Average Error: 0.2 → 0.1
Time: 6.7s
Precision: 64
\[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right|\]
↓
\[\left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \left|x\right|, \left|x\right|, \mathsf{fma}\left(\left|x\right|, 2, {\left(\left|x\right|\right)}^{3} \cdot \frac{2}{3}\right)\right) + \left(\frac{\frac{1}{\sqrt{\pi}} \cdot 1}{21} \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right|\]
\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 \mathsf{fma}\left(\left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \left|x\right|, \left|x\right|, \mathsf{fma}\left(\left|x\right|, 2, {\left(\left|x\right|\right)}^{3} \cdot \frac{2}{3}\right)\right) + \left(\frac{\frac{1}{\sqrt{\pi}} \cdot 1}{21} \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right|
double f(double x) {
        double r163703 = 1.0;
        double r163704 = atan2(1.0, 0.0);
        double r163705 = sqrt(r163704);
        double r163706 = r163703 / r163705;
        double r163707 = 2.0;
        double r163708 = x;
        double r163709 = fabs(r163708);
        double r163710 = r163707 * r163709;
        double r163711 = 3.0;
        double r163712 = r163707 / r163711;
        double r163713 = r163709 * r163709;
        double r163714 = r163713 * r163709;
        double r163715 = r163712 * r163714;
        double r163716 = r163710 + r163715;
        double r163717 = 5.0;
        double r163718 = r163703 / r163717;
        double r163719 = r163714 * r163709;
        double r163720 = r163719 * r163709;
        double r163721 = r163718 * r163720;
        double r163722 = r163716 + r163721;
        double r163723 = 21.0;
        double r163724 = r163703 / r163723;
        double r163725 = r163720 * r163709;
        double r163726 = r163725 * r163709;
        double r163727 = r163724 * r163726;
        double r163728 = r163722 + r163727;
        double r163729 = r163706 * r163728;
        double r163730 = fabs(r163729);
        return r163730;
}
↓
double f(double x) {
        double r163731 = 1.0;
        double r163732 = atan2(1.0, 0.0);
        double r163733 = sqrt(r163732);
        double r163734 = r163731 / r163733;
        double r163735 = 5.0;
        double r163736 = r163731 / r163735;
        double r163737 = x;
        double r163738 = fabs(r163737);
        double r163739 = 3.0;
        double r163740 = pow(r163738, r163739);
        double r163741 = r163736 * r163740;
        double r163742 = r163741 * r163738;
        double r163743 = 2.0;
        double r163744 = 3.0;
        double r163745 = r163743 / r163744;
        double r163746 = r163740 * r163745;
        double r163747 = fma(r163738, r163743, r163746);
        double r163748 = fma(r163742, r163738, r163747);
        double r163749 = r163734 * r163748;
        double r163750 = r163734 * r163731;
        double r163751 = 21.0;
        double r163752 = r163750 / r163751;
        double r163753 = 6.0;
        double r163754 = pow(r163738, r163753);
        double r163755 = r163752 * r163754;
        double r163756 = r163755 * r163738;
        double r163757 = r163749 + r163756;
        double r163758 = fabs(r163757);
        return r163758;
}

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 distribute-lft-in0.2

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

  4. Simplified0.2

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

  5. Simplified0.2

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \left|x\right|, \left|x\right|, \mathsf{fma}\left(\left|x\right|, 2, {\left(\left|x\right|\right)}^{3} \cdot \frac{2}{3}\right)\right) + \color{blue}{\left(\left(\frac{1}{\sqrt{\pi}} \cdot \frac{1}{21}\right) \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|}\right|\]
  6. Using strategy rm

  7. Applied associate-*r/0.1

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \left|x\right|, \left|x\right|, \mathsf{fma}\left(\left|x\right|, 2, {\left(\left|x\right|\right)}^{3} \cdot \frac{2}{3}\right)\right) + \left(\color{blue}{\frac{\frac{1}{\sqrt{\pi}} \cdot 1}{21}} \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right|\]

  8. Final simplification0.1

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{3}\right) \cdot \left|x\right|, \left|x\right|, \mathsf{fma}\left(\left|x\right|, 2, {\left(\left|x\right|\right)}^{3} \cdot \frac{2}{3}\right)\right) + \left(\frac{\frac{1}{\sqrt{\pi}} \cdot 1}{21} \cdot {\left(\left|x\right|\right)}^{6}\right) \cdot \left|x\right|\right|\]

Reproduce

herbie shell --seed 2020025 +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)))))))