Toniolo and Linder, Equation (10+)

Average Error: 32.0 → 21.7

Time: 17.8s

Precision: 64

Math

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

↓

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

C

double f(double t, double l, double k) {
        double r123484 = 2.0;
        double r123485 = t;
        double r123486 = 3.0;
        double r123487 = pow(r123485, r123486);
        double r123488 = l;
        double r123489 = r123488 * r123488;
        double r123490 = r123487 / r123489;
        double r123491 = k;
        double r123492 = sin(r123491);
        double r123493 = r123490 * r123492;
        double r123494 = tan(r123491);
        double r123495 = r123493 * r123494;
        double r123496 = 1.0;
        double r123497 = r123491 / r123485;
        double r123498 = pow(r123497, r123484);
        double r123499 = r123496 + r123498;
        double r123500 = r123499 + r123496;
        double r123501 = r123495 * r123500;
        double r123502 = r123484 / r123501;
        return r123502;
}

↓

double f(double t, double l, double k) {
        double r123503 = 1.0;
        double r123504 = sqrt(r123503);
        double r123505 = t;
        double r123506 = cbrt(r123505);
        double r123507 = 3.0;
        double r123508 = pow(r123506, r123507);
        double r123509 = r123504 / r123508;
        double r123510 = 2.0;
        double r123511 = k;
        double r123512 = sin(r123511);
        double r123513 = r123508 * r123512;
        double r123514 = r123510 / r123513;
        double r123515 = l;
        double r123516 = r123514 * r123515;
        double r123517 = r123516 / r123508;
        double r123518 = r123509 * r123517;
        double r123519 = tan(r123511);
        double r123520 = r123518 / r123519;
        double r123521 = 2.0;
        double r123522 = 1.0;
        double r123523 = r123511 / r123505;
        double r123524 = pow(r123523, r123510);
        double r123525 = fma(r123521, r123522, r123524);
        double r123526 = r123515 / r123525;
        double r123527 = r123520 * r123526;
        return r123527;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Initial program 32.0

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

2. Simplified 32.3

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

4. Applied *-un-lft-identity 32.3

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

5. Applied times-frac 31.3

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

6. Applied associate-*r* 28.8

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

7. Simplified 27.5

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

9. Applied add-cube-cbrt 27.7

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

10. Applied unpow-prod-down 27.8

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

11. Applied associate-*l* 26.7

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

13. Applied *-un-lft-identity 26.7

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

14. Applied times-frac 26.6

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

15. Applied associate-*l* 23.9

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

17. Applied unpow-prod-down 23.9

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

18. Applied add-sqr-sqrt 23.9

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

19. Applied times-frac 23.7

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

20. Applied associate-*l* 21.7

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

21. Simplified 21.7

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

22. Final simplification 21.7

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

Reproduce

herbie shell --seed 2020027 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))