Average Error: 13.9 → 13.9
Time: 2.7m
Precision: 64
Internal Precision: 576
\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[\log \left(\sqrt{e^{(\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left((\left((\left(\frac{1.061405429}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*} + -1.453152027\right) \cdot \left(\frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}\right) + 1.421413741)_*\right) \cdot \left(\frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}\right) + -0.284496736)_*\right) \cdot \left(\frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*} \cdot \frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}\right) + \left(\frac{0.254829592}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}\right))_*\right) + 1)_*}}\right) + e^{\log \left(\log \left(\sqrt{e^{(\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left((\left((\left(\frac{1.061405429}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*} + -1.453152027\right) \cdot \left(\frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}\right) + 1.421413741)_*\right) \cdot \left(\frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}\right) + -0.284496736)_*\right) \cdot \left(\frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*} \cdot \frac{1}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}\right) + \left(\frac{0.254829592}{(\left(\left|x\right|\right) \cdot 0.3275911 + 1)_*}\right))_*\right) + 1)_*}}\right)\right)}\]

Error

Bits error versus x

Derivation

  1. Initial program 13.9

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

  2. Applied simplify13.9

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

  4. Applied add-log-exp13.9

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

  6. Applied add-sqr-sqrt13.9

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

  7. Applied log-prod13.9

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

  9. Applied add-exp-log13.9

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

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed '#(1071852389 864846987 1238109217 3425890003 4124793586 650694553)' +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)))))))