Average Error: 0.2 → 0.2
Time: 30.1s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left({\left(\left|x\right|\right)}^{5} \cdot \frac{1}{5} + \left(\frac{2}{3} \cdot \left(\left|x\right| \cdot \left|x\right|\right) + 2\right) \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right|\]
\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(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left({\left(\left|x\right|\right)}^{5} \cdot \frac{1}{5} + \left(\frac{2}{3} \cdot \left(\left|x\right| \cdot \left|x\right|\right) + 2\right) \cdot \left|x\right|\right)\right) \cdot \sqrt{\frac{1}{\pi}}\right|
double f(double x) {
        double r6011918 = 1.0;
        double r6011919 = atan2(1.0, 0.0);
        double r6011920 = sqrt(r6011919);
        double r6011921 = r6011918 / r6011920;
        double r6011922 = 2.0;
        double r6011923 = x;
        double r6011924 = fabs(r6011923);
        double r6011925 = r6011922 * r6011924;
        double r6011926 = 3.0;
        double r6011927 = r6011922 / r6011926;
        double r6011928 = r6011924 * r6011924;
        double r6011929 = r6011928 * r6011924;
        double r6011930 = r6011927 * r6011929;
        double r6011931 = r6011925 + r6011930;
        double r6011932 = 5.0;
        double r6011933 = r6011918 / r6011932;
        double r6011934 = r6011929 * r6011924;
        double r6011935 = r6011934 * r6011924;
        double r6011936 = r6011933 * r6011935;
        double r6011937 = r6011931 + r6011936;
        double r6011938 = 21.0;
        double r6011939 = r6011918 / r6011938;
        double r6011940 = r6011935 * r6011924;
        double r6011941 = r6011940 * r6011924;
        double r6011942 = r6011939 * r6011941;
        double r6011943 = r6011937 + r6011942;
        double r6011944 = r6011921 * r6011943;
        double r6011945 = fabs(r6011944);
        return r6011945;
}
↓
double f(double x) {
        double r6011946 = 0.047619047619047616;
        double r6011947 = x;
        double r6011948 = fabs(r6011947);
        double r6011949 = 7.0;
        double r6011950 = pow(r6011948, r6011949);
        double r6011951 = r6011946 * r6011950;
        double r6011952 = 5.0;
        double r6011953 = pow(r6011948, r6011952);
        double r6011954 = 0.2;
        double r6011955 = r6011953 * r6011954;
        double r6011956 = 0.6666666666666666;
        double r6011957 = r6011948 * r6011948;
        double r6011958 = r6011956 * r6011957;
        double r6011959 = 2.0;
        double r6011960 = r6011958 + r6011959;
        double r6011961 = r6011960 * r6011948;
        double r6011962 = r6011955 + r6011961;
        double r6011963 = r6011951 + r6011962;
        double r6011964 = 1.0;
        double r6011965 = atan2(1.0, 0.0);
        double r6011966 = r6011964 / r6011965;
        double r6011967 = sqrt(r6011966);
        double r6011968 = r6011963 * r6011967;
        double r6011969 = fabs(r6011968);
        return r6011969;
}

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. Taylor expanded around 0 0.2

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

  3. Simplified0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019158 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  (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)))))))