Average Error: 0.2 → 0.2
Time: 21.4s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\frac{1}{5} \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\left|x\right| \cdot 2 + \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right) + \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)\right) \cdot \frac{1}{21}\right)\right|\]
\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|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\frac{1}{5} \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\left|x\right| \cdot 2 + \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right) + \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right)\right) \cdot \frac{1}{21}\right)\right|
double f(double x) {
        double r3982846 = 1.0;
        double r3982847 = atan2(1.0, 0.0);
        double r3982848 = sqrt(r3982847);
        double r3982849 = r3982846 / r3982848;
        double r3982850 = 2.0;
        double r3982851 = x;
        double r3982852 = fabs(r3982851);
        double r3982853 = r3982850 * r3982852;
        double r3982854 = 3.0;
        double r3982855 = r3982850 / r3982854;
        double r3982856 = r3982852 * r3982852;
        double r3982857 = r3982856 * r3982852;
        double r3982858 = r3982855 * r3982857;
        double r3982859 = r3982853 + r3982858;
        double r3982860 = 5.0;
        double r3982861 = r3982846 / r3982860;
        double r3982862 = r3982857 * r3982852;
        double r3982863 = r3982862 * r3982852;
        double r3982864 = r3982861 * r3982863;
        double r3982865 = r3982859 + r3982864;
        double r3982866 = 21.0;
        double r3982867 = r3982846 / r3982866;
        double r3982868 = r3982863 * r3982852;
        double r3982869 = r3982868 * r3982852;
        double r3982870 = r3982867 * r3982869;
        double r3982871 = r3982865 + r3982870;
        double r3982872 = r3982849 * r3982871;
        double r3982873 = fabs(r3982872);
        return r3982873;
}
↓
double f(double x) {
        double r3982874 = 1.0;
        double r3982875 = atan2(1.0, 0.0);
        double r3982876 = sqrt(r3982875);
        double r3982877 = r3982874 / r3982876;
        double r3982878 = 5.0;
        double r3982879 = r3982874 / r3982878;
        double r3982880 = x;
        double r3982881 = fabs(r3982880);
        double r3982882 = r3982881 * r3982881;
        double r3982883 = r3982882 * r3982881;
        double r3982884 = r3982883 * r3982881;
        double r3982885 = r3982881 * r3982884;
        double r3982886 = r3982879 * r3982885;
        double r3982887 = 2.0;
        double r3982888 = r3982881 * r3982887;
        double r3982889 = 3.0;
        double r3982890 = r3982887 / r3982889;
        double r3982891 = r3982883 * r3982890;
        double r3982892 = r3982888 + r3982891;
        double r3982893 = r3982886 + r3982892;
        double r3982894 = r3982881 * r3982885;
        double r3982895 = r3982881 * r3982894;
        double r3982896 = 21.0;
        double r3982897 = r3982874 / r3982896;
        double r3982898 = r3982895 * r3982897;
        double r3982899 = r3982893 + r3982898;
        double r3982900 = r3982877 * r3982899;
        double r3982901 = fabs(r3982900);
        return r3982901;
}

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. Final simplification0.2

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

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))