Average Error: 13.6 → 2.1
Time: 35.1s
Precision: 64
Internal Precision: 576
\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[\sqrt{(\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{\frac{\frac{0.254829592}{\sqrt{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{\sqrt{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{\sqrt{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}}{\sqrt[3]{\sqrt{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}\right))_*\right) + 1)_*} \cdot \sqrt{\left(1 + \frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_* \cdot (0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) + \left(e^{\left|x\right| \cdot \left(-\left|x\right|\right)} \cdot \left(\frac{1.453152027}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{4}} - \frac{\frac{1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_* \cdot (0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) - \left(\frac{1.061405429}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{5}} + \frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot e^{\left|x\right| \cdot \left(-\left|x\right|\right)}\right)}\]

Error

Bits error versus x

Derivation

  1. Initial program 13.6

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

  2. Initial simplification13.6

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

  4. Applied add-sqr-sqrt13.6

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

  5. Applied associate-/r*13.6

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

  7. Applied add-cube-cbrt13.6

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

  8. Applied associate-/r*13.6

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

  10. Applied add-sqr-sqrt13.6

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

  11. Taylor expanded around -inf 2.1

    \[\leadsto \sqrt{\color{blue}{\left(1.453152027 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911 \cdot \left|x\right| + 1\right)}^{4}} + \left(0.284496736 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}} + 1\right)\right) - \left(1.421413741 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911 \cdot \left|x\right| + 1\right)}^{3}} + \left(1.061405429 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot {\left(0.3275911 \cdot \left|x\right| + 1\right)}^{5}} + 0.254829592 \cdot \frac{1}{e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(0.3275911 \cdot \left|x\right| + 1\right)}\right)\right)}} \cdot \sqrt{(\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{\frac{\frac{0.254829592}{\sqrt{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{\sqrt{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{\sqrt{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}}{\sqrt[3]{\sqrt{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}\right))_*\right) + 1)_*}\]

  12. Simplified2.1

    \[\leadsto \sqrt{\color{blue}{\left(\frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_* \cdot (0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + 1\right) + \left(e^{\left(-\left|x\right|\right) \cdot \left|x\right|} \cdot \left(\frac{1.453152027}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{4}} - \frac{\frac{1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_* \cdot (0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) - e^{\left(-\left|x\right|\right) \cdot \left|x\right|} \cdot \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + \frac{1.061405429}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{5}}\right)\right)}} \cdot \sqrt{(\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{\frac{\frac{0.254829592}{\sqrt{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{\sqrt{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{\sqrt{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}}{\sqrt[3]{\sqrt{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}\right))_*\right) + 1)_*}\]

  13. Final simplification2.1

    \[\leadsto \sqrt{(\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{\frac{\frac{0.254829592}{\sqrt{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}{\sqrt[3]{\sqrt{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}} \cdot \sqrt[3]{\sqrt{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}}{\sqrt[3]{\sqrt{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}}\right))_*\right) + 1)_*} \cdot \sqrt{\left(1 + \frac{\frac{0.284496736}{e^{\left|x\right| \cdot \left|x\right|}}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_* \cdot (0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) + \left(e^{\left|x\right| \cdot \left(-\left|x\right|\right)} \cdot \left(\frac{1.453152027}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{4}} - \frac{\frac{1.421413741}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_* \cdot (0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) - \left(\frac{1.061405429}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{5}} + \frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot e^{\left|x\right| \cdot \left(-\left|x\right|\right)}\right)}\]

Runtime

Time bar (total: 35.1s)Debug logProfile

herbie shell --seed 2018254 +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)))))))