Jmat.Real.erfi, branch x less than or equal to 0.5

Average Error: 0.2 → 0.2

Time: 18.8s

Precision: 64

Math

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

↓

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

C

double f(double x) {
        double r214769 = 1.0;
        double r214770 = atan2(1.0, 0.0);
        double r214771 = sqrt(r214770);
        double r214772 = r214769 / r214771;
        double r214773 = 2.0;
        double r214774 = x;
        double r214775 = fabs(r214774);
        double r214776 = r214773 * r214775;
        double r214777 = 3.0;
        double r214778 = r214773 / r214777;
        double r214779 = r214775 * r214775;
        double r214780 = r214779 * r214775;
        double r214781 = r214778 * r214780;
        double r214782 = r214776 + r214781;
        double r214783 = 5.0;
        double r214784 = r214769 / r214783;
        double r214785 = r214780 * r214775;
        double r214786 = r214785 * r214775;
        double r214787 = r214784 * r214786;
        double r214788 = r214782 + r214787;
        double r214789 = 21.0;
        double r214790 = r214769 / r214789;
        double r214791 = r214786 * r214775;
        double r214792 = r214791 * r214775;
        double r214793 = r214790 * r214792;
        double r214794 = r214788 + r214793;
        double r214795 = r214772 * r214794;
        double r214796 = fabs(r214795);
        return r214796;
}

↓

double f(double x) {
        double r214797 = 1.0;
        double r214798 = r214797 / r214797;
        double r214799 = 1.0;
        double r214800 = atan2(1.0, 0.0);
        double r214801 = sqrt(r214800);
        double r214802 = r214799 / r214801;
        double r214803 = sqrt(r214799);
        double r214804 = 21.0;
        double r214805 = sqrt(r214804);
        double r214806 = r214803 / r214805;
        double r214807 = x;
        double r214808 = fabs(r214807);
        double r214809 = 6.0;
        double r214810 = pow(r214808, r214809);
        double r214811 = r214806 * r214810;
        double r214812 = r214806 * r214811;
        double r214813 = 5.0;
        double r214814 = r214799 / r214813;
        double r214815 = 3.0;
        double r214816 = pow(r214808, r214815);
        double r214817 = r214814 * r214816;
        double r214818 = r214817 * r214808;
        double r214819 = 2.0;
        double r214820 = 3.0;
        double r214821 = r214819 / r214820;
        double r214822 = r214816 * r214821;
        double r214823 = fma(r214808, r214819, r214822);
        double r214824 = fma(r214818, r214808, r214823);
        double r214825 = fma(r214812, r214808, r214824);
        double r214826 = r214802 * r214825;
        double r214827 = r214798 * r214826;
        double r214828 = fabs(r214827);
        return r214828;
}

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 add-sqr-sqrt 0.2

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

4. Applied add-sqr-sqrt 0.2

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

5. Applied times-frac 0.2

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

6. Applied associate-*l* 0.2

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

7. Simplified 0.2

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

9. Applied *-un-lft-identity 0.2

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

10. Applied *-un-lft-identity 0.2

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

11. Applied times-frac 0.2

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

12. Applied associate-*l* 0.2

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

13. Simplified 0.2

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

14. Final simplification 0.2

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

Reproduce

herbie shell --seed 2020001 +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)))))))