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

Average Error: 0.4 → 0.3

Time: 25.8s

Precision: 64

Math

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

↓

\[\left(\left(\frac{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{{1}^{6} - {v}^{12}}}{t}}{\sqrt{\left({1}^{4} - {3}^{4} \cdot {v}^{8}\right) \cdot 2}} \cdot \sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}}\right) \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left({1}^{3} + {v}^{6}\right)\right)\right) \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]

C

double f(double v, double t) {
        double r177241 = 1.0;
        double r177242 = 5.0;
        double r177243 = v;
        double r177244 = r177243 * r177243;
        double r177245 = r177242 * r177244;
        double r177246 = r177241 - r177245;
        double r177247 = atan2(1.0, 0.0);
        double r177248 = t;
        double r177249 = r177247 * r177248;
        double r177250 = 2.0;
        double r177251 = 3.0;
        double r177252 = r177251 * r177244;
        double r177253 = r177241 - r177252;
        double r177254 = r177250 * r177253;
        double r177255 = sqrt(r177254);
        double r177256 = r177249 * r177255;
        double r177257 = r177241 - r177244;
        double r177258 = r177256 * r177257;
        double r177259 = r177246 / r177258;
        return r177259;
}

↓

double f(double v, double t) {
        double r177260 = 1.0;
        double r177261 = 5.0;
        double r177262 = v;
        double r177263 = r177262 * r177262;
        double r177264 = r177261 * r177263;
        double r177265 = r177260 - r177264;
        double r177266 = atan2(1.0, 0.0);
        double r177267 = r177265 / r177266;
        double r177268 = 6.0;
        double r177269 = pow(r177260, r177268);
        double r177270 = 12.0;
        double r177271 = pow(r177262, r177270);
        double r177272 = r177269 - r177271;
        double r177273 = r177267 / r177272;
        double r177274 = t;
        double r177275 = r177273 / r177274;
        double r177276 = 4.0;
        double r177277 = pow(r177260, r177276);
        double r177278 = 3.0;
        double r177279 = pow(r177278, r177276);
        double r177280 = 8.0;
        double r177281 = pow(r177262, r177280);
        double r177282 = r177279 * r177281;
        double r177283 = r177277 - r177282;
        double r177284 = 2.0;
        double r177285 = r177283 * r177284;
        double r177286 = sqrt(r177285);
        double r177287 = r177275 / r177286;
        double r177288 = r177260 * r177260;
        double r177289 = r177278 * r177278;
        double r177290 = pow(r177262, r177276);
        double r177291 = r177289 * r177290;
        double r177292 = r177288 + r177291;
        double r177293 = sqrt(r177292);
        double r177294 = r177287 * r177293;
        double r177295 = r177278 * r177263;
        double r177296 = r177260 + r177295;
        double r177297 = sqrt(r177296);
        double r177298 = 3.0;
        double r177299 = pow(r177260, r177298);
        double r177300 = pow(r177262, r177268);
        double r177301 = r177299 + r177300;
        double r177302 = r177297 * r177301;
        double r177303 = r177294 * r177302;
        double r177304 = r177263 * r177263;
        double r177305 = r177260 * r177263;
        double r177306 = r177304 + r177305;
        double r177307 = r177288 + r177306;
        double r177308 = r177303 * r177307;
        return r177308;
}

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 associate-*l* 0.4

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

5. Applied flip3-- 0.4

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

6. Applied associate-*r/ 0.4

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

7. Applied associate-/r/ 0.4

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

8. Simplified 0.3

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

10. Applied flip-- 0.3

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

11. Applied flip-- 0.3

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

12. Applied associate-*r/ 0.3

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

13. Applied sqrt-div 0.3

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

14. Applied associate-*r/ 0.3

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

15. Applied frac-times 0.3

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

16. Applied associate-/r/ 0.3

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

17. Simplified 0.3

    \[\leadsto \left(\color{blue}{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left({1}^{6} - {v}^{12}\right) \cdot \left(t \cdot \sqrt{\left(1 \cdot 1 - \left(3 \cdot 3\right) \cdot {v}^{4}\right) \cdot 2}\right)}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left({1}^{3} + {v}^{6}\right)\right)\right) \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]
18. Using strategy rm

19. Applied flip-- 0.3

    \[\leadsto \left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left({1}^{6} - {v}^{12}\right) \cdot \left(t \cdot \sqrt{\color{blue}{\frac{\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right) \cdot \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right)}{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}}} \cdot 2}\right)} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left({1}^{3} + {v}^{6}\right)\right)\right) \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]

20. Applied associate-*l/ 0.3

    \[\leadsto \left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left({1}^{6} - {v}^{12}\right) \cdot \left(t \cdot \sqrt{\color{blue}{\frac{\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right) \cdot \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right)\right) \cdot 2}{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}}}}\right)} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left({1}^{3} + {v}^{6}\right)\right)\right) \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]

21. Applied sqrt-div 0.3

    \[\leadsto \left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left({1}^{6} - {v}^{12}\right) \cdot \left(t \cdot \color{blue}{\frac{\sqrt{\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right) \cdot \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right)\right) \cdot 2}}{\sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}}}}\right)} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left({1}^{3} + {v}^{6}\right)\right)\right) \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]

22. Applied associate-*r/ 0.3

    \[\leadsto \left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left({1}^{6} - {v}^{12}\right) \cdot \color{blue}{\frac{t \cdot \sqrt{\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right) \cdot \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right)\right) \cdot 2}}{\sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}}}}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left({1}^{3} + {v}^{6}\right)\right)\right) \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]

23. Applied associate-*r/ 0.3

    \[\leadsto \left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\color{blue}{\frac{\left({1}^{6} - {v}^{12}\right) \cdot \left(t \cdot \sqrt{\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right) \cdot \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right)\right) \cdot 2}\right)}{\sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}}}}} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left({1}^{3} + {v}^{6}\right)\right)\right) \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]

24. Applied associate-/r/ 0.3

    \[\leadsto \left(\color{blue}{\left(\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{\left({1}^{6} - {v}^{12}\right) \cdot \left(t \cdot \sqrt{\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right) \cdot \left(\left(3 \cdot 3\right) \cdot {v}^{4}\right)\right) \cdot 2}\right)} \cdot \sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}}\right)} \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left({1}^{3} + {v}^{6}\right)\right)\right) \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]

25. Simplified 0.3

    \[\leadsto \left(\left(\color{blue}{\frac{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{{1}^{6} - {v}^{12}}}{t}}{\sqrt{\left({1}^{4} - {3}^{4} \cdot {v}^{8}\right) \cdot 2}}} \cdot \sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}}\right) \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left({1}^{3} + {v}^{6}\right)\right)\right) \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]

26. Final simplification 0.3

    \[\leadsto \left(\left(\frac{\frac{\frac{\frac{1 - 5 \cdot \left(v \cdot v\right)}{\pi}}{{1}^{6} - {v}^{12}}}{t}}{\sqrt{\left({1}^{4} - {3}^{4} \cdot {v}^{8}\right) \cdot 2}} \cdot \sqrt{1 \cdot 1 + \left(3 \cdot 3\right) \cdot {v}^{4}}\right) \cdot \left(\sqrt{1 + 3 \cdot \left(v \cdot v\right)} \cdot \left({1}^{3} + {v}^{6}\right)\right)\right) \cdot \left(1 \cdot 1 + \left(\left(v \cdot v\right) \cdot \left(v \cdot v\right) + 1 \cdot \left(v \cdot v\right)\right)\right)\]

Reproduce

herbie shell --seed 2019306 
(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)))))