Average Error: 13.8 → 13.8
Time: 4.6m
Precision: 64
Internal Precision: 576
\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[\frac{\left(\left(\left(\left(\frac{1.061405429}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{5}} + \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{1.453152027}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{4}} + \frac{0.284496736}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_* \cdot (0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right)\right) + \frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \frac{1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left(\left(\left(\left(\frac{1.061405429}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{5}} + \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{1.453152027}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{4}} + \frac{0.284496736}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_* \cdot (0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right)\right) + \frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \frac{1}{e^{\left|x\right| \cdot \left|x\right|}}\right) - 1}{\left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + \left(\left(\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)_*} + \frac{1.061405429}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{5}}\right) - \left(\frac{1.453152027}{{\left((0.3275911 \cdot \left(\left|x\right|\right) + 1)_*\right)}^{4}} + \frac{0.284496736}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_* \cdot (0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right)\right)\right) \cdot \frac{-1}{e^{\left|x\right| \cdot \left|x\right|}} - 1}\]

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. Taylor expanded around 0 14.5

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

  4. Applied simplify13.8

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

  6. Applied fma-udef13.8

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

  8. Applied flip-+13.8

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

  9. Applied simplify13.8

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

Runtime

Time bar (total: 4.6m)Debug logProfile

herbie shell --seed '#(1071373924 2949776965 1885069702 3247780810 90874544 2263903749)' +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)))))))