Average Error: 32.1 → 19.5
Time: 24.5s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\frac{2}{\left(\left(\tan k \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}{\ell}\right)\right) \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \ell\]
\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{2}{\left(\left(\tan k \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}\right) \cdot \left({\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3} \cdot \sin k}{\ell}\right)\right) \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \ell
double f(double t, double l, double k) {
        double r96300 = 2.0;
        double r96301 = t;
        double r96302 = 3.0;
        double r96303 = pow(r96301, r96302);
        double r96304 = l;
        double r96305 = r96304 * r96304;
        double r96306 = r96303 / r96305;
        double r96307 = k;
        double r96308 = sin(r96307);
        double r96309 = r96306 * r96308;
        double r96310 = tan(r96307);
        double r96311 = r96309 * r96310;
        double r96312 = 1.0;
        double r96313 = r96307 / r96301;
        double r96314 = pow(r96313, r96300);
        double r96315 = r96312 + r96314;
        double r96316 = r96315 + r96312;
        double r96317 = r96311 * r96316;
        double r96318 = r96300 / r96317;
        return r96318;
}
↓
double f(double t, double l, double k) {
        double r96319 = 2.0;
        double r96320 = k;
        double r96321 = tan(r96320);
        double r96322 = t;
        double r96323 = cbrt(r96322);
        double r96324 = r96323 * r96323;
        double r96325 = 3.0;
        double r96326 = 2.0;
        double r96327 = r96325 / r96326;
        double r96328 = pow(r96324, r96327);
        double r96329 = r96321 * r96328;
        double r96330 = pow(r96323, r96325);
        double r96331 = sin(r96320);
        double r96332 = r96330 * r96331;
        double r96333 = l;
        double r96334 = r96332 / r96333;
        double r96335 = r96328 * r96334;
        double r96336 = r96329 * r96335;
        double r96337 = 1.0;
        double r96338 = r96320 / r96322;
        double r96339 = pow(r96338, r96319);
        double r96340 = fma(r96326, r96337, r96339);
        double r96341 = r96336 * r96340;
        double r96342 = r96319 / r96341;
        double r96343 = r96342 * r96333;
        return r96343;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Initial program 32.1

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

  2. Simplified27.6

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

  4. Applied add-cube-cbrt27.9

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

  5. Applied unpow-prod-down27.9

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

  6. Applied associate-*l*26.8

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

  8. Applied *-un-lft-identity26.8

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

  9. Applied times-frac23.9

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

  10. Simplified23.9

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

  12. Applied sqr-pow23.9

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

  13. Applied associate-*l*21.7

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

  15. Applied associate-*r*19.5

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

  16. Final simplification19.5

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

Reproduce

herbie shell --seed 2020045 +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))))