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

Average Error: 0.4 → 0.4

Time: 7.7s

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 r309753 = 1.0;
        double r309754 = 5.0;
        double r309755 = v;
        double r309756 = r309755 * r309755;
        double r309757 = r309754 * r309756;
        double r309758 = r309753 - r309757;
        double r309759 = atan2(1.0, 0.0);
        double r309760 = t;
        double r309761 = r309759 * r309760;
        double r309762 = 2.0;
        double r309763 = 3.0;
        double r309764 = r309763 * r309756;
        double r309765 = r309753 - r309764;
        double r309766 = r309762 * r309765;
        double r309767 = sqrt(r309766);
        double r309768 = r309761 * r309767;
        double r309769 = r309753 - r309756;
        double r309770 = r309768 * r309769;
        double r309771 = r309758 / r309770;
        return r309771;
}

↓

double f(double v, double t) {
        double r309772 = 1.0;
        double r309773 = 5.0;
        double r309774 = v;
        double r309775 = r309774 * r309774;
        double r309776 = r309773 * r309775;
        double r309777 = r309772 - r309776;
        double r309778 = atan2(1.0, 0.0);
        double r309779 = t;
        double r309780 = r309778 * r309779;
        double r309781 = r309777 / r309780;
        double r309782 = 2.0;
        double r309783 = 3.0;
        double r309784 = pow(r309772, r309783);
        double r309785 = 3.0;
        double r309786 = r309785 * r309775;
        double r309787 = pow(r309786, r309783);
        double r309788 = r309784 - r309787;
        double r309789 = r309782 * r309788;
        double r309790 = sqrt(r309789);
        double r309791 = 4.0;
        double r309792 = pow(r309774, r309791);
        double r309793 = fma(r309772, r309772, r309792);
        double r309794 = r309790 * r309793;
        double r309795 = r309772 * r309772;
        double r309796 = r309775 * r309775;
        double r309797 = r309795 - r309796;
        double r309798 = r309794 * r309797;
        double r309799 = r309781 / r309798;
        double r309800 = r309786 * r309786;
        double r309801 = r309772 * r309786;
        double r309802 = r309800 + r309801;
        double r309803 = r309795 + r309802;
        double r309804 = sqrt(r309803);
        double r309805 = r309795 + r309796;
        double r309806 = r309804 * r309805;
        double r309807 = r309799 * r309806;
        double r309808 = r309772 + r309775;
        double r309809 = r309807 * r309808;
        return r309809;
}

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)))))