Average Error: 13.8 → 1.9
Time: 3.3m
Precision: 64
Internal Precision: 128
\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[\sqrt{(\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + 1)_*} \cdot e^{\log \left(\sqrt{\left(\frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{4}} \cdot 1.453152027 + \left(0.284496736 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}} + 1\right)\right) - \left(1.061405429 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{5}} + \left(\frac{e^{-{\left(\left|x\right|\right)}^{2}}}{0.3275911 \cdot \left|x\right| + 1} \cdot 0.254829592 + 1.421413741 \cdot \frac{e^{-{\left(\left|x\right|\right)}^{2}}}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{3}}\right)\right)}\right)}\]

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. Using strategy rm

  3. Applied add-log-exp13.7

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

  5. Applied add-exp-log13.7

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

  7. Applied add-sqr-sqrt13.7

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

  8. Applied log-prod13.7

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

  9. Applied exp-sum13.7

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

  10. Simplified10.1

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

  11. Taylor expanded around -inf 1.9

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

  12. Final simplification1.9

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

Runtime

Time bar (total: 3.3m)Debug logProfile

herbie shell --seed 2018349 +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)))))))