Average Error: 13.6 → 13.6
Time: 16.6s
Precision: 64
\[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
↓
\[{e}^{\left(\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\left(\frac{1.42141374100000006}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + 0.25482959199999999\right) + \left(\frac{1.0614054289999999}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}} - \left(\frac{0.284496735999999972}{0.32759110000000002 \cdot \left|x\right| + 1} + \frac{1.45315202700000001}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}}\right)\right)\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)\right)}\]
1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}
↓
{e}^{\left(\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\left(\frac{1.42141374100000006}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + 0.25482959199999999\right) + \left(\frac{1.0614054289999999}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}} - \left(\frac{0.284496735999999972}{0.32759110000000002 \cdot \left|x\right| + 1} + \frac{1.45315202700000001}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}}\right)\right)\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)\right)}
double f(double x) {
        double r213013 = 1.0;
        double r213014 = 0.3275911;
        double r213015 = x;
        double r213016 = fabs(r213015);
        double r213017 = r213014 * r213016;
        double r213018 = r213013 + r213017;
        double r213019 = r213013 / r213018;
        double r213020 = 0.254829592;
        double r213021 = -0.284496736;
        double r213022 = 1.421413741;
        double r213023 = -1.453152027;
        double r213024 = 1.061405429;
        double r213025 = r213019 * r213024;
        double r213026 = r213023 + r213025;
        double r213027 = r213019 * r213026;
        double r213028 = r213022 + r213027;
        double r213029 = r213019 * r213028;
        double r213030 = r213021 + r213029;
        double r213031 = r213019 * r213030;
        double r213032 = r213020 + r213031;
        double r213033 = r213019 * r213032;
        double r213034 = r213016 * r213016;
        double r213035 = -r213034;
        double r213036 = exp(r213035);
        double r213037 = r213033 * r213036;
        double r213038 = r213013 - r213037;
        return r213038;
}
↓
double f(double x) {
        double r213039 = exp(1.0);
        double r213040 = 1.0;
        double r213041 = 0.3275911;
        double r213042 = x;
        double r213043 = fabs(r213042);
        double r213044 = r213041 * r213043;
        double r213045 = r213040 + r213044;
        double r213046 = r213040 / r213045;
        double r213047 = 1.421413741;
        double r213048 = r213044 + r213040;
        double r213049 = 2.0;
        double r213050 = pow(r213048, r213049);
        double r213051 = r213047 / r213050;
        double r213052 = 0.254829592;
        double r213053 = r213051 + r213052;
        double r213054 = 1.061405429;
        double r213055 = 4.0;
        double r213056 = pow(r213048, r213055);
        double r213057 = r213054 / r213056;
        double r213058 = 0.284496736;
        double r213059 = r213058 / r213048;
        double r213060 = 1.453152027;
        double r213061 = 3.0;
        double r213062 = pow(r213048, r213061);
        double r213063 = r213060 / r213062;
        double r213064 = r213059 + r213063;
        double r213065 = r213057 - r213064;
        double r213066 = r213053 + r213065;
        double r213067 = r213046 * r213066;
        double r213068 = pow(r213043, r213049);
        double r213069 = -r213068;
        double r213070 = exp(r213069);
        double r213071 = r213067 * r213070;
        double r213072 = r213040 - r213071;
        double r213073 = log(r213072);
        double r213074 = pow(r213039, r213073);
        return r213074;
}

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

    \[1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
  2. Using strategy rm

  3. Applied add-cbrt-cube13.6

    \[\leadsto 1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \color{blue}{\sqrt[3]{\left(\left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right) \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)\right) \cdot \left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)}}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

  4. Simplified13.6

    \[\leadsto 1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \sqrt[3]{\color{blue}{{\left(0.25482959199999999 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-0.284496735999999972 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(1.42141374100000006 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(-1.45315202700000001 + \frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot 1.0614054289999999\right)\right)\right)\right)}^{3}}}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

