Average Error: 32.2 → 13.8
Time: 48.8s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\left(\frac{\frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}}{\sin k} \cdot \frac{\frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}}{\left(\frac{1}{\frac{\ell}{t}} \cdot \sin k\right) \cdot t}\right) \cdot \left(\cos k \cdot \frac{\ell}{t}\right)\]
\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)}
↓
\left(\frac{\frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}}{\sin k} \cdot \frac{\frac{\sqrt{2}}{\sqrt{\mathsf{fma}\left(\left(\frac{k}{t}\right), \left(\frac{k}{t}\right), 2\right)}}}{\left(\frac{1}{\frac{\ell}{t}} \cdot \sin k\right) \cdot t}\right) \cdot \left(\cos k \cdot \frac{\ell}{t}\right)
double f(double t, double l, double k) {
        double r1181335 = 2.0;
        double r1181336 = t;
        double r1181337 = 3.0;
        double r1181338 = pow(r1181336, r1181337);
        double r1181339 = l;
        double r1181340 = r1181339 * r1181339;
        double r1181341 = r1181338 / r1181340;
        double r1181342 = k;
        double r1181343 = sin(r1181342);
        double r1181344 = r1181341 * r1181343;
        double r1181345 = tan(r1181342);
        double r1181346 = r1181344 * r1181345;
        double r1181347 = 1.0;
        double r1181348 = r1181342 / r1181336;
        double r1181349 = pow(r1181348, r1181335);
        double r1181350 = r1181347 + r1181349;
        double r1181351 = r1181350 + r1181347;
        double r1181352 = r1181346 * r1181351;
        double r1181353 = r1181335 / r1181352;
        return r1181353;
}
↓
double f(double t, double l, double k) {
        double r1181354 = 2.0;
        double r1181355 = sqrt(r1181354);
        double r1181356 = k;
        double r1181357 = t;
        double r1181358 = r1181356 / r1181357;
        double r1181359 = fma(r1181358, r1181358, r1181354);
        double r1181360 = sqrt(r1181359);
        double r1181361 = r1181355 / r1181360;
        double r1181362 = sin(r1181356);
        double r1181363 = r1181361 / r1181362;
        double r1181364 = 1.0;
        double r1181365 = l;
        double r1181366 = r1181365 / r1181357;
        double r1181367 = r1181364 / r1181366;
        double r1181368 = r1181367 * r1181362;
        double r1181369 = r1181368 * r1181357;
        double r1181370 = r1181361 / r1181369;
        double r1181371 = r1181363 * r1181370;
        double r1181372 = cos(r1181356);
        double r1181373 = r1181372 * r1181366;
        double r1181374 = r1181371 * r1181373;
        return r1181374;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Initial program 32.2

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

  2. Simplified24.5

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

  4. Applied associate-*l*20.4

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

  6. Applied *-un-lft-identity20.4

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

  7. Applied times-frac19.7

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

  8. Applied associate-*r*17.5

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

  10. Applied tan-quot17.5

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

  11. Applied associate-*l/17.5

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

  12. Applied frac-times16.6

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

  13. Applied associate-*r/15.4

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

  14. Applied associate-/r/14.0

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

  16. Applied add-sqr-sqrt14.1

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

  17. Applied add-sqr-sqrt14.1

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

  18. Applied times-frac14.1

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

  19. Applied times-frac13.8

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

  20. Final simplification13.8

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

Reproduce

herbie shell --seed 2019128 +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))))