Average Error: 14.0 → 14.0
Time: 3.6m
Precision: 64
Internal Precision: 384
\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{{-0.284496736}^{3} + {\left(\frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)} + \frac{1.421413741}{1 + 0.3275911 \cdot \left|x\right|}\right)}^{3}}{-0.284496736 \cdot \left(-0.284496736 + \left(-0.284496736 \cdot 0.3275911\right) \cdot \left|x\right|\right) + \left(\left(1.421413741 + \frac{-1.453152027}{\left|x\right| \cdot 0.3275911 + 1}\right) + \frac{\frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}}{\left|x\right| \cdot 0.3275911 + 1}\right) \cdot \left(\frac{-1.453152027 + \frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)} - \left(-0.284496736 - \frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1}\right)\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

Error

Bits error versus x

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 flip3-+14.0

    \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\frac{{-0.284496736}^{3} + {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)}^{3}}{-0.284496736 \cdot -0.284496736 + \left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right) \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right) - -0.284496736 \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)}}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
  4. Applied frac-times14.0

    \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \color{blue}{\frac{1 \cdot \left({-0.284496736}^{3} + {\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)}^{3}\right)}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(-0.284496736 \cdot -0.284496736 + \left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right) \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right) - -0.284496736 \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right)}}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
  5. Applied simplify14.0

    \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{\color{blue}{{-0.284496736}^{3} + {\left(\frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)} + \frac{1.421413741}{1 + 0.3275911 \cdot \left|x\right|}\right)}^{3}}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(-0.284496736 \cdot -0.284496736 + \left(\left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right) \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right) - -0.284496736 \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
  6. Applied simplify14.0

    \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{{-0.284496736}^{3} + {\left(\frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)} + \frac{1.421413741}{1 + 0.3275911 \cdot \left|x\right|}\right)}^{3}}{\color{blue}{\left(-0.284496736 \cdot -0.284496736 + \left|x\right| \cdot \left(\left(-0.284496736 \cdot -0.284496736\right) \cdot 0.3275911\right)\right) + \left(\frac{\frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1} + -1.453152027}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)} - \left(-0.284496736 - \frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1}\right)\right) \cdot \left(\left(\frac{-1.453152027}{\left|x\right| \cdot 0.3275911 + 1} + 1.421413741\right) + \frac{1}{\left|x\right| \cdot 0.3275911 + 1} \cdot \frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}\right)}}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
  7. Applied simplify14.0

    \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{{-0.284496736}^{3} + {\left(\frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot \left(1 + 0.3275911 \cdot \left|x\right|\right)} + \frac{1.421413741}{1 + 0.3275911 \cdot \left|x\right|}\right)}^{3}}{\color{blue}{-0.284496736 \cdot \left(-0.284496736 + \left(-0.284496736 \cdot 0.3275911\right) \cdot \left|x\right|\right)} + \left(\frac{\frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1} + -1.453152027}{\left(\left|x\right| \cdot 0.3275911 + 1\right) \cdot \left(\left|x\right| \cdot 0.3275911 + 1\right)} - \left(-0.284496736 - \frac{1.421413741}{\left|x\right| \cdot 0.3275911 + 1}\right)\right) \cdot \left(\left(\frac{-1.453152027}{\left|x\right| \cdot 0.3275911 + 1} + 1.421413741\right) + \frac{1}{\left|x\right| \cdot 0.3275911 + 1} \cdot \frac{1.061405429}{\left|x\right| \cdot 0.3275911 + 1}\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
  8. Applied simplify14.0

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

Runtime

Time bar (total: 3.6m)Debug logProfile

herbie shell --seed '#(1063027428 1192549564 1443466578 604016274 3637110559 1698629644)' 
(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)))))))