Average Error: 0.2 → 0.2
Time: 1.1m
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 r2888833 = 1.0;
        double r2888834 = atan2(1.0, 0.0);
        double r2888835 = sqrt(r2888834);
        double r2888836 = r2888833 / r2888835;
        double r2888837 = 2.0;
        double r2888838 = x;
        double r2888839 = fabs(r2888838);
        double r2888840 = r2888837 * r2888839;
        double r2888841 = 3.0;
        double r2888842 = r2888837 / r2888841;
        double r2888843 = r2888839 * r2888839;
        double r2888844 = r2888843 * r2888839;
        double r2888845 = r2888842 * r2888844;
        double r2888846 = r2888840 + r2888845;
        double r2888847 = 5.0;
        double r2888848 = r2888833 / r2888847;
        double r2888849 = r2888844 * r2888839;
        double r2888850 = r2888849 * r2888839;
        double r2888851 = r2888848 * r2888850;
        double r2888852 = r2888846 + r2888851;
        double r2888853 = 21.0;
        double r2888854 = r2888833 / r2888853;
        double r2888855 = r2888850 * r2888839;
        double r2888856 = r2888855 * r2888839;
        double r2888857 = r2888854 * r2888856;
        double r2888858 = r2888852 + r2888857;
        double r2888859 = r2888836 * r2888858;
        double r2888860 = fabs(r2888859);
        return r2888860;
}
↓
double f(double x) {
        double r2888861 = 1.0;
        double r2888862 = atan2(1.0, 0.0);
        double r2888863 = sqrt(r2888862);
        double r2888864 = r2888861 / r2888863;
        double r2888865 = 0.2;
        double r2888866 = x;
        double r2888867 = fabs(r2888866);
        double r2888868 = r2888867 * r2888867;
        double r2888869 = r2888868 * r2888867;
        double r2888870 = r2888869 * r2888867;
        double r2888871 = r2888867 * r2888870;
        double r2888872 = r2888865 * r2888871;
        double r2888873 = 2.0;
        double r2888874 = r2888867 * r2888873;
        double r2888875 = 0.6666666666666666;
        double r2888876 = r2888869 * r2888875;
        double r2888877 = r2888874 + r2888876;
        double r2888878 = r2888872 + r2888877;
        double r2888879 = r2888867 * r2888871;
        double r2888880 = r2888867 * r2888879;
        double r2888881 = 0.047619047619047616;
        double r2888882 = r2888880 * r2888881;
        double r2888883 = r2888878 + r2888882;
        double r2888884 = r2888864 * r2888883;
        double r2888885 = fabs(r2888884);
        return r2888885;
}

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 2019156 
(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)))))))