Average Error: 32.9 → 15.6
Time: 1.3m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\frac{\frac{\frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}}}{\tan k}\]
\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{\frac{2}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\frac{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}}}{\tan k}
double f(double t, double l, double k) {
        double r227987 = 2.0;
        double r227988 = t;
        double r227989 = 3.0;
        double r227990 = pow(r227988, r227989);
        double r227991 = l;
        double r227992 = r227991 * r227991;
        double r227993 = r227990 / r227992;
        double r227994 = k;
        double r227995 = sin(r227994);
        double r227996 = r227993 * r227995;
        double r227997 = tan(r227994);
        double r227998 = r227996 * r227997;
        double r227999 = 1.0;
        double r228000 = r227994 / r227988;
        double r228001 = pow(r228000, r227987);
        double r228002 = r227999 + r228001;
        double r228003 = r228002 + r227999;
        double r228004 = r227998 * r228003;
        double r228005 = r227987 / r228004;
        return r228005;
}
↓
double f(double t, double l, double k) {
        double r228006 = 2.0;
        double r228007 = t;
        double r228008 = cbrt(r228007);
        double r228009 = r228008 * r228008;
        double r228010 = 3.0;
        double r228011 = 2.0;
        double r228012 = r228010 / r228011;
        double r228013 = pow(r228009, r228012);
        double r228014 = pow(r228008, r228010);
        double r228015 = l;
        double r228016 = r228014 / r228015;
        double r228017 = k;
        double r228018 = sin(r228017);
        double r228019 = r228016 * r228018;
        double r228020 = r228013 * r228019;
        double r228021 = r228006 / r228020;
        double r228022 = 1.0;
        double r228023 = r228017 / r228007;
        double r228024 = pow(r228023, r228006);
        double r228025 = fma(r228011, r228022, r228024);
        double r228026 = r228015 / r228013;
        double r228027 = r228025 / r228026;
        double r228028 = r228021 / r228027;
        double r228029 = tan(r228017);
        double r228030 = r228028 / r228029;
        return r228030;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Initial program 32.9

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

  2. Simplified32.9

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

  4. Applied add-cube-cbrt33.0

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

  5. Applied unpow-prod-down33.0

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

  6. Applied times-frac25.3

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

  8. Applied sqr-pow25.3

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

  9. Applied associate-/l*20.5

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

  11. Applied associate-*l/20.5

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

  12. Applied associate-*l/18.3

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

  13. Applied associate-/r/18.2

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

  14. Applied associate-/l*16.7

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

  16. Applied associate-*l*15.6

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

  17. Final simplification15.6

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

Reproduce

herbie shell --seed 2019212 +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))))