Average Error: 0.2 → 0.2
Time: 28.7s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\left(1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\left(\left|x\right|\right)}^{3}, \mathsf{fma}\left(0.2000000000000000111022302462515654042363, {\left(\left|x\right|\right)}^{5}, \mathsf{fma}\left(2, \left|x\right|, 0.04761904761904761640423089374962728470564 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right|\]
\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|\left(1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\left(\left|x\right|\right)}^{3}, \mathsf{fma}\left(0.2000000000000000111022302462515654042363, {\left(\left|x\right|\right)}^{5}, \mathsf{fma}\left(2, \left|x\right|, 0.04761904761904761640423089374962728470564 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right|
double f(double x) {
        double r230697 = 1.0;
        double r230698 = atan2(1.0, 0.0);
        double r230699 = sqrt(r230698);
        double r230700 = r230697 / r230699;
        double r230701 = 2.0;
        double r230702 = x;
        double r230703 = fabs(r230702);
        double r230704 = r230701 * r230703;
        double r230705 = 3.0;
        double r230706 = r230701 / r230705;
        double r230707 = r230703 * r230703;
        double r230708 = r230707 * r230703;
        double r230709 = r230706 * r230708;
        double r230710 = r230704 + r230709;
        double r230711 = 5.0;
        double r230712 = r230697 / r230711;
        double r230713 = r230708 * r230703;
        double r230714 = r230713 * r230703;
        double r230715 = r230712 * r230714;
        double r230716 = r230710 + r230715;
        double r230717 = 21.0;
        double r230718 = r230697 / r230717;
        double r230719 = r230714 * r230703;
        double r230720 = r230719 * r230703;
        double r230721 = r230718 * r230720;
        double r230722 = r230716 + r230721;
        double r230723 = r230700 * r230722;
        double r230724 = fabs(r230723);
        return r230724;
}
↓
double f(double x) {
        double r230725 = 1.0;
        double r230726 = 1.0;
        double r230727 = atan2(1.0, 0.0);
        double r230728 = r230726 / r230727;
        double r230729 = sqrt(r230728);
        double r230730 = r230725 * r230729;
        double r230731 = 0.6666666666666666;
        double r230732 = x;
        double r230733 = fabs(r230732);
        double r230734 = 3.0;
        double r230735 = pow(r230733, r230734);
        double r230736 = 0.2;
        double r230737 = 5.0;
        double r230738 = pow(r230733, r230737);
        double r230739 = 2.0;
        double r230740 = 0.047619047619047616;
        double r230741 = 7.0;
        double r230742 = pow(r230733, r230741);
        double r230743 = r230740 * r230742;
        double r230744 = fma(r230739, r230733, r230743);
        double r230745 = fma(r230736, r230738, r230744);
        double r230746 = fma(r230731, r230735, r230745);
        double r230747 = r230730 * r230746;
        double r230748 = fabs(r230747);
        return r230748;
}

Error

Bits error versus x

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

    \[\leadsto \left|\color{blue}{1 \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666296592325124947819858789 \cdot {\left(\left|x\right|\right)}^{3} + \left(0.2000000000000000111022302462515654042363 \cdot {\left(\left|x\right|\right)}^{5} + \left(2 \cdot \left|x\right| + 0.04761904761904761640423089374962728470564 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right)}\right|\]

  3. Simplified0.2

    \[\leadsto \left|\color{blue}{\left(1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\left(\left|x\right|\right)}^{3}, \mathsf{fma}\left(0.2000000000000000111022302462515654042363, {\left(\left|x\right|\right)}^{5}, \mathsf{fma}\left(2, \left|x\right|, 0.04761904761904761640423089374962728470564 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)}\right|\]

  4. Final simplification0.2

    \[\leadsto \left|\left(1 \cdot \sqrt{\frac{1}{\pi}}\right) \cdot \mathsf{fma}\left(0.6666666666666666296592325124947819858789, {\left(\left|x\right|\right)}^{3}, \mathsf{fma}\left(0.2000000000000000111022302462515654042363, {\left(\left|x\right|\right)}^{5}, \mathsf{fma}\left(2, \left|x\right|, 0.04761904761904761640423089374962728470564 \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right|\]

Reproduce

herbie shell --seed 2019326 +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)))))))