Average Error: 13.8 → 13.8
Time: 1.7m
Precision: 64
Internal Precision: 576
\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[e^{\log \left((\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}} \cdot \frac{\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((e^{\log_* (1 + (\left((\left(-1.453152027 + \frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) + 1.421413741)_*\right) \cdot \left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) + -0.284496736)_*)} - 1)^*\right) + \left((\left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) + 1)_*\right))_*\right)}\]

Error

Bits error versus x

Derivation

  1. Initial program 13.8

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

  2. Applied simplify13.8

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

  4. Applied add-cube-cbrt13.8

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

  5. Taylor expanded around 0 13.8

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

  6. Applied simplify13.8

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

  8. Applied expm1-log1p-u13.8

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

  10. Applied add-exp-log13.8

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed 2018201 +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)))))))