Average Error: 47.9 → 11.1
Time: 31.9s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\cos k}{\frac{\sin k}{\ell}}\right) \cdot \frac{\ell}{\sin k}\right)\]
\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)}
↓
2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\cos k}{\frac{\sin k}{\ell}}\right) \cdot \frac{\ell}{\sin k}\right)
double f(double t, double l, double k) {
        double r117307 = 2.0;
        double r117308 = t;
        double r117309 = 3.0;
        double r117310 = pow(r117308, r117309);
        double r117311 = l;
        double r117312 = r117311 * r117311;
        double r117313 = r117310 / r117312;
        double r117314 = k;
        double r117315 = sin(r117314);
        double r117316 = r117313 * r117315;
        double r117317 = tan(r117314);
        double r117318 = r117316 * r117317;
        double r117319 = 1.0;
        double r117320 = r117314 / r117308;
        double r117321 = pow(r117320, r117307);
        double r117322 = r117319 + r117321;
        double r117323 = r117322 - r117319;
        double r117324 = r117318 * r117323;
        double r117325 = r117307 / r117324;
        return r117325;
}
↓
double f(double t, double l, double k) {
        double r117326 = 2.0;
        double r117327 = 1.0;
        double r117328 = k;
        double r117329 = 2.0;
        double r117330 = r117326 / r117329;
        double r117331 = pow(r117328, r117330);
        double r117332 = t;
        double r117333 = 1.0;
        double r117334 = pow(r117332, r117333);
        double r117335 = r117331 * r117334;
        double r117336 = r117331 * r117335;
        double r117337 = r117327 / r117336;
        double r117338 = pow(r117337, r117333);
        double r117339 = cos(r117328);
        double r117340 = sin(r117328);
        double r117341 = l;
        double r117342 = r117340 / r117341;
        double r117343 = r117339 / r117342;
        double r117344 = r117338 * r117343;
        double r117345 = r117341 / r117340;
        double r117346 = r117344 * r117345;
        double r117347 = r117326 * r117346;
        return r117347;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 47.9

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

  2. Simplified40.1

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

  3. Taylor expanded around inf 21.8

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

  5. Applied sqr-pow21.8

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

  6. Applied associate-*l*19.8

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

  8. Applied unpow219.8

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

  9. Applied associate-/r*19.5

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

  10. Simplified17.9

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

  12. Applied *-un-lft-identity17.9

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

  13. Applied associate-/r/17.7

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

  14. Applied times-frac16.5

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

  15. Applied associate-*r*11.1

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

  16. Simplified11.1

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

  17. Final simplification11.1

    \[\leadsto 2 \cdot \left(\left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \frac{\cos k}{\frac{\sin k}{\ell}}\right) \cdot \frac{\ell}{\sin k}\right)\]

Reproduce

herbie shell --seed 2020002 +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))))