Average Error: 0.2 → 0.2
Time: 4.4s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left|{\left(\left|x\right|\right)}^{\frac{3}{2}}\right| \cdot \left|{\left(\left|x\right|\right)}^{\frac{3}{2}}\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\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(\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(\left|x\right|\right)}^{\frac{3}{2}}\right| \cdot \left|{\left(\left|x\right|\right)}^{\frac{3}{2}}\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|
double f(double x) {
        double r97778 = 1.0;
        double r97779 = atan2(1.0, 0.0);
        double r97780 = sqrt(r97779);
        double r97781 = r97778 / r97780;
        double r97782 = 2.0;
        double r97783 = x;
        double r97784 = fabs(r97783);
        double r97785 = r97782 * r97784;
        double r97786 = 3.0;
        double r97787 = r97782 / r97786;
        double r97788 = r97784 * r97784;
        double r97789 = r97788 * r97784;
        double r97790 = r97787 * r97789;
        double r97791 = r97785 + r97790;
        double r97792 = 5.0;
        double r97793 = r97778 / r97792;
        double r97794 = r97789 * r97784;
        double r97795 = r97794 * r97784;
        double r97796 = r97793 * r97795;
        double r97797 = r97791 + r97796;
        double r97798 = 21.0;
        double r97799 = r97778 / r97798;
        double r97800 = r97795 * r97784;
        double r97801 = r97800 * r97784;
        double r97802 = r97799 * r97801;
        double r97803 = r97797 + r97802;
        double r97804 = r97781 * r97803;
        double r97805 = fabs(r97804);
        return r97805;
}
↓
double f(double x) {
        double r97806 = 1.0;
        double r97807 = atan2(1.0, 0.0);
        double r97808 = sqrt(r97807);
        double r97809 = r97806 / r97808;
        double r97810 = 2.0;
        double r97811 = x;
        double r97812 = fabs(r97811);
        double r97813 = r97810 * r97812;
        double r97814 = 3.0;
        double r97815 = r97810 / r97814;
        double r97816 = r97812 * r97812;
        double r97817 = r97816 * r97812;
        double r97818 = r97815 * r97817;
        double r97819 = r97813 + r97818;
        double r97820 = 5.0;
        double r97821 = r97806 / r97820;
        double r97822 = r97817 * r97812;
        double r97823 = r97822 * r97812;
        double r97824 = r97821 * r97823;
        double r97825 = r97819 + r97824;
        double r97826 = 21.0;
        double r97827 = r97806 / r97826;
        double r97828 = 1.5;
        double r97829 = pow(r97812, r97828);
        double r97830 = fabs(r97829);
        double r97831 = r97830 * r97830;
        double r97832 = r97831 * r97812;
        double r97833 = r97832 * r97812;
        double r97834 = r97833 * r97812;
        double r97835 = r97834 * r97812;
        double r97836 = r97827 * r97835;
        double r97837 = r97825 + r97836;
        double r97838 = r97809 * r97837;
        double r97839 = fabs(r97838);
        return r97839;
}

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 add-sqr-sqrt0.2

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

  4. Simplified0.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  (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)))))))