Toniolo and Linder, Equation (10+)

Average Error: 32.5 → 13.4

Time: 1.3m

Precision: 64

Math

\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

↓

\[\frac{\frac{1}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}} \cdot \frac{\ell \cdot \left(\sqrt[3]{\ell} \cdot \frac{2}{{\left(\sqrt[3]{t}\right)}^{3}}\right)}{\tan k \cdot \left(\left(\left(\sqrt[3]{\sin k} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\]

C

double f(double t, double l, double k) {
        double r175453 = 2.0;
        double r175454 = t;
        double r175455 = 3.0;
        double r175456 = pow(r175454, r175455);
        double r175457 = l;
        double r175458 = r175457 * r175457;
        double r175459 = r175456 / r175458;
        double r175460 = k;
        double r175461 = sin(r175460);
        double r175462 = r175459 * r175461;
        double r175463 = tan(r175460);
        double r175464 = r175462 * r175463;
        double r175465 = 1.0;
        double r175466 = r175460 / r175454;
        double r175467 = pow(r175466, r175453);
        double r175468 = r175465 + r175467;
        double r175469 = r175468 + r175465;
        double r175470 = r175464 * r175469;
        double r175471 = r175453 / r175470;
        return r175471;
}

↓

double f(double t, double l, double k) {
        double r175472 = 1.0;
        double r175473 = t;
        double r175474 = cbrt(r175473);
        double r175475 = 3.0;
        double r175476 = pow(r175474, r175475);
        double r175477 = l;
        double r175478 = cbrt(r175477);
        double r175479 = r175476 / r175478;
        double r175480 = r175472 / r175479;
        double r175481 = k;
        double r175482 = sin(r175481);
        double r175483 = cbrt(r175482);
        double r175484 = r175483 * r175483;
        double r175485 = r175480 / r175484;
        double r175486 = 2.0;
        double r175487 = r175486 / r175476;
        double r175488 = r175478 * r175487;
        double r175489 = r175477 * r175488;
        double r175490 = tan(r175481);
        double r175491 = r175483 * r175479;
        double r175492 = 2.0;
        double r175493 = 1.0;
        double r175494 = r175481 / r175473;
        double r175495 = pow(r175494, r175486);
        double r175496 = fma(r175492, r175493, r175495);
        double r175497 = cbrt(r175496);
        double r175498 = r175497 * r175497;
        double r175499 = r175491 * r175498;
        double r175500 = r175499 * r175497;
        double r175501 = r175490 * r175500;
        double r175502 = r175489 / r175501;
        double r175503 = r175485 * r175502;
        return r175503;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Initial program 32.5

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

2. Simplified 32.6

   \[\leadsto \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]
3. Using strategy rm

4. Applied add-cube-cbrt 32.7

   \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

5. Applied add-cube-cbrt 32.8

   \[\leadsto \frac{\frac{2}{\frac{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

6. Applied unpow-prod-down 32.8

   \[\leadsto \frac{\frac{2}{\frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

7. Applied times-frac 30.2

   \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

8. Applied *-un-lft-identity 30.2

   \[\leadsto \frac{\frac{\color{blue}{1 \cdot 2}}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

9. Applied times-frac 30.1

   \[\leadsto \frac{\color{blue}{\frac{1}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \frac{2}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

10. Applied times-frac 24.6

    \[\leadsto \color{blue}{\left(\frac{\frac{1}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\sin k} \cdot \frac{\frac{2}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\tan k}\right)} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]

11. Applied associate-*l* 22.4

    \[\leadsto \color{blue}{\frac{\frac{1}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\sin k} \cdot \left(\frac{\frac{2}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\]

12. Simplified 23.1

    \[\leadsto \frac{\frac{1}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\sin k} \cdot \color{blue}{\frac{\left(\frac{2}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}\right) \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}}\]
13. Using strategy rm

14. Applied add-cube-cbrt 23.1

    \[\leadsto \frac{\frac{1}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\color{blue}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\sin k}}} \cdot \frac{\left(\frac{2}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}\right) \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

15. Applied unpow-prod-down 23.1

    \[\leadsto \frac{\frac{1}{\frac{\color{blue}{{\left(\sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\sin k}} \cdot \frac{\left(\frac{2}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}\right) \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

16. Applied times-frac 19.4

    \[\leadsto \frac{\frac{1}{\color{blue}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\sin k}} \cdot \frac{\left(\frac{2}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}\right) \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

