Average Error: 31.6 → 12.5
Time: 4.9m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\frac{\sqrt{\sqrt{2}}}{\sqrt[3]{t}} \cdot \left(\frac{\frac{\sqrt{\sqrt{2}}}{\frac{\sqrt[3]{t}}{\frac{\ell}{t}}}}{\tan k} \cdot \frac{\frac{\sqrt{2}}{\sqrt[3]{\frac{\sqrt[3]{t}}{\frac{\ell}{t}} \cdot \sin k} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{t}}{\frac{\ell}{t}} \cdot \sin k} \cdot \sqrt[3]{\frac{\sqrt[3]{t}}{\frac{\ell}{t}} \cdot \sin k}\right)}}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\right)\]
\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{\sqrt{\sqrt{2}}}{\sqrt[3]{t}} \cdot \left(\frac{\frac{\sqrt{\sqrt{2}}}{\frac{\sqrt[3]{t}}{\frac{\ell}{t}}}}{\tan k} \cdot \frac{\frac{\sqrt{2}}{\sqrt[3]{\frac{\sqrt[3]{t}}{\frac{\ell}{t}} \cdot \sin k} \cdot \left(\sqrt[3]{\frac{\sqrt[3]{t}}{\frac{\ell}{t}} \cdot \sin k} \cdot \sqrt[3]{\frac{\sqrt[3]{t}}{\frac{\ell}{t}} \cdot \sin k}\right)}}{\mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 2\right)}\right)
double f(double t, double l, double k) {
        double r6012160 = 2.0;
        double r6012161 = t;
        double r6012162 = 3.0;
        double r6012163 = pow(r6012161, r6012162);
        double r6012164 = l;
        double r6012165 = r6012164 * r6012164;
        double r6012166 = r6012163 / r6012165;
        double r6012167 = k;
        double r6012168 = sin(r6012167);
        double r6012169 = r6012166 * r6012168;
        double r6012170 = tan(r6012167);
        double r6012171 = r6012169 * r6012170;
        double r6012172 = 1.0;
        double r6012173 = r6012167 / r6012161;
        double r6012174 = pow(r6012173, r6012160);
        double r6012175 = r6012172 + r6012174;
        double r6012176 = r6012175 + r6012172;
        double r6012177 = r6012171 * r6012176;
        double r6012178 = r6012160 / r6012177;
        return r6012178;
}
↓
double f(double t, double l, double k) {
        double r6012179 = 2.0;
        double r6012180 = sqrt(r6012179);
        double r6012181 = sqrt(r6012180);
        double r6012182 = t;
        double r6012183 = cbrt(r6012182);
        double r6012184 = r6012181 / r6012183;
        double r6012185 = l;
        double r6012186 = r6012185 / r6012182;
        double r6012187 = r6012183 / r6012186;
        double r6012188 = r6012181 / r6012187;
        double r6012189 = k;
        double r6012190 = tan(r6012189);
        double r6012191 = r6012188 / r6012190;
        double r6012192 = sin(r6012189);
        double r6012193 = r6012187 * r6012192;
        double r6012194 = cbrt(r6012193);
        double r6012195 = r6012194 * r6012194;
        double r6012196 = r6012194 * r6012195;
        double r6012197 = r6012180 / r6012196;
        double r6012198 = r6012189 / r6012182;
        double r6012199 = fma(r6012198, r6012198, r6012179);
        double r6012200 = r6012197 / r6012199;
        double r6012201 = r6012191 * r6012200;
        double r6012202 = r6012184 * r6012201;
        return r6012202;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Initial program 31.6

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

  2. Simplified20.3

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

  4. Applied add-cube-cbrt20.5

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

  5. Applied times-frac19.8

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

  6. Applied associate-*l*16.7

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

  8. Applied add-sqr-sqrt16.8

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

  9. Applied times-frac16.6

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

  10. Applied times-frac13.3

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

  12. Applied *-un-lft-identity13.3

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

  13. Applied *-un-lft-identity13.3

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

  14. Applied times-frac13.3

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

  15. Applied add-sqr-sqrt13.3

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

  16. Applied sqrt-prod13.3

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

  17. Applied times-frac13.2

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

  18. Applied times-frac12.4

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

  19. Applied associate-*l*12.4

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

  21. Applied add-cube-cbrt12.5

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

  22. Final simplification12.5

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

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))