Average Error: 13.6 → 2.0
Time: 16.6s
Precision: binary64
\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[\sqrt{e^{\log \left(1 - \frac{0.254829592 + \frac{\frac{1.421413741}{1 + 0.3275911 \cdot \left|x\right|} + \left(\frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} + -0.284496736\right)}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{{\left(\left|x\right|\right)}^{2}}}\right)}} \cdot \sqrt{1 - \frac{\sqrt[3]{{\left(0.254829592 + \frac{\frac{1.421413741}{1 + 0.3275911 \cdot \left|x\right|} + \left(\frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} + -0.284496736\right)}{1 + 0.3275911 \cdot \left|x\right|}\right)}^{3}}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{{\left(\left|x\right|\right)}^{2}}}}\]
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{e^{\log \left(1 - \frac{0.254829592 + \frac{\frac{1.421413741}{1 + 0.3275911 \cdot \left|x\right|} + \left(\frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} + -0.284496736\right)}{1 + 0.3275911 \cdot \left|x\right|}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{{\left(\left|x\right|\right)}^{2}}}\right)}} \cdot \sqrt{1 - \frac{\sqrt[3]{{\left(0.254829592 + \frac{\frac{1.421413741}{1 + 0.3275911 \cdot \left|x\right|} + \left(\frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} + -0.284496736\right)}{1 + 0.3275911 \cdot \left|x\right|}\right)}^{3}}}{\left(1 + 0.3275911 \cdot \left|x\right|\right) \cdot e^{{\left(\left|x\right|\right)}^{2}}}}
(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
 (*
  (sqrt
   (exp
    (log
     (-
      1.0
      (/
       (+
        0.254829592
        (/
         (+
          (/ 1.421413741 (+ 1.0 (* 0.3275911 (fabs x))))
          (+
           (/
            (+ -1.453152027 (/ 1.061405429 (+ 1.0 (* 0.3275911 (fabs x)))))
            (pow (+ 1.0 (* 0.3275911 (fabs x))) 2.0))
           -0.284496736))
         (+ 1.0 (* 0.3275911 (fabs x)))))
       (* (+ 1.0 (* 0.3275911 (fabs x))) (exp (pow (fabs x) 2.0))))))))
  (sqrt
   (-
    1.0
    (/
     (cbrt
      (pow
       (+
        0.254829592
        (/
         (+
          (/ 1.421413741 (+ 1.0 (* 0.3275911 (fabs x))))
          (+
           (/
            (+ -1.453152027 (/ 1.061405429 (+ 1.0 (* 0.3275911 (fabs x)))))
            (pow (+ 1.0 (* 0.3275911 (fabs x))) 2.0))
           -0.284496736))
         (+ 1.0 (* 0.3275911 (fabs x)))))
       3.0))
     (* (+ 1.0 (* 0.3275911 (fabs x))) (exp (pow (fabs x) 2.0))))))))
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 sqrt(exp(log(1.0 - ((0.254829592 + (((1.421413741 / (1.0 + (0.3275911 * fabs(x)))) + (((-1.453152027 + (1.061405429 / (1.0 + (0.3275911 * fabs(x))))) / pow((1.0 + (0.3275911 * fabs(x))), 2.0)) + -0.284496736)) / (1.0 + (0.3275911 * fabs(x))))) / ((1.0 + (0.3275911 * fabs(x))) * exp(pow(fabs(x), 2.0))))))) * sqrt(1.0 - (cbrt(pow((0.254829592 + (((1.421413741 / (1.0 + (0.3275911 * fabs(x)))) + (((-1.453152027 + (1.061405429 / (1.0 + (0.3275911 * fabs(x))))) / pow((1.0 + (0.3275911 * fabs(x))), 2.0)) + -0.284496736)) / (1.0 + (0.3275911 * fabs(x))))), 3.0)) / ((1.0 + (0.3275911 * fabs(x))) * exp(pow(fabs(x), 2.0)))));
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.6

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

  2. Simplified13.6

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

  3. Taylor expanded around 0 13.6

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

  4. Simplified13.6

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

  6. Applied add-sqr-sqrt_binary64_57513.6

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

  7. Simplified13.6

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

  8. Simplified13.6

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

  10. Applied add-cbrt-cube_binary64_5622.2

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

  11. Simplified2.2

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

  13. Applied add-exp-log_binary64_5542.0

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

  14. Final simplification2.0

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

Reproduce

herbie shell --seed 2020233 
(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)))))))