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

Average Error: 0.4 → 0.1

Time: 27.6s

Precision: 64

Math

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

↓

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

C

double f(double v, double t) {
        double r5785244 = 1.0;
        double r5785245 = 5.0;
        double r5785246 = v;
        double r5785247 = r5785246 * r5785246;
        double r5785248 = r5785245 * r5785247;
        double r5785249 = r5785244 - r5785248;
        double r5785250 = atan2(1.0, 0.0);
        double r5785251 = t;
        double r5785252 = r5785250 * r5785251;
        double r5785253 = 2.0;
        double r5785254 = 3.0;
        double r5785255 = r5785254 * r5785247;
        double r5785256 = r5785244 - r5785255;
        double r5785257 = r5785253 * r5785256;
        double r5785258 = sqrt(r5785257);
        double r5785259 = r5785252 * r5785258;
        double r5785260 = r5785244 - r5785247;
        double r5785261 = r5785259 * r5785260;
        double r5785262 = r5785249 / r5785261;
        return r5785262;
}

↓

double f(double v, double t) {
        double r5785263 = 1.0;
        double r5785264 = 5.0;
        double r5785265 = v;
        double r5785266 = r5785265 * r5785265;
        double r5785267 = r5785264 * r5785266;
        double r5785268 = r5785263 - r5785267;
        double r5785269 = atan2(1.0, 0.0);
        double r5785270 = 2.0;
        double r5785271 = r5785263 * r5785263;
        double r5785272 = 3.0;
        double r5785273 = r5785266 * r5785272;
        double r5785274 = r5785273 * r5785273;
        double r5785275 = r5785271 - r5785274;
        double r5785276 = r5785270 * r5785275;
        double r5785277 = sqrt(r5785276);
        double r5785278 = r5785269 * r5785277;
        double r5785279 = r5785268 / r5785278;
        double r5785280 = fma(r5785266, r5785264, r5785263);
        double r5785281 = r5785280 * r5785264;
        double r5785282 = fma(r5785266, r5785281, r5785271);
        double r5785283 = r5785279 * r5785282;
        double r5785284 = t;
        double r5785285 = r5785283 / r5785284;
        double r5785286 = r5785263 - r5785266;
        double r5785287 = r5785263 + r5785267;
        double r5785288 = r5785267 * r5785287;
        double r5785289 = fma(r5785263, r5785263, r5785288);
        double r5785290 = r5785286 * r5785289;
        double r5785291 = r5785285 / r5785290;
        double r5785292 = r5785263 + r5785273;
        double r5785293 = sqrt(r5785292);
        double r5785294 = r5785291 * r5785293;
        return r5785294;
}

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 flip3-- 0.4

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

4. Applied associate-/l/ 0.4

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

5. Simplified 0.4

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

7. Applied flip-- 0.4

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

8. Applied associate-*r/ 0.4

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

9. Applied sqrt-div 0.4

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

10. Applied associate-*r/ 0.4

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

11. Applied associate-*r/ 0.4

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

12. Applied associate-*l/ 0.4

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

13. Applied associate-/r/ 0.4

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

14. Simplified 0.4

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

16. Applied cube-unmult 0.4

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

17. Applied cube-unmult 0.4

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

18. Applied difference-cubes 0.3

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

19. Applied times-frac 0.3

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

20. Simplified 0.3

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

22. Applied associate-*l/ 0.1

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

23. Final simplification 0.1

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

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (v t)
  :name "Falkner and Boettcher, Equation (20:1,3)"
  (/ (- 1.0 (* 5.0 (* v v))) (* (* (* PI t) (sqrt (* 2.0 (- 1.0 (* 3.0 (* v v)))))) (- 1.0 (* v v)))))