Average Error: 32.7 → 21.9
Time: 18.7s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\frac{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \left(\frac{\sqrt{2}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k} \cdot \ell\right)\right)}{\tan k} \cdot \frac{\ell}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}\]
\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}{{\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)}
double f(double t, double l, double k) {
        double r143684 = 2.0;
        double r143685 = t;
        double r143686 = 3.0;
        double r143687 = pow(r143685, r143686);
        double r143688 = l;
        double r143689 = r143688 * r143688;
        double r143690 = r143687 / r143689;
        double r143691 = k;
        double r143692 = sin(r143691);
        double r143693 = r143690 * r143692;
        double r143694 = tan(r143691);
        double r143695 = r143693 * r143694;
        double r143696 = 1.0;
        double r143697 = r143691 / r143685;
        double r143698 = pow(r143697, r143684);
        double r143699 = r143696 + r143698;
        double r143700 = r143699 + r143696;
        double r143701 = r143695 * r143700;
        double r143702 = r143684 / r143701;
        return r143702;
}
↓
double f(double t, double l, double k) {
        double r143703 = 1.0;
        double r143704 = t;
        double r143705 = cbrt(r143704);
        double r143706 = 3.0;
        double r143707 = pow(r143705, r143706);
        double r143708 = r143703 / r143707;
        double r143709 = 2.0;
        double r143710 = sqrt(r143709);
        double r143711 = r143710 / r143707;
        double r143712 = k;
        double r143713 = sin(r143712);
        double r143714 = r143707 * r143713;
        double r143715 = r143710 / r143714;
        double r143716 = l;
        double r143717 = r143715 * r143716;
        double r143718 = r143711 * r143717;
        double r143719 = r143708 * r143718;
        double r143720 = tan(r143712);
        double r143721 = r143719 / r143720;
        double r143722 = 2.0;
        double r143723 = 1.0;
        double r143724 = r143712 / r143704;
        double r143725 = pow(r143724, r143709);
        double r143726 = fma(r143722, r143723, r143725);
        double r143727 = r143716 / r143726;
        double r143728 = r143721 * r143727;
        return r143728;
}

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. Simplified32.7

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

  4. Applied *-un-lft-identity32.7

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

  5. Applied times-frac31.9

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

  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. Simplified28.1

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

  9. Applied add-cube-cbrt28.4

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

  10. Applied unpow-prod-down28.4

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

  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-sqrt27.2

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

  14. Applied times-frac27.1

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

  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-down24.1

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

  18. Applied *-un-lft-identity24.1

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

  19. Applied times-frac23.9

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

  20. Applied associate-*l*21.9

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

  21. Final simplification21.9

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

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))))