Toniolo and Linder, Equation (10+)

Average Error: 33.1 → 12.4

Time: 3.2m

Precision: 64

Math

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

↓

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

C

double f(double t, double l, double k) {
        double r8516582 = 2.0;
        double r8516583 = t;
        double r8516584 = 3.0;
        double r8516585 = pow(r8516583, r8516584);
        double r8516586 = l;
        double r8516587 = r8516586 * r8516586;
        double r8516588 = r8516585 / r8516587;
        double r8516589 = k;
        double r8516590 = sin(r8516589);
        double r8516591 = r8516588 * r8516590;
        double r8516592 = tan(r8516589);
        double r8516593 = r8516591 * r8516592;
        double r8516594 = 1.0;
        double r8516595 = r8516589 / r8516583;
        double r8516596 = pow(r8516595, r8516582);
        double r8516597 = r8516594 + r8516596;
        double r8516598 = r8516597 + r8516594;
        double r8516599 = r8516593 * r8516598;
        double r8516600 = r8516582 / r8516599;
        return r8516600;
}

↓

double f(double t, double l, double k) {
        double r8516601 = 1.0;
        double r8516602 = t;
        double r8516603 = cbrt(r8516602);
        double r8516604 = 3.0;
        double r8516605 = pow(r8516603, r8516604);
        double r8516606 = l;
        double r8516607 = cbrt(r8516606);
        double r8516608 = r8516605 / r8516607;
        double r8516609 = r8516601 / r8516608;
        double r8516610 = k;
        double r8516611 = sin(r8516610);
        double r8516612 = cbrt(r8516611);
        double r8516613 = r8516612 * r8516612;
        double r8516614 = r8516609 / r8516613;
        double r8516615 = r8516609 / r8516612;
        double r8516616 = 2.0;
        double r8516617 = 1.0;
        double r8516618 = r8516610 / r8516602;
        double r8516619 = 2.0;
        double r8516620 = pow(r8516618, r8516619);
        double r8516621 = fma(r8516616, r8516617, r8516620);
        double r8516622 = cbrt(r8516621);
        double r8516623 = r8516622 * r8516622;
        double r8516624 = r8516601 / r8516623;
        double r8516625 = r8516619 / r8516608;
        double r8516626 = tan(r8516610);
        double r8516627 = r8516625 / r8516626;
        double r8516628 = r8516624 * r8516627;
        double r8516629 = r8516606 / r8516622;
        double r8516630 = r8516628 * r8516629;
        double r8516631 = r8516615 * r8516630;
        double r8516632 = r8516614 * r8516631;
        return r8516632;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Initial program 33.1

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

2. Simplified 32.7

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

4. Applied add-cube-cbrt 32.8

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

5. Applied add-cube-cbrt 32.9

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

6. Applied unpow-prod-down 32.9

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

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 24.8

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

11. Applied associate-*l* 22.8

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

13. Applied add-cube-cbrt 22.8

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

14. Applied unpow-prod-down 22.8

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

15. Applied times-frac 18.8

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

16. Applied *-un-lft-identity 18.8

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

17. Applied times-frac 18.5

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

18. Applied times-frac 15.8

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

19. Applied associate-*l* 13.0

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

21. Applied add-cube-cbrt 13.0

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

22. Applied *-un-lft-identity 13.0

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

23. Applied times-frac 13.0

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

24. Applied associate-*r* 12.4

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

25. Final simplification 12.4

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

Reproduce

herbie shell --seed 2019174 +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))))