Average Error: 0.2 → 0.2
Time: 42.8s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\sqrt{\frac{1}{\pi}} \cdot \mathsf{fma}\left(\left(\left|x\right| \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right), \frac{1}{5}, \mathsf{fma}\left(2, \left|x\right|, \mathsf{fma}\left(\left|x\right| \cdot \left|x\right|, \left|x\right| \cdot \frac{2}{3}, \frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right|\]
\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 \mathsf{fma}\left(\left(\left|x\right| \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right), \frac{1}{5}, \mathsf{fma}\left(2, \left|x\right|, \mathsf{fma}\left(\left|x\right| \cdot \left|x\right|, \left|x\right| \cdot \frac{2}{3}, \frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7}\right)\right)\right)\right|
double f(double x) {
        double r6572673 = 1.0;
        double r6572674 = atan2(1.0, 0.0);
        double r6572675 = sqrt(r6572674);
        double r6572676 = r6572673 / r6572675;
        double r6572677 = 2.0;
        double r6572678 = x;
        double r6572679 = fabs(r6572678);
        double r6572680 = r6572677 * r6572679;
        double r6572681 = 3.0;
        double r6572682 = r6572677 / r6572681;
        double r6572683 = r6572679 * r6572679;
        double r6572684 = r6572683 * r6572679;
        double r6572685 = r6572682 * r6572684;
        double r6572686 = r6572680 + r6572685;
        double r6572687 = 5.0;
        double r6572688 = r6572673 / r6572687;
        double r6572689 = r6572684 * r6572679;
        double r6572690 = r6572689 * r6572679;
        double r6572691 = r6572688 * r6572690;
        double r6572692 = r6572686 + r6572691;
        double r6572693 = 21.0;
        double r6572694 = r6572673 / r6572693;
        double r6572695 = r6572690 * r6572679;
        double r6572696 = r6572695 * r6572679;
        double r6572697 = r6572694 * r6572696;
        double r6572698 = r6572692 + r6572697;
        double r6572699 = r6572676 * r6572698;
        double r6572700 = fabs(r6572699);
        return r6572700;
}
↓
double f(double x) {
        double r6572701 = 1.0;
        double r6572702 = atan2(1.0, 0.0);
        double r6572703 = r6572701 / r6572702;
        double r6572704 = sqrt(r6572703);
        double r6572705 = x;
        double r6572706 = fabs(r6572705);
        double r6572707 = r6572706 * r6572706;
        double r6572708 = r6572706 * r6572707;
        double r6572709 = r6572708 * r6572707;
        double r6572710 = 0.2;
        double r6572711 = 2.0;
        double r6572712 = 0.6666666666666666;
        double r6572713 = r6572706 * r6572712;
        double r6572714 = 0.047619047619047616;
        double r6572715 = 7.0;
        double r6572716 = pow(r6572706, r6572715);
        double r6572717 = r6572714 * r6572716;
        double r6572718 = fma(r6572707, r6572713, r6572717);
        double r6572719 = fma(r6572711, r6572706, r6572718);
        double r6572720 = fma(r6572709, r6572710, r6572719);
        double r6572721 = r6572704 * r6572720;
        double r6572722 = fabs(r6572721);
        return r6572722;
}

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. Using strategy rm

  3. Applied associate-*l/0.6

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

  4. Simplified0.6

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

  5. Taylor expanded around -inf 0.2

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

  6. Simplified0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(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)))))))