Average Error: 46.7 → 7.3
Time: 4.8m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\left(2 \cdot \frac{\cos k}{\sin k \cdot k}\right) \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\frac{\ell}{t}}{k}\]
\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)}
↓
\left(\left(2 \cdot \frac{\cos k}{\sin k \cdot k}\right) \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\frac{\ell}{t}}{k}
double f(double t, double l, double k) {
        double r6670437 = 2.0;
        double r6670438 = t;
        double r6670439 = 3.0;
        double r6670440 = pow(r6670438, r6670439);
        double r6670441 = l;
        double r6670442 = r6670441 * r6670441;
        double r6670443 = r6670440 / r6670442;
        double r6670444 = k;
        double r6670445 = sin(r6670444);
        double r6670446 = r6670443 * r6670445;
        double r6670447 = tan(r6670444);
        double r6670448 = r6670446 * r6670447;
        double r6670449 = 1.0;
        double r6670450 = r6670444 / r6670438;
        double r6670451 = pow(r6670450, r6670437);
        double r6670452 = r6670449 + r6670451;
        double r6670453 = r6670452 - r6670449;
        double r6670454 = r6670448 * r6670453;
        double r6670455 = r6670437 / r6670454;
        return r6670455;
}
↓
double f(double t, double l, double k) {
        double r6670456 = 2.0;
        double r6670457 = k;
        double r6670458 = cos(r6670457);
        double r6670459 = sin(r6670457);
        double r6670460 = r6670459 * r6670457;
        double r6670461 = r6670458 / r6670460;
        double r6670462 = r6670456 * r6670461;
        double r6670463 = l;
        double r6670464 = r6670463 / r6670459;
        double r6670465 = r6670462 * r6670464;
        double r6670466 = t;
        double r6670467 = r6670463 / r6670466;
        double r6670468 = r6670467 / r6670457;
        double r6670469 = r6670465 * r6670468;
        return r6670469;
}

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 46.7

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

  2. Simplified30.2

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

  4. Applied times-frac30.1

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

  5. Applied times-frac19.9

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

  7. Applied div-inv19.9

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

  8. Applied *-un-lft-identity19.9

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

  9. Applied times-frac19.0

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

  10. Applied times-frac13.1

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

  11. Applied associate-*l*11.6

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

  12. Taylor expanded around -inf 7.3

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

  13. Taylor expanded around -inf 7.3

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

  14. Final simplification7.3

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

Reproduce

herbie shell --seed 2019134 +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))))