Error

Derivation

1. Initial program 13.7

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

3. Applied flip3--13.7

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

4. Applied simplify13.7

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

6. Applied add-sqr-sqrt13.0

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

7. Applied simplify13.0

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

8. Applied simplify13.0

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

10. Applied expm1-log1p-u13.0

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

Runtime

Time bar (total: 3.5m)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +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)))))))