Average Error: 48.2 → 11.5
Time: 1.4m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\left(2 \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}\right)}\right)}^{1}\right) \cdot \left(\frac{1}{\sin k} \cdot \left(\ell \cdot \cos k\right)\right)\right) \cdot \frac{1}{\frac{\sin k}{\ell}}\]
\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)}
↓
\left(\left(2 \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}\right)}\right)}^{1}\right) \cdot \left(\frac{1}{\sin k} \cdot \left(\ell \cdot \cos k\right)\right)\right) \cdot \frac{1}{\frac{\sin k}{\ell}}
double f(double t, double l, double k) {
        double r8413303 = 2.0;
        double r8413304 = t;
        double r8413305 = 3.0;
        double r8413306 = pow(r8413304, r8413305);
        double r8413307 = l;
        double r8413308 = r8413307 * r8413307;
        double r8413309 = r8413306 / r8413308;
        double r8413310 = k;
        double r8413311 = sin(r8413310);
        double r8413312 = r8413309 * r8413311;
        double r8413313 = tan(r8413310);
        double r8413314 = r8413312 * r8413313;
        double r8413315 = 1.0;
        double r8413316 = r8413310 / r8413304;
        double r8413317 = pow(r8413316, r8413303);
        double r8413318 = r8413315 + r8413317;
        double r8413319 = r8413318 - r8413315;
        double r8413320 = r8413314 * r8413319;
        double r8413321 = r8413303 / r8413320;
        return r8413321;
}
↓
double f(double t, double l, double k) {
        double r8413322 = 2.0;
        double r8413323 = 1.0;
        double r8413324 = k;
        double r8413325 = 2.0;
        double r8413326 = r8413322 / r8413325;
        double r8413327 = pow(r8413324, r8413326);
        double r8413328 = t;
        double r8413329 = 1.0;
        double r8413330 = pow(r8413328, r8413329);
        double r8413331 = r8413330 * r8413327;
        double r8413332 = r8413327 * r8413331;
        double r8413333 = r8413323 / r8413332;
        double r8413334 = pow(r8413333, r8413329);
        double r8413335 = r8413322 * r8413334;
        double r8413336 = sin(r8413324);
        double r8413337 = r8413323 / r8413336;
        double r8413338 = l;
        double r8413339 = cos(r8413324);
        double r8413340 = r8413338 * r8413339;
        double r8413341 = r8413337 * r8413340;
        double r8413342 = r8413335 * r8413341;
        double r8413343 = r8413336 / r8413338;
        double r8413344 = r8413323 / r8413343;
        double r8413345 = r8413342 * r8413344;
        return r8413345;
}

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 48.2

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

  2. Simplified40.3

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

  3. Taylor expanded around inf 21.9

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

  4. Simplified20.4

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

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

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

  7. Applied times-frac20.2

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

  8. Applied associate-*l*16.1

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

  10. Applied sqr-pow16.1

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

  11. Applied associate-*r*11.5

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

  13. Applied div-inv11.5

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

  14. Applied *-un-lft-identity11.5

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

  15. Applied times-frac11.5

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

  16. Simplified11.5

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

  17. Final simplification11.5

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

Reproduce

herbie shell --seed 2019169 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))