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

Average Error: 0.2 → 0.2

Time: 23.2s

Precision: 64

Math

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

↓

\[\left|\mathsf{fma}\left({\left(\left|x\right|\right)}^{4} \cdot \left|x\right|, \left(\left|x\right| \cdot \frac{\sqrt{1}}{\sqrt{21}}\right) \cdot \left(\left|x\right| \cdot \frac{\sqrt{1}}{\sqrt{21}}\right), \left(\mathsf{fma}\left(\left|x\right| \cdot \left|x\right|, \frac{2}{3}, \frac{\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot 1}{5}\right) + 2\right) \cdot \left|x\right|\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]

C

double f(double x) {
        double r4531856 = 1.0;
        double r4531857 = atan2(1.0, 0.0);
        double r4531858 = sqrt(r4531857);
        double r4531859 = r4531856 / r4531858;
        double r4531860 = 2.0;
        double r4531861 = x;
        double r4531862 = fabs(r4531861);
        double r4531863 = r4531860 * r4531862;
        double r4531864 = 3.0;
        double r4531865 = r4531860 / r4531864;
        double r4531866 = r4531862 * r4531862;
        double r4531867 = r4531866 * r4531862;
        double r4531868 = r4531865 * r4531867;
        double r4531869 = r4531863 + r4531868;
        double r4531870 = 5.0;
        double r4531871 = r4531856 / r4531870;
        double r4531872 = r4531867 * r4531862;
        double r4531873 = r4531872 * r4531862;
        double r4531874 = r4531871 * r4531873;
        double r4531875 = r4531869 + r4531874;
        double r4531876 = 21.0;
        double r4531877 = r4531856 / r4531876;
        double r4531878 = r4531873 * r4531862;
        double r4531879 = r4531878 * r4531862;
        double r4531880 = r4531877 * r4531879;
        double r4531881 = r4531875 + r4531880;
        double r4531882 = r4531859 * r4531881;
        double r4531883 = fabs(r4531882);
        return r4531883;
}

↓

double f(double x) {
        double r4531884 = x;
        double r4531885 = fabs(r4531884);
        double r4531886 = 4.0;
        double r4531887 = pow(r4531885, r4531886);
        double r4531888 = r4531887 * r4531885;
        double r4531889 = 1.0;
        double r4531890 = sqrt(r4531889);
        double r4531891 = 21.0;
        double r4531892 = sqrt(r4531891);
        double r4531893 = r4531890 / r4531892;
        double r4531894 = r4531885 * r4531893;
        double r4531895 = r4531894 * r4531894;
        double r4531896 = r4531885 * r4531885;
        double r4531897 = 2.0;
        double r4531898 = 3.0;
        double r4531899 = r4531897 / r4531898;
        double r4531900 = r4531896 * r4531896;
        double r4531901 = r4531900 * r4531889;
        double r4531902 = 5.0;
        double r4531903 = r4531901 / r4531902;
        double r4531904 = fma(r4531896, r4531899, r4531903);
        double r4531905 = r4531904 + r4531897;
        double r4531906 = r4531905 * r4531885;
        double r4531907 = fma(r4531888, r4531895, r4531906);
        double r4531908 = atan2(1.0, 0.0);
        double r4531909 = sqrt(r4531908);
        double r4531910 = r4531889 / r4531909;
        double r4531911 = r4531907 * r4531910;
        double r4531912 = fabs(r4531911);
        return r4531912;
}

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 pow1 0.2

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

4. Applied pow1 0.2

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

5. Applied pow1 0.2

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

6. Applied pow-prod-up 0.2

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

7. Applied pow-prod-up 0.2

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

8. Simplified 0.2

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

9. Simplified 0.2

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

11. Applied pow1 0.2

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

12. Applied pow1 0.2

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

13. Applied pow-prod-up 0.2

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

14. Applied pow1 0.2

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

15. Applied pow1 0.2

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

16. Applied pow-prod-up 0.2

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

17. Applied pow-prod-up 0.2

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

18. Simplified 0.2

    \[\leadsto \left|\mathsf{fma}\left({\left(\left|x\right|\right)}^{\color{blue}{4}} \cdot \left|x\right|, \left(\left|x\right| \cdot \left|x\right|\right) \cdot \frac{1}{21}, \left|x\right| \cdot \left(\mathsf{fma}\left(\left|x\right| \cdot \left|x\right|, \frac{2}{3}, \frac{1 \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)}{5}\right) + 2\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]
19. Using strategy rm

20. Applied add-sqr-sqrt 0.2

    \[\leadsto \left|\mathsf{fma}\left({\left(\left|x\right|\right)}^{4} \cdot \left|x\right|, \left(\left|x\right| \cdot \left|x\right|\right) \cdot \frac{1}{\color{blue}{\sqrt{21} \cdot \sqrt{21}}}, \left|x\right| \cdot \left(\mathsf{fma}\left(\left|x\right| \cdot \left|x\right|, \frac{2}{3}, \frac{1 \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)}{5}\right) + 2\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]

21. Applied add-sqr-sqrt 0.2

    \[\leadsto \left|\mathsf{fma}\left({\left(\left|x\right|\right)}^{4} \cdot \left|x\right|, \left(\left|x\right| \cdot \left|x\right|\right) \cdot \frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\sqrt{21} \cdot \sqrt{21}}, \left|x\right| \cdot \left(\mathsf{fma}\left(\left|x\right| \cdot \left|x\right|, \frac{2}{3}, \frac{1 \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)}{5}\right) + 2\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]

22. Applied times-frac 0.2

    \[\leadsto \left|\mathsf{fma}\left({\left(\left|x\right|\right)}^{4} \cdot \left|x\right|, \left(\left|x\right| \cdot \left|x\right|\right) \cdot \color{blue}{\left(\frac{\sqrt{1}}{\sqrt{21}} \cdot \frac{\sqrt{1}}{\sqrt{21}}\right)}, \left|x\right| \cdot \left(\mathsf{fma}\left(\left|x\right| \cdot \left|x\right|, \frac{2}{3}, \frac{1 \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)}{5}\right) + 2\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]

23. Applied unswap-sqr 0.2

    \[\leadsto \left|\mathsf{fma}\left({\left(\left|x\right|\right)}^{4} \cdot \left|x\right|, \color{blue}{\left(\left|x\right| \cdot \frac{\sqrt{1}}{\sqrt{21}}\right) \cdot \left(\left|x\right| \cdot \frac{\sqrt{1}}{\sqrt{21}}\right)}, \left|x\right| \cdot \left(\mathsf{fma}\left(\left|x\right| \cdot \left|x\right|, \frac{2}{3}, \frac{1 \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right)}{5}\right) + 2\right)\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]

24. Final simplification 0.2

    \[\leadsto \left|\mathsf{fma}\left({\left(\left|x\right|\right)}^{4} \cdot \left|x\right|, \left(\left|x\right| \cdot \frac{\sqrt{1}}{\sqrt{21}}\right) \cdot \left(\left|x\right| \cdot \frac{\sqrt{1}}{\sqrt{21}}\right), \left(\mathsf{fma}\left(\left|x\right| \cdot \left|x\right|, \frac{2}{3}, \frac{\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left(\left|x\right| \cdot \left|x\right|\right)\right) \cdot 1}{5}\right) + 2\right) \cdot \left|x\right|\right) \cdot \frac{1}{\sqrt{\pi}}\right|\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))