Falkner and Boettcher, Equation (20:1,3)

Average Error: 0.4 → 0.4

Time: 7.4s

Precision: 64

Math

\[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]

↓

\[\left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi \cdot t}}{\left(\sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)} \cdot \mathsf{fma}\left(1, 1, {v}^{4}\right)\right) \cdot \left(1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)} \cdot \left(\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left(1 \cdot 1 + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right)\right) \cdot \left(1 + v \cdot v\right)\]

C

double f(double v, double t) {
        double r398860 = 1.0;
        double r398861 = 5.0;
        double r398862 = v;
        double r398863 = r398862 * r398862;
        double r398864 = r398861 * r398863;
        double r398865 = r398860 - r398864;
        double r398866 = atan2(1.0, 0.0);
        double r398867 = t;
        double r398868 = r398866 * r398867;
        double r398869 = 2.0;
        double r398870 = 3.0;
        double r398871 = r398870 * r398863;
        double r398872 = r398860 - r398871;
        double r398873 = r398869 * r398872;
        double r398874 = sqrt(r398873);
        double r398875 = r398868 * r398874;
        double r398876 = r398860 - r398863;
        double r398877 = r398875 * r398876;
        double r398878 = r398865 / r398877;
        return r398878;
}

↓

double f(double v, double t) {
        double r398879 = 1.0;
        double r398880 = 5.0;
        double r398881 = v;
        double r398882 = r398881 * r398881;
        double r398883 = r398880 * r398882;
        double r398884 = r398879 - r398883;
        double r398885 = atan2(1.0, 0.0);
        double r398886 = t;
        double r398887 = r398885 * r398886;
        double r398888 = r398884 / r398887;
        double r398889 = 2.0;
        double r398890 = 3.0;
        double r398891 = pow(r398879, r398890);
        double r398892 = 3.0;
        double r398893 = r398892 * r398882;
        double r398894 = pow(r398893, r398890);
        double r398895 = r398891 - r398894;
        double r398896 = r398889 * r398895;
        double r398897 = sqrt(r398896);
        double r398898 = 4.0;
        double r398899 = pow(r398881, r398898);
        double r398900 = fma(r398879, r398879, r398899);
        double r398901 = r398897 * r398900;
        double r398902 = r398879 * r398879;
        double r398903 = r398882 * r398882;
        double r398904 = r398902 - r398903;
        double r398905 = r398901 * r398904;
        double r398906 = r398888 / r398905;
        double r398907 = r398893 * r398893;
        double r398908 = r398879 * r398893;
        double r398909 = r398907 + r398908;
        double r398910 = r398902 + r398909;
        double r398911 = sqrt(r398910);
        double r398912 = r398902 + r398903;
        double r398913 = r398911 * r398912;
        double r398914 = r398906 * r398913;
        double r398915 = r398879 + r398882;
        double r398916 = r398914 * r398915;
        return r398916;
}

Error

Bits error versus v

Bits error versus t

Derivation

1. Initial program 0.4

   \[\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 - v \cdot v\right)}\]
2. Using strategy rm

3. Applied flip-- 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \color{blue}{\frac{1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)}{1 + v \cdot v}}}\]

4. Applied associate-*r/ 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\frac{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}{1 + v \cdot v}}}\]

5. Applied associate-/r/ 0.4

   \[\leadsto \color{blue}{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \left(1 \cdot 1 - \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)} \cdot \left(1 + v \cdot v\right)}\]
6. Using strategy rm

7. Applied flip-- 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left(1 - 3 \cdot \left(v \cdot v\right)\right)}\right) \cdot \color{blue}{\frac{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}{1 \cdot 1 + \left(v \cdot v\right) \cdot \left(v \cdot v\right)}}} \cdot \left(1 + v \cdot v\right)\]

8. Applied flip3-- 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \color{blue}{\frac{{1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}}{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}}\right) \cdot \frac{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}{1 \cdot 1 + \left(v \cdot v\right) \cdot \left(v \cdot v\right)}} \cdot \left(1 + v \cdot v\right)\]

9. Applied associate-*r/ 0.4

   \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{\color{blue}{\frac{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}}\right) \cdot \frac{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}{1 \cdot 1 + \left(v \cdot v\right) \cdot \left(v \cdot v\right)}} \cdot \left(1 + v \cdot v\right)\]

10. Applied sqrt-div 0.4

    \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \color{blue}{\frac{\sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}}\right) \cdot \frac{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}{1 \cdot 1 + \left(v \cdot v\right) \cdot \left(v \cdot v\right)}} \cdot \left(1 + v \cdot v\right)\]

11. Applied associate-*r/ 0.4

    \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\frac{\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}}{\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)}}} \cdot \frac{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}{1 \cdot 1 + \left(v \cdot v\right) \cdot \left(v \cdot v\right)}} \cdot \left(1 + v \cdot v\right)\]

12. Applied frac-times 0.4

    \[\leadsto \frac{1 - 5 \cdot \left(v \cdot v\right)}{\color{blue}{\frac{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right)}{\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left(1 \cdot 1 + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)}}} \cdot \left(1 + v \cdot v\right)\]

13. Applied associate-/r/ 0.4

    \[\leadsto \color{blue}{\left(\frac{1 - 5 \cdot \left(v \cdot v\right)}{\left(\left(\pi \cdot t\right) \cdot \sqrt{2 \cdot \left({1}^{3} - {\left(3 \cdot \left(v \cdot v\right)\right)}^{3}\right)}\right) \cdot \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right) \cdot \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right)} \cdot \left(\sqrt{1 \cdot 1 + \left(\left(3 \cdot \left(v \cdot v\right)\right) \cdot \left(3 \cdot \left(v \cdot v\right)\right) + 1 \cdot \left(3 \cdot \left(v \cdot v\right)\right)\right)} \cdot \left(1 \cdot 1 + \left(v \cdot v\right) \cdot \left(v \cdot v\right)\right)\right)\right)} \cdot \left(1 + v \cdot v\right)\]

14. Simplified 0.4

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

15. Final simplification 0.4

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  :precision binary64
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))