Toniolo and Linder, Equation (10+)

Average Error: 32.7 → 21.9

Time: 18.0s

Precision: 64

Math

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

↓

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

C

double f(double t, double l, double k) {
        double r137480 = 2.0;
        double r137481 = t;
        double r137482 = 3.0;
        double r137483 = pow(r137481, r137482);
        double r137484 = l;
        double r137485 = r137484 * r137484;
        double r137486 = r137483 / r137485;
        double r137487 = k;
        double r137488 = sin(r137487);
        double r137489 = r137486 * r137488;
        double r137490 = tan(r137487);
        double r137491 = r137489 * r137490;
        double r137492 = 1.0;
        double r137493 = r137487 / r137481;
        double r137494 = pow(r137493, r137480);
        double r137495 = r137492 + r137494;
        double r137496 = r137495 + r137492;
        double r137497 = r137491 * r137496;
        double r137498 = r137480 / r137497;
        return r137498;
}

↓

double f(double t, double l, double k) {
        double r137499 = 1.0;
        double r137500 = sqrt(r137499);
        double r137501 = t;
        double r137502 = cbrt(r137501);
        double r137503 = 3.0;
        double r137504 = pow(r137502, r137503);
        double r137505 = r137500 / r137504;
        double r137506 = 2.0;
        double r137507 = sqrt(r137506);
        double r137508 = r137507 / r137504;
        double r137509 = k;
        double r137510 = sin(r137509);
        double r137511 = r137504 * r137510;
        double r137512 = r137507 / r137511;
        double r137513 = l;
        double r137514 = r137512 * r137513;
        double r137515 = r137508 * r137514;
        double r137516 = r137505 * r137515;
        double r137517 = tan(r137509);
        double r137518 = r137516 / r137517;
        double r137519 = 2.0;
        double r137520 = 1.0;
        double r137521 = r137509 / r137501;
        double r137522 = pow(r137521, r137506);
        double r137523 = fma(r137519, r137520, r137522);
        double r137524 = r137513 / r137523;
        double r137525 = r137518 * r137524;
        return r137525;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Initial program 32.7

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

2. Simplified 32.7

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

4. Applied *-un-lft-identity 32.7

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

5. Applied times-frac 31.9

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

6. Applied associate-*r* 29.4

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

7. Simplified 28.1

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

9. Applied add-cube-cbrt 28.4

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

10. Applied unpow-prod-down 28.4

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

11. Applied associate-*l* 27.2

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

13. Applied add-sqr-sqrt 27.2

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

14. Applied times-frac 27.1

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

15. Applied associate-*l* 24.1

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

17. Applied unpow-prod-down 24.1

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

18. Applied *-un-lft-identity 24.1

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

19. Applied sqrt-prod 24.1

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

20. Applied times-frac 23.9

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

21. Applied associate-*l* 21.9

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

22. Final simplification 21.9

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

Reproduce

herbie shell --seed 2020056 +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))))