Average Error: 46.3 → 6.9
Time: 1.9m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\frac{\ell}{t} \cdot \left(\frac{\cos k}{\sin k \cdot k} \cdot 2\right)\right) \cdot \frac{\ell}{\sin k \cdot k}\]
\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(\frac{\ell}{t} \cdot \left(\frac{\cos k}{\sin k \cdot k} \cdot 2\right)\right) \cdot \frac{\ell}{\sin k \cdot k}
double f(double t, double l, double k) {
        double r2455679 = 2.0;
        double r2455680 = t;
        double r2455681 = 3.0;
        double r2455682 = pow(r2455680, r2455681);
        double r2455683 = l;
        double r2455684 = r2455683 * r2455683;
        double r2455685 = r2455682 / r2455684;
        double r2455686 = k;
        double r2455687 = sin(r2455686);
        double r2455688 = r2455685 * r2455687;
        double r2455689 = tan(r2455686);
        double r2455690 = r2455688 * r2455689;
        double r2455691 = 1.0;
        double r2455692 = r2455686 / r2455680;
        double r2455693 = pow(r2455692, r2455679);
        double r2455694 = r2455691 + r2455693;
        double r2455695 = r2455694 - r2455691;
        double r2455696 = r2455690 * r2455695;
        double r2455697 = r2455679 / r2455696;
        return r2455697;
}
↓
double f(double t, double l, double k) {
        double r2455698 = l;
        double r2455699 = t;
        double r2455700 = r2455698 / r2455699;
        double r2455701 = k;
        double r2455702 = cos(r2455701);
        double r2455703 = sin(r2455701);
        double r2455704 = r2455703 * r2455701;
        double r2455705 = r2455702 / r2455704;
        double r2455706 = 2.0;
        double r2455707 = r2455705 * r2455706;
        double r2455708 = r2455700 * r2455707;
        double r2455709 = r2455698 / r2455704;
        double r2455710 = r2455708 * r2455709;
        return r2455710;
}

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 46.3

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

  2. Simplified29.5

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

  4. Applied times-frac19.6

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

  6. Applied *-un-lft-identity19.6

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

  7. Applied *-un-lft-identity19.6

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

  8. Applied times-frac18.9

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

  9. Applied times-frac13.0

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

  10. Applied associate-*r*11.7

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

  11. Taylor expanded around inf 10.9

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

  12. Taylor expanded around inf 6.9

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

  13. Final simplification6.9

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

Reproduce

herbie shell --seed 2019155 +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))))