Average Error: 0.2 → 0.2
Time: 43.5s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\sqrt{\frac{1}{\pi}} \cdot \left({\left(\left|x\right|\right)}^{5} \cdot \frac{1}{5} + \left(2 \cdot \left|x\right| + \left(\left(\left|x\right| \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \frac{2}{3} + \frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right|\]
\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|\sqrt{\frac{1}{\pi}} \cdot \left({\left(\left|x\right|\right)}^{5} \cdot \frac{1}{5} + \left(2 \cdot \left|x\right| + \left(\left(\left|x\right| \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \frac{2}{3} + \frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right|
double f(double x) {
        double r3175792 = 1.0;
        double r3175793 = atan2(1.0, 0.0);
        double r3175794 = sqrt(r3175793);
        double r3175795 = r3175792 / r3175794;
        double r3175796 = 2.0;
        double r3175797 = x;
        double r3175798 = fabs(r3175797);
        double r3175799 = r3175796 * r3175798;
        double r3175800 = 3.0;
        double r3175801 = r3175796 / r3175800;
        double r3175802 = r3175798 * r3175798;
        double r3175803 = r3175802 * r3175798;
        double r3175804 = r3175801 * r3175803;
        double r3175805 = r3175799 + r3175804;
        double r3175806 = 5.0;
        double r3175807 = r3175792 / r3175806;
        double r3175808 = r3175803 * r3175798;
        double r3175809 = r3175808 * r3175798;
        double r3175810 = r3175807 * r3175809;
        double r3175811 = r3175805 + r3175810;
        double r3175812 = 21.0;
        double r3175813 = r3175792 / r3175812;
        double r3175814 = r3175809 * r3175798;
        double r3175815 = r3175814 * r3175798;
        double r3175816 = r3175813 * r3175815;
        double r3175817 = r3175811 + r3175816;
        double r3175818 = r3175795 * r3175817;
        double r3175819 = fabs(r3175818);
        return r3175819;
}
↓
double f(double x) {
        double r3175820 = 1.0;
        double r3175821 = atan2(1.0, 0.0);
        double r3175822 = r3175820 / r3175821;
        double r3175823 = sqrt(r3175822);
        double r3175824 = x;
        double r3175825 = fabs(r3175824);
        double r3175826 = 5.0;
        double r3175827 = pow(r3175825, r3175826);
        double r3175828 = 0.2;
        double r3175829 = r3175827 * r3175828;
        double r3175830 = 2.0;
        double r3175831 = r3175830 * r3175825;
        double r3175832 = r3175825 * r3175825;
        double r3175833 = r3175825 * r3175832;
        double r3175834 = 0.6666666666666666;
        double r3175835 = r3175833 * r3175834;
        double r3175836 = 0.047619047619047616;
        double r3175837 = 7.0;
        double r3175838 = pow(r3175825, r3175837);
        double r3175839 = r3175836 * r3175838;
        double r3175840 = r3175835 + r3175839;
        double r3175841 = r3175831 + r3175840;
        double r3175842 = r3175829 + r3175841;
        double r3175843 = r3175823 * r3175842;
        double r3175844 = fabs(r3175843);
        return r3175844;
}

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 -inf 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

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019151 
(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)))))))