17. Applied add-sqr-sqrt 19.4

    \[\leadsto \frac{\frac{\color{blue}{\sqrt{1} \cdot \sqrt{1}}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\sin k}} \cdot \frac{\left(\frac{2}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}\right) \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

18. Applied times-frac 19.2

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{1}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}} \cdot \frac{\sqrt{1}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\sin k}} \cdot \frac{\left(\frac{2}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}\right) \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

19. Applied times-frac 16.9

    \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt{1}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}} \cdot \frac{\frac{\sqrt{1}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sqrt[3]{\sin k}}\right)} \cdot \frac{\left(\frac{2}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}\right) \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

20. Applied associate-*l* 14.0

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{1}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}} \cdot \left(\frac{\frac{\sqrt{1}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sqrt[3]{\sin k}} \cdot \frac{\left(\frac{2}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \sqrt[3]{\ell}\right) \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\right)}\]

21. Simplified 15.0

    \[\leadsto \frac{\frac{\sqrt{1}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}} \cdot \color{blue}{\frac{\frac{1}{\sqrt[3]{\sin k} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}} \cdot \left(\frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(\sqrt[3]{\ell} \cdot \frac{2}{{\left(\sqrt[3]{t}\right)}^{3}}\right)\right)}{\tan k}}\]
22. Using strategy rm

23. Applied associate-*l/ 14.6

    \[\leadsto \frac{\frac{\sqrt{1}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}} \cdot \frac{\frac{1}{\sqrt[3]{\sin k} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}} \cdot \color{blue}{\frac{\ell \cdot \left(\sqrt[3]{\ell} \cdot \frac{2}{{\left(\sqrt[3]{t}\right)}^{3}}\right)}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}{\tan k}\]

24. Applied frac-times 14.4

    \[\leadsto \frac{\frac{\sqrt{1}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}} \cdot \frac{\color{blue}{\frac{1 \cdot \left(\ell \cdot \left(\sqrt[3]{\ell} \cdot \frac{2}{{\left(\sqrt[3]{t}\right)}^{3}}\right)\right)}{\left(\sqrt[3]{\sin k} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}\right) \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}{\tan k}\]

25. Applied associate-/l/ 13.3

    \[\leadsto \frac{\frac{\sqrt{1}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}} \cdot \color{blue}{\frac{1 \cdot \left(\ell \cdot \left(\sqrt[3]{\ell} \cdot \frac{2}{{\left(\sqrt[3]{t}\right)}^{3}}\right)\right)}{\tan k \cdot \left(\left(\sqrt[3]{\sin k} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}\right) \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)\right)}}\]

26. Simplified 13.3

    \[\leadsto \frac{\frac{\sqrt{1}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}} \cdot \frac{1 \cdot \left(\ell \cdot \left(\sqrt[3]{\ell} \cdot \frac{2}{{\left(\sqrt[3]{t}\right)}^{3}}\right)\right)}{\color{blue}{\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\sin k}\right) \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \tan k}}\]
27. Using strategy rm

28. Applied add-cube-cbrt 13.4

    \[\leadsto \frac{\frac{\sqrt{1}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}} \cdot \frac{1 \cdot \left(\ell \cdot \left(\sqrt[3]{\ell} \cdot \frac{2}{{\left(\sqrt[3]{t}\right)}^{3}}\right)\right)}{\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\sin k}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\right) \cdot \tan k}\]

29. Applied associate-*r* 13.4

    \[\leadsto \frac{\frac{\sqrt{1}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}} \cdot \frac{1 \cdot \left(\ell \cdot \left(\sqrt[3]{\ell} \cdot \frac{2}{{\left(\sqrt[3]{t}\right)}^{3}}\right)\right)}{\color{blue}{\left(\left(\left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\sin k}\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \cdot \tan k}\]

30. Final simplification 13.4

    \[\leadsto \frac{\frac{1}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}} \cdot \frac{\ell \cdot \left(\sqrt[3]{\ell} \cdot \frac{2}{{\left(\sqrt[3]{t}\right)}^{3}}\right)}{\tan k \cdot \left(\left(\left(\sqrt[3]{\sin k} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\sqrt[3]{\ell}}\right) \cdot \left(\sqrt[3]{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right) \cdot \sqrt[3]{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\right)}\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))