqrt{\frac{1}{1 - \left(0.3275911 \cdot \left|x\right|\right) \cdot \left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right)})} - 1)^* \cdot 1.061405429\right) \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 1.421413741\right) + -0.284496736\right) \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} + 0.254829592\right) \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\right) \cdot e^{\left(-\left|x\right|\right) \cdot \left|x\right|}\right) + \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(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left((e^{\log_* (1 + \frac{1}{1 - \left(0.3275911 \cdot \left|x\right|\right) \cdot \left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right))} - 1)^* \cdot 1.061405429 + -1.453152027\right) + 1.421413741\right) + -0.284496736\right)\right)\right) \cdot e^{\left(-\left|x\right|\right) \cdot \left|x\right|}\right) + 1}\]

Error

Bits error versus x

Derivation

  1. Initial program 14.2

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

  3. Applied flip-+14.2

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

  4. Applied associate-/r/14.2

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

  6. Applied expm1-log1p-u14.2

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

  8. Applied flip3--14.2

    \[\leadsto \color{blue}{\frac{{1}^{3} - {\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 + (e^{\log_* (1 + \frac{1}{1 \cdot 1 - \left(0.3275911 \cdot \left|x\right|\right) \cdot \left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right))} - 1)^* \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}^{3}}{1 \cdot 1 + \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 + (e^{\log_* (1 + \frac{1}{1 \cdot 1 - \left(0.3275911 \cdot \left|x\right|\right) \cdot \left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right))} - 1)^* \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 + (e^{\log_* (1 + \frac{1}{1 \cdot 1 - \left(0.3275911 \cdot \left|x\right|\right) \cdot \left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right))} - 1)^* \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right) + 1 \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 + (e^{\log_* (1 + \frac{1}{1 \cdot 1 - \left(0.3275911 \cdot \left|x\right|\right) \cdot \left(0.3275911 \cdot \left|x\right|\right)} \cdot \left(1 - 0.3275911 \cdot \left|x\right|\right))} - 1)^* \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-sqr-sqrt14.2

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

  11. Final simplification14.2

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

Reproduce

herbie shell --seed 2019094 +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)))))))