Average Error: 32.0 → 11.4
Time: 1.1m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\frac{\left(\frac{\sqrt{\sqrt{2}}}{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \left(\frac{\sqrt{2}}{\sin k} \cdot \frac{\frac{\sqrt{\sqrt{2}}}{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}}{t}\right)\right) \cdot \frac{\ell}{t}}{\frac{\tan k}{\frac{\ell}{t}}}\]
\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{\left(\frac{\sqrt{\sqrt{2}}}{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}} \cdot \left(\frac{\sqrt{2}}{\sin k} \cdot \frac{\frac{\sqrt{\sqrt{2}}}{\sqrt{2 + \frac{k}{t} \cdot \frac{k}{t}}}}{t}\right)\right) \cdot \frac{\ell}{t}}{\frac{\tan k}{\frac{\ell}{t}}}
double f(double t, double l, double k) {
        double r3196788 = 2.0;
        double r3196789 = t;
        double r3196790 = 3.0;
        double r3196791 = pow(r3196789, r3196790);
        double r3196792 = l;
        double r3196793 = r3196792 * r3196792;
        double r3196794 = r3196791 / r3196793;
        double r3196795 = k;
        double r3196796 = sin(r3196795);
        double r3196797 = r3196794 * r3196796;
        double r3196798 = tan(r3196795);
        double r3196799 = r3196797 * r3196798;
        double r3196800 = 1.0;
        double r3196801 = r3196795 / r3196789;
        double r3196802 = pow(r3196801, r3196788);
        double r3196803 = r3196800 + r3196802;
        double r3196804 = r3196803 + r3196800;
        double r3196805 = r3196799 * r3196804;
        double r3196806 = r3196788 / r3196805;
        return r3196806;
}
↓
double f(double t, double l, double k) {
        double r3196807 = 2.0;
        double r3196808 = sqrt(r3196807);
        double r3196809 = sqrt(r3196808);
        double r3196810 = k;
        double r3196811 = t;
        double r3196812 = r3196810 / r3196811;
        double r3196813 = r3196812 * r3196812;
        double r3196814 = r3196807 + r3196813;
        double r3196815 = sqrt(r3196814);
        double r3196816 = r3196809 / r3196815;
        double r3196817 = sin(r3196810);
        double r3196818 = r3196808 / r3196817;
        double r3196819 = r3196816 / r3196811;
        double r3196820 = r3196818 * r3196819;
        double r3196821 = r3196816 * r3196820;
        double r3196822 = l;
        double r3196823 = r3196822 / r3196811;
        double r3196824 = r3196821 * r3196823;
        double r3196825 = tan(r3196810);
        double r3196826 = r3196825 / r3196823;
        double r3196827 = r3196824 / r3196826;
        return r3196827;
}

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 32.0

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

  2. Simplified17.0

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

  4. Applied associate-*l/16.2

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

  5. Applied associate-*r/14.9

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

  6. Applied associate-/r/13.5

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

  8. Applied *-un-lft-identity13.5

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

  9. Applied add-sqr-sqrt13.6

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

  10. Applied times-frac13.6

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

  11. Applied times-frac13.3

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

  12. Simplified13.3

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

  14. Applied add-sqr-sqrt13.2

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

  15. Applied add-sqr-sqrt13.3

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

  16. Applied times-frac13.2

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

  17. Applied times-frac11.9

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

  18. Applied associate-*r*11.9

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

  20. Applied associate-*r/12.3

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

  21. Applied associate-*l/11.4

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

  22. Final simplification11.4

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

Reproduce

herbie shell --seed 2019138 
(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))))