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

Average Error: 0.4 → 0.3

Time: 5.0m

Precision: 64

Math

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

↓

\[\frac{\frac{\frac{\mathsf{fma}\left(\left(v \cdot v\right), -5, 1\right)}{1 - v}}{\pi}}{\mathsf{fma}\left(v, \left(t \cdot \sqrt{\mathsf{fma}\left(\left(-6 \cdot v\right), v, 2\right)}\right), \left(t \cdot \sqrt{\mathsf{fma}\left(\left(-6 \cdot v\right), v, 2\right)}\right)\right)}\]

C

double f(double v, double t) {
        double r34313847 = 1.0;
        double r34313848 = 5.0;
        double r34313849 = v;
        double r34313850 = r34313849 * r34313849;
        double r34313851 = r34313848 * r34313850;
        double r34313852 = r34313847 - r34313851;
        double r34313853 = atan2(1.0, 0.0);
        double r34313854 = t;
        double r34313855 = r34313853 * r34313854;
        double r34313856 = 2.0;
        double r34313857 = 3.0;
        double r34313858 = r34313857 * r34313850;
        double r34313859 = r34313847 - r34313858;
        double r34313860 = r34313856 * r34313859;
        double r34313861 = sqrt(r34313860);
        double r34313862 = r34313855 * r34313861;
        double r34313863 = r34313847 - r34313850;
        double r34313864 = r34313862 * r34313863;
        double r34313865 = r34313852 / r34313864;
        return r34313865;
}

↓

double f(double v, double t) {
        double r34313866 = v;
        double r34313867 = r34313866 * r34313866;
        double r34313868 = -5.0;
        double r34313869 = 1.0;
        double r34313870 = fma(r34313867, r34313868, r34313869);
        double r34313871 = r34313869 - r34313866;
        double r34313872 = r34313870 / r34313871;
        double r34313873 = atan2(1.0, 0.0);
        double r34313874 = r34313872 / r34313873;
        double r34313875 = t;
        double r34313876 = -6.0;
        double r34313877 = r34313876 * r34313866;
        double r34313878 = 2.0;
        double r34313879 = fma(r34313877, r34313866, r34313878);
        double r34313880 = sqrt(r34313879);
        double r34313881 = r34313875 * r34313880;
        double r34313882 = fma(r34313866, r34313881, r34313881);
        double r34313883 = r34313874 / r34313882;
        return r34313883;
}

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. Simplified 0.4

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

4. Applied add-sqr-sqrt 0.4

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

5. Applied difference-of-squares 0.4

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

6. Applied *-un-lft-identity 0.4

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

7. Applied times-frac 0.4

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

8. Applied times-frac 0.5

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

9. Simplified 0.5

   \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(v, \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right), \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right)\right)}} \cdot \frac{\frac{\mathsf{fma}\left(-5, \left(v \cdot v\right), 1\right)}{\sqrt{1} - v}}{t \cdot \pi}\]

10. Simplified 0.3

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

12. Applied add-cube-cbrt 0.3

    \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(v, \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right), \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(v, \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right), \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(v, \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right), \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right)\right)}}} \cdot \frac{\frac{\mathsf{fma}\left(\left(v \cdot v\right), -5, 1\right)}{\pi - v \cdot \pi}}{t}\]

13. Applied *-un-lft-identity 0.3

    \[\leadsto \frac{\color{blue}{1 \cdot 1}}{\left(\sqrt[3]{\mathsf{fma}\left(v, \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right), \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(v, \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right), \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(v, \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right), \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right)\right)}} \cdot \frac{\frac{\mathsf{fma}\left(\left(v \cdot v\right), -5, 1\right)}{\pi - v \cdot \pi}}{t}\]

14. Applied times-frac 0.3

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

16. Applied frac-times 0.3

    \[\leadsto \color{blue}{\frac{1 \cdot 1}{\left(\sqrt[3]{\mathsf{fma}\left(v, \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right), \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(v, \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right), \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(v, \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right), \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right)\right)}}} \cdot \frac{\frac{\mathsf{fma}\left(\left(v \cdot v\right), -5, 1\right)}{\pi - v \cdot \pi}}{t}\]

17. Applied frac-times 0.3

    \[\leadsto \color{blue}{\frac{\left(1 \cdot 1\right) \cdot \frac{\mathsf{fma}\left(\left(v \cdot v\right), -5, 1\right)}{\pi - v \cdot \pi}}{\left(\left(\sqrt[3]{\mathsf{fma}\left(v, \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right), \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(v, \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right), \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(v, \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right), \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right)\right)}\right) \cdot t}}\]

18. Simplified 0.3

    \[\leadsto \frac{\color{blue}{\frac{\frac{\mathsf{fma}\left(\left(v \cdot v\right), -5, 1\right)}{1 - v}}{\pi}}}{\left(\left(\sqrt[3]{\mathsf{fma}\left(v, \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right), \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(v, \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right), \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(v, \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right), \left(\sqrt{\mathsf{fma}\left(v, \left(v \cdot -6\right), 2\right)}\right)\right)}\right) \cdot t}\]

19. Simplified 0.3

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\left(v \cdot v\right), -5, 1\right)}{1 - v}}{\pi}}{\color{blue}{\mathsf{fma}\left(v, \left(\sqrt{\mathsf{fma}\left(\left(v \cdot -6\right), v, 2\right)} \cdot t\right), \left(\sqrt{\mathsf{fma}\left(\left(v \cdot -6\right), v, 2\right)} \cdot t\right)\right)}}\]

20. Final simplification 0.3

    \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\left(v \cdot v\right), -5, 1\right)}{1 - v}}{\pi}}{\mathsf{fma}\left(v, \left(t \cdot \sqrt{\mathsf{fma}\left(\left(-6 \cdot v\right), v, 2\right)}\right), \left(t \cdot \sqrt{\mathsf{fma}\left(\left(-6 \cdot v\right), v, 2\right)}\right)\right)}\]

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1 (* 5 (* v v))) (* (* (* PI t) (sqrt (* 2 (- 1 (* 3 (* v v)))))) (- 1 (* v v)))))