Average Error: 13.9 → 13.9
Time: 40.0s
Precision: binary64
\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[1 - \frac{\left(\left(0.254829592 + \frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right) + \frac{\frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}}{{\left(\sqrt[3]{1 + 0.3275911 \cdot \left|x\right|}\right)}^{3}}\right) + \frac{-0.284496736}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{\left|x\right| \cdot \left|x\right|}}\]
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|}
↓
1 - \frac{\left(\left(0.254829592 + \frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}\right) + \frac{\frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}}}{{\left(\sqrt[3]{1 + 0.3275911 \cdot \left|x\right|}\right)}^{3}}\right) + \frac{-0.284496736}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{\left|x\right| \cdot \left|x\right|}}
(FPCore (x)
 :precision binary64
 (-
  1.0
  (*
   (*
    (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
    (+
     0.254829592
     (*
      (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
      (+
       -0.284496736
       (*
        (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
        (+
         1.421413741
         (*
          (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
          (+
           -1.453152027
           (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) 1.061405429)))))))))
   (exp (- (* (fabs x) (fabs x)))))))
↓
(FPCore (x)
 :precision binary64
 (-
  1.0
  (/
   (+
    (+
     (+ 0.254829592 (/ 1.421413741 (pow (+ 1.0 (* 0.3275911 (fabs x))) 2.0)))
     (/
      (/
       (+ -1.453152027 (/ 1.061405429 (+ 1.0 (* 0.3275911 (fabs x)))))
       (pow (+ 1.0 (* 0.3275911 (fabs x))) 2.0))
      (pow (cbrt (+ 1.0 (* 0.3275911 (fabs x)))) 3.0)))
    (/ -0.284496736 (+ 1.0 (* 0.3275911 (fabs x)))))
   (* (+ 1.0 (* 0.3275911 (fabs x))) (exp (* (fabs x) (fabs x)))))))
double code(double x) {
	return 1.0 - (((1.0 / (1.0 + (0.3275911 * fabs(x)))) * (0.254829592 + ((1.0 / (1.0 + (0.3275911 * fabs(x)))) * (-0.284496736 + ((1.0 / (1.0 + (0.3275911 * fabs(x)))) * (1.421413741 + ((1.0 / (1.0 + (0.3275911 * fabs(x)))) * (-1.453152027 + ((1.0 / (1.0 + (0.3275911 * fabs(x)))) * 1.061405429))))))))) * exp(-(fabs(x) * fabs(x))));
}
↓
double code(double x) {
	return 1.0 - ((((0.254829592 + (1.421413741 / pow((1.0 + (0.3275911 * fabs(x))), 2.0))) + (((-1.453152027 + (1.061405429 / (1.0 + (0.3275911 * fabs(x))))) / pow((1.0 + (0.3275911 * fabs(x))), 2.0)) / pow(cbrt(1.0 + (0.3275911 * fabs(x))), 3.0))) + (-0.284496736 / (1.0 + (0.3275911 * fabs(x))))) / ((1.0 + (0.3275911 * fabs(x))) * exp(fabs(x) * fabs(x))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified13.9

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

  3. Taylor expanded around 0 14.7

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

  4. Simplified13.9

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

  6. Applied add-cube-cbrt_binary6413.9

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

  7. Applied unpow-prod-down_binary6413.9

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

  8. Applied associate-/r*_binary6413.9

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

  9. Simplified13.9

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

  10. Final simplification13.9

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

Reproduce

herbie shell --seed 2020274 
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 1.0 (* (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))