Average Error: 0.2 → 0.2
Time: 1.3m
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left(\left(\frac{1}{5} \cdot \left(\left|x\right| \cdot \left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \left(\left|x\right| \cdot 2 + \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right) + \frac{1}{21} \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left(\left|x\right| \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \left|x\right|\right)\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]
\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(\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) + \frac{1}{21} \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot \left(\left(\left|x\right| \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \left|x\right|\right)\right)\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|
double f(double x) {
        double r11607751 = 1.0;
        double r11607752 = atan2(1.0, 0.0);
        double r11607753 = sqrt(r11607752);
        double r11607754 = r11607751 / r11607753;
        double r11607755 = 2.0;
        double r11607756 = x;
        double r11607757 = fabs(r11607756);
        double r11607758 = r11607755 * r11607757;
        double r11607759 = 3.0;
        double r11607760 = r11607755 / r11607759;
        double r11607761 = r11607757 * r11607757;
        double r11607762 = r11607761 * r11607757;
        double r11607763 = r11607760 * r11607762;
        double r11607764 = r11607758 + r11607763;
        double r11607765 = 5.0;
        double r11607766 = r11607751 / r11607765;
        double r11607767 = r11607762 * r11607757;
        double r11607768 = r11607767 * r11607757;
        double r11607769 = r11607766 * r11607768;
        double r11607770 = r11607764 + r11607769;
        double r11607771 = 21.0;
        double r11607772 = r11607751 / r11607771;
        double r11607773 = r11607768 * r11607757;
        double r11607774 = r11607773 * r11607757;
        double r11607775 = r11607772 * r11607774;
        double r11607776 = r11607770 + r11607775;
        double r11607777 = r11607754 * r11607776;
        double r11607778 = fabs(r11607777);
        return r11607778;
}
↓
double f(double x) {
        double r11607779 = 0.2;
        double r11607780 = x;
        double r11607781 = fabs(r11607780);
        double r11607782 = r11607781 * r11607781;
        double r11607783 = r11607782 * r11607781;
        double r11607784 = r11607783 * r11607781;
        double r11607785 = r11607781 * r11607784;
        double r11607786 = r11607779 * r11607785;
        double r11607787 = 2.0;
        double r11607788 = r11607781 * r11607787;
        double r11607789 = 0.6666666666666666;
        double r11607790 = r11607783 * r11607789;
        double r11607791 = r11607788 + r11607790;
        double r11607792 = r11607786 + r11607791;
        double r11607793 = 0.047619047619047616;
        double r11607794 = 3.0;
        double r11607795 = pow(r11607781, r11607794);
        double r11607796 = r11607781 * r11607795;
        double r11607797 = r11607796 * r11607781;
        double r11607798 = r11607781 * r11607797;
        double r11607799 = r11607781 * r11607798;
        double r11607800 = r11607793 * r11607799;
        double r11607801 = r11607792 + r11607800;
        double r11607802 = 1.0;
        double r11607803 = atan2(1.0, 0.0);
        double r11607804 = sqrt(r11607803);
        double r11607805 = r11607802 / r11607804;
        double r11607806 = r11607801 * r11607805;
        double r11607807 = fabs(r11607806);
        return r11607807;
}

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 pow30.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2019120 
(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)))))))