Average Error: 0.2 → 0.2
Time: 35.5s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{6004799503160661}{9007199254740992} \cdot {\left(\left|x\right|\right)}^{3} + \left(\frac{3602879701896397}{18014398509481984} \cdot {\left(\left|x\right|\right)}^{5} + 2 \cdot \left|x\right|\right)\right) + \frac{1 \cdot {\left(\left|x\right|\right)}^{7}}{21}\right)\right) \cdot 1\right|\]
\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|
↓
\left|\left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{6004799503160661}{9007199254740992} \cdot {\left(\left|x\right|\right)}^{3} + \left(\frac{3602879701896397}{18014398509481984} \cdot {\left(\left|x\right|\right)}^{5} + 2 \cdot \left|x\right|\right)\right) + \frac{1 \cdot {\left(\left|x\right|\right)}^{7}}{21}\right)\right) \cdot 1\right|
double f(double x) {
        double r127880 = 1.0;
        double r127881 = atan2(1.0, 0.0);
        double r127882 = sqrt(r127881);
        double r127883 = r127880 / r127882;
        double r127884 = 2.0;
        double r127885 = x;
        double r127886 = fabs(r127885);
        double r127887 = r127884 * r127886;
        double r127888 = 3.0;
        double r127889 = r127884 / r127888;
        double r127890 = r127886 * r127886;
        double r127891 = r127890 * r127886;
        double r127892 = r127889 * r127891;
        double r127893 = r127887 + r127892;
        double r127894 = 5.0;
        double r127895 = r127880 / r127894;
        double r127896 = r127891 * r127886;
        double r127897 = r127896 * r127886;
        double r127898 = r127895 * r127897;
        double r127899 = r127893 + r127898;
        double r127900 = 21.0;
        double r127901 = r127880 / r127900;
        double r127902 = r127897 * r127886;
        double r127903 = r127902 * r127886;
        double r127904 = r127901 * r127903;
        double r127905 = r127899 + r127904;
        double r127906 = r127883 * r127905;
        double r127907 = fabs(r127906);
        return r127907;
}
↓
double f(double x) {
        double r127908 = 1.0;
        double r127909 = atan2(1.0, 0.0);
        double r127910 = r127908 / r127909;
        double r127911 = sqrt(r127910);
        double r127912 = 6004799503160661.0;
        double r127913 = 9007199254740992.0;
        double r127914 = r127912 / r127913;
        double r127915 = x;
        double r127916 = fabs(r127915);
        double r127917 = 3.0;
        double r127918 = pow(r127916, r127917);
        double r127919 = r127914 * r127918;
        double r127920 = 3602879701896397.0;
        double r127921 = 18014398509481984.0;
        double r127922 = r127920 / r127921;
        double r127923 = 5.0;
        double r127924 = pow(r127916, r127923);
        double r127925 = r127922 * r127924;
        double r127926 = 2.0;
        double r127927 = r127926 * r127916;
        double r127928 = r127925 + r127927;
        double r127929 = r127919 + r127928;
        double r127930 = 1.0;
        double r127931 = 7.0;
        double r127932 = pow(r127916, r127931);
        double r127933 = r127930 * r127932;
        double r127934 = 21.0;
        double r127935 = r127933 / r127934;
        double r127936 = r127929 + r127935;
        double r127937 = r127911 * r127936;
        double r127938 = r127937 * r127930;
        double r127939 = fabs(r127938);
        return r127939;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.2

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{\color{blue}{\sqrt{21} \cdot \sqrt{21}}} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]

  4. Applied *-un-lft-identity0.2

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{\color{blue}{1 \cdot 1}}{\sqrt{21} \cdot \sqrt{21}} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]

  5. Applied times-frac0.2

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \color{blue}{\left(\frac{1}{\sqrt{21}} \cdot \frac{1}{\sqrt{21}}\right)} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]

  6. Applied associate-*l*0.2

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \color{blue}{\frac{1}{\sqrt{21}} \cdot \left(\frac{1}{\sqrt{21}} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)}\right)\right|\]

  7. Simplified0.2

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{\sqrt{21}} \cdot \color{blue}{\left(\left(\frac{1}{\sqrt{21}} \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right)}\right)\right|\]

  8. Taylor expanded around 0 0.2

    \[\leadsto \left|\color{blue}{1 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666296592325124947819858789 \cdot {\left(\left|x\right|\right)}^{3} + \left(1 \cdot \frac{{\left(\left|x\right|\right)}^{7}}{{\left(\sqrt{21}\right)}^{2}} + \left(0.2000000000000000111022302462515654042363 \cdot {\left(\left|x\right|\right)}^{5} + 2 \cdot \left|x\right|\right)\right)\right)\right)}\right|\]

  9. Simplified0.2

    \[\leadsto \left|\color{blue}{\left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{6004799503160661}{9007199254740992} \cdot {\left(\left|x\right|\right)}^{3} + \left(\frac{3602879701896397}{18014398509481984} \cdot {\left(\left|x\right|\right)}^{5} + 2 \cdot \left|x\right|\right)\right) + \frac{1 \cdot {\left(\left|x\right|\right)}^{7}}{21}\right)\right) \cdot 1}\right|\]

  10. Final simplification0.2

    \[\leadsto \left|\left(\sqrt{\frac{1}{\pi}} \cdot \left(\left(\frac{6004799503160661}{9007199254740992} \cdot {\left(\left|x\right|\right)}^{3} + \left(\frac{3602879701896397}{18014398509481984} \cdot {\left(\left|x\right|\right)}^{5} + 2 \cdot \left|x\right|\right)\right) + \frac{1 \cdot {\left(\left|x\right|\right)}^{7}}{21}\right)\right) \cdot 1\right|\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  (fabs (* (/ 1 (sqrt PI)) (+ (+ (+ (* 2 (fabs x)) (* (/ 2 3) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1 5) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1 21) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))