Average Error: 46.8 → 1.7
Time: 1.4m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{\frac{-2}{\frac{k}{\ell}}}{\frac{k}{\ell} \cdot \left(\sin k \cdot t\right)}}{-\tan k}\]
\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{\frac{\frac{-2}{\frac{k}{\ell}}}{\frac{k}{\ell} \cdot \left(\sin k \cdot t\right)}}{-\tan k}
double f(double t, double l, double k) {
        double r2454446 = 2.0;
        double r2454447 = t;
        double r2454448 = 3.0;
        double r2454449 = pow(r2454447, r2454448);
        double r2454450 = l;
        double r2454451 = r2454450 * r2454450;
        double r2454452 = r2454449 / r2454451;
        double r2454453 = k;
        double r2454454 = sin(r2454453);
        double r2454455 = r2454452 * r2454454;
        double r2454456 = tan(r2454453);
        double r2454457 = r2454455 * r2454456;
        double r2454458 = 1.0;
        double r2454459 = r2454453 / r2454447;
        double r2454460 = pow(r2454459, r2454446);
        double r2454461 = r2454458 + r2454460;
        double r2454462 = r2454461 - r2454458;
        double r2454463 = r2454457 * r2454462;
        double r2454464 = r2454446 / r2454463;
        return r2454464;
}
↓
double f(double t, double l, double k) {
        double r2454465 = -2.0;
        double r2454466 = k;
        double r2454467 = l;
        double r2454468 = r2454466 / r2454467;
        double r2454469 = r2454465 / r2454468;
        double r2454470 = sin(r2454466);
        double r2454471 = t;
        double r2454472 = r2454470 * r2454471;
        double r2454473 = r2454468 * r2454472;
        double r2454474 = r2454469 / r2454473;
        double r2454475 = tan(r2454466);
        double r2454476 = -r2454475;
        double r2454477 = r2454474 / r2454476;
        return r2454477;
}

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 46.8

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

  2. Simplified29.9

    \[\leadsto \color{blue}{\frac{\frac{2}{\frac{\frac{k}{t} \cdot \frac{k}{t}}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{t}} \cdot \sin k}}{\tan k}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity29.9

    \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t} \cdot \frac{k}{t}}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\color{blue}{1 \cdot t}}} \cdot \sin k}}{\tan k}\]

  5. Applied times-frac29.3

    \[\leadsto \frac{\frac{2}{\frac{\frac{k}{t} \cdot \frac{k}{t}}{\color{blue}{\frac{\frac{\ell}{t}}{1} \cdot \frac{\frac{\ell}{t}}{t}}} \cdot \sin k}}{\tan k}\]

  6. Applied times-frac18.4

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

  7. Simplified18.4

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

  8. Simplified11.4

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

  9. Taylor expanded around -inf 11.3

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

  10. Taylor expanded around -inf 2.8

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

  12. Applied frac-2neg2.8

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

  13. Simplified1.7

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

  14. Final simplification1.7

    \[\leadsto \frac{\frac{\frac{-2}{\frac{k}{\ell}}}{\frac{k}{\ell} \cdot \left(\sin k \cdot t\right)}}{-\tan k}\]

Reproduce

herbie shell --seed 2019133 +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))))