Average Error: 46.7 → 7.5
Time: 4.5m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\left(2 \cdot \frac{\cos k}{k \cdot \sin k}\right) \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\frac{1}{\frac{t}{\ell}}}{k}\]
\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}{k \cdot \sin k}\right) \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\frac{1}{\frac{t}{\ell}}}{k}
double f(double t, double l, double k) {
        double r5808762 = 2.0;
        double r5808763 = t;
        double r5808764 = 3.0;
        double r5808765 = pow(r5808763, r5808764);
        double r5808766 = l;
        double r5808767 = r5808766 * r5808766;
        double r5808768 = r5808765 / r5808767;
        double r5808769 = k;
        double r5808770 = sin(r5808769);
        double r5808771 = r5808768 * r5808770;
        double r5808772 = tan(r5808769);
        double r5808773 = r5808771 * r5808772;
        double r5808774 = 1.0;
        double r5808775 = r5808769 / r5808763;
        double r5808776 = pow(r5808775, r5808762);
        double r5808777 = r5808774 + r5808776;
        double r5808778 = r5808777 - r5808774;
        double r5808779 = r5808773 * r5808778;
        double r5808780 = r5808762 / r5808779;
        return r5808780;
}
↓
double f(double t, double l, double k) {
        double r5808781 = 2.0;
        double r5808782 = k;
        double r5808783 = cos(r5808782);
        double r5808784 = sin(r5808782);
        double r5808785 = r5808782 * r5808784;
        double r5808786 = r5808783 / r5808785;
        double r5808787 = r5808781 * r5808786;
        double r5808788 = l;
        double r5808789 = r5808788 / r5808784;
        double r5808790 = r5808787 * r5808789;
        double r5808791 = 1.0;
        double r5808792 = t;
        double r5808793 = r5808792 / r5808788;
        double r5808794 = r5808791 / r5808793;
        double r5808795 = r5808794 / r5808782;
        double r5808796 = r5808790 * r5808795;
        return r5808796;
}

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. Simplified30.5

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

  4. Applied *-un-lft-identity30.5

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

  5. Applied times-frac30.5

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

  6. Applied times-frac30.4

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

  7. Applied times-frac20.3

    \[\leadsto \color{blue}{\frac{\frac{\frac{1}{\frac{t}{\ell} \cdot \frac{t}{\ell}}}{\sin k}}{\frac{k}{t}} \cdot \frac{\frac{\frac{2}{t}}{\tan k}}{\frac{k}{t}}}\]
  8. Using strategy rm

  9. Applied div-inv20.3

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

  10. Applied *-un-lft-identity20.3

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

  11. Applied *-un-lft-identity20.3

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

  12. Applied times-frac20.2

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

  13. Applied times-frac19.3

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

  14. Applied times-frac13.2

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

  15. Applied associate-*l*12.0

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

  16. Taylor expanded around -inf 7.6

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

  17. Taylor expanded around inf 7.5

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

  18. Final simplification7.5

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

Reproduce

herbie shell --seed 2019152 
(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))))