  5. Taylor expanded around 0 14.3

    \[\leadsto 1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \color{blue}{\left(\left(1.0614054289999999 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}} + \left(1.42141374100000006 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + 0.25482959199999999\right)\right) - \left(1.45315202700000001 \cdot \frac{1}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}} + 0.284496735999999972 \cdot \frac{1}{0.32759110000000002 \cdot \left|x\right| + 1}\right)\right)}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

  6. Simplified13.6

    \[\leadsto 1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \color{blue}{\left(\left(\frac{1.42141374100000006}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + 0.25482959199999999\right) + \left(\frac{1.0614054289999999}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}} - \left(\frac{0.284496735999999972}{0.32759110000000002 \cdot \left|x\right| + 1} + \frac{1.45315202700000001}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}}\right)\right)\right)}\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]
  7. Using strategy rm

  8. Applied add-exp-log13.6

    \[\leadsto \color{blue}{e^{\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\left(\frac{1.42141374100000006}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + 0.25482959199999999\right) + \left(\frac{1.0614054289999999}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}} - \left(\frac{0.284496735999999972}{0.32759110000000002 \cdot \left|x\right| + 1} + \frac{1.45315202700000001}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}}\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)}}\]

  9. Simplified13.6

    \[\leadsto e^{\color{blue}{\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\left(\frac{1.42141374100000006}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + 0.25482959199999999\right) + \left(\frac{1.0614054289999999}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}} - \left(\frac{0.284496735999999972}{0.32759110000000002 \cdot \left|x\right| + 1} + \frac{1.45315202700000001}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}}\right)\right)\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)}}\]
  10. Using strategy rm

  11. Applied pow113.6

    \[\leadsto e^{\log \color{blue}{\left({\left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\left(\frac{1.42141374100000006}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + 0.25482959199999999\right) + \left(\frac{1.0614054289999999}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}} - \left(\frac{0.284496735999999972}{0.32759110000000002 \cdot \left|x\right| + 1} + \frac{1.45315202700000001}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}}\right)\right)\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)}^{1}\right)}}\]

  12. Applied log-pow13.6

    \[\leadsto e^{\color{blue}{1 \cdot \log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\left(\frac{1.42141374100000006}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + 0.25482959199999999\right) + \left(\frac{1.0614054289999999}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}} - \left(\frac{0.284496735999999972}{0.32759110000000002 \cdot \left|x\right| + 1} + \frac{1.45315202700000001}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}}\right)\right)\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)}}\]

  13. Applied exp-prod13.6

    \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\left(\frac{1.42141374100000006}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + 0.25482959199999999\right) + \left(\frac{1.0614054289999999}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}} - \left(\frac{0.284496735999999972}{0.32759110000000002 \cdot \left|x\right| + 1} + \frac{1.45315202700000001}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}}\right)\right)\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)\right)}}\]

  14. Simplified13.6

    \[\leadsto {\color{blue}{e}}^{\left(\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\left(\frac{1.42141374100000006}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + 0.25482959199999999\right) + \left(\frac{1.0614054289999999}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}} - \left(\frac{0.284496735999999972}{0.32759110000000002 \cdot \left|x\right| + 1} + \frac{1.45315202700000001}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}}\right)\right)\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)\right)}\]

  15. Final simplification13.6

    \[\leadsto {e}^{\left(\log \left(1 - \left(\frac{1}{1 + 0.32759110000000002 \cdot \left|x\right|} \cdot \left(\left(\frac{1.42141374100000006}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{2}} + 0.25482959199999999\right) + \left(\frac{1.0614054289999999}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{4}} - \left(\frac{0.284496735999999972}{0.32759110000000002 \cdot \left|x\right| + 1} + \frac{1.45315202700000001}{{\left(0.32759110000000002 \cdot \left|x\right| + 1\right)}^{3}}\right)\right)\right)\right) \cdot e^{-{\left(\left|x\right|\right)}^{2}}\right)\right)}\]

Reproduce

herbie shell --seed 2020043 
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 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)))))))