Average Error: 13.9 → 13.9
Time: 36.7s
Precision: binary64
\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[\sqrt[3]{1 - \frac{\sqrt[3]{{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|} + -1.453152027}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}\right)}^{3}}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{x \cdot x}}} \cdot \left(\sqrt[3]{1 - \frac{\sqrt[3]{{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|} + -1.453152027}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}\right)}^{3}}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{x \cdot x}}} \cdot \sqrt[3]{1 - \frac{\sqrt[3]{{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|} + -1.453152027}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}\right)}^{3}}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{x \cdot x}}}\right)\]
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|}
↓
\sqrt[3]{1 - \frac{\sqrt[3]{{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|} + -1.453152027}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}\right)}^{3}}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{x \cdot x}}} \cdot \left(\sqrt[3]{1 - \frac{\sqrt[3]{{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|} + -1.453152027}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}\right)}^{3}}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{x \cdot x}}} \cdot \sqrt[3]{1 - \frac{\sqrt[3]{{\left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|} + -1.453152027}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}\right)}^{3}}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{x \cdot 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
 (*
  (cbrt
   (-
    1.0
    (/
     (cbrt
      (pow
       (+
        0.254829592
        (/
         (+
          -0.284496736
          (/
           (+
            1.421413741
            (/
             (+ (/ 1.061405429 (+ 1.0 (* 0.3275911 (fabs x)))) -1.453152027)
             (+ 1.0 (* 0.3275911 (fabs x)))))
           (+ 1.0 (* 0.3275911 (fabs x)))))
         (+ 1.0 (* 0.3275911 (fabs x)))))
       3.0))
     (* (+ 1.0 (* 0.3275911 (fabs x))) (exp (* x x))))))
  (*
   (cbrt
    (-
     1.0
     (/
      (cbrt
       (pow
        (+
         0.254829592
         (/
          (+
           -0.284496736
           (/
            (+
             1.421413741
             (/
              (+ (/ 1.061405429 (+ 1.0 (* 0.3275911 (fabs x)))) -1.453152027)
              (+ 1.0 (* 0.3275911 (fabs x)))))
            (+ 1.0 (* 0.3275911 (fabs x)))))
          (+ 1.0 (* 0.3275911 (fabs x)))))
        3.0))
      (* (+ 1.0 (* 0.3275911 (fabs x))) (exp (* x x))))))
   (cbrt
    (-
     1.0
     (/
      (cbrt
       (pow
        (+
         0.254829592
         (/
          (+
           -0.284496736
           (/
            (+
             1.421413741
             (/
              (+ (/ 1.061405429 (+ 1.0 (* 0.3275911 (fabs x)))) -1.453152027)
              (+ 1.0 (* 0.3275911 (fabs x)))))
            (+ 1.0 (* 0.3275911 (fabs x)))))
          (+ 1.0 (* 0.3275911 (fabs x)))))
        3.0))
      (* (+ 1.0 (* 0.3275911 (fabs x))) (exp (* x 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 cbrt(1.0 - (cbrt(pow((0.254829592 + ((-0.284496736 + ((1.421413741 + (((1.061405429 / (1.0 + (0.3275911 * fabs(x)))) + -1.453152027) / (1.0 + (0.3275911 * fabs(x))))) / (1.0 + (0.3275911 * fabs(x))))) / (1.0 + (0.3275911 * fabs(x))))), 3.0)) / ((1.0 + (0.3275911 * fabs(x))) * exp(x * x)))) * (cbrt(1.0 - (cbrt(pow((0.254829592 + ((-0.284496736 + ((1.421413741 + (((1.061405429 / (1.0 + (0.3275911 * fabs(x)))) + -1.453152027) / (1.0 + (0.3275911 * fabs(x))))) / (1.0 + (0.3275911 * fabs(x))))) / (1.0 + (0.3275911 * fabs(x))))), 3.0)) / ((1.0 + (0.3275911 * fabs(x))) * exp(x * x)))) * cbrt(1.0 - (cbrt(pow((0.254829592 + ((-0.284496736 + ((1.421413741 + (((1.061405429 / (1.0 + (0.3275911 * fabs(x)))) + -1.453152027) / (1.0 + (0.3275911 * fabs(x))))) / (1.0 + (0.3275911 * fabs(x))))) / (1.0 + (0.3275911 * fabs(x))))), 3.0)) / ((1.0 + (0.3275911 * fabs(x))) * exp(x * 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

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

  4. Applied add-cbrt-cube_binary6413.9

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

  5. Simplified13.9

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

  7. Applied add-cube-cbrt_binary6413.9

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

  8. Final simplification13.9

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

Alternatives

Reproduce

herbie shell --seed 2021118 
(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)))))))