Average Error: 46.9 → 1.3
Time: 4.0m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{\frac{2}{\frac{\frac{k}{\ell}}{\frac{1}{t}}}}{\sin k}}{\frac{k}{\ell} \cdot \tan k}\]
\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{\frac{k}{\ell}}{\frac{1}{t}}}}{\sin k}}{\frac{k}{\ell} \cdot \tan k}
double f(double t, double l, double k) {
        double r8416127 = 2.0;
        double r8416128 = t;
        double r8416129 = 3.0;
        double r8416130 = pow(r8416128, r8416129);
        double r8416131 = l;
        double r8416132 = r8416131 * r8416131;
        double r8416133 = r8416130 / r8416132;
        double r8416134 = k;
        double r8416135 = sin(r8416134);
        double r8416136 = r8416133 * r8416135;
        double r8416137 = tan(r8416134);
        double r8416138 = r8416136 * r8416137;
        double r8416139 = 1.0;
        double r8416140 = r8416134 / r8416128;
        double r8416141 = pow(r8416140, r8416127);
        double r8416142 = r8416139 + r8416141;
        double r8416143 = r8416142 - r8416139;
        double r8416144 = r8416138 * r8416143;
        double r8416145 = r8416127 / r8416144;
        return r8416145;
}
↓
double f(double t, double l, double k) {
        double r8416146 = 2.0;
        double r8416147 = k;
        double r8416148 = l;
        double r8416149 = r8416147 / r8416148;
        double r8416150 = 1.0;
        double r8416151 = t;
        double r8416152 = r8416150 / r8416151;
        double r8416153 = r8416149 / r8416152;
        double r8416154 = r8416146 / r8416153;
        double r8416155 = sin(r8416147);
        double r8416156 = r8416154 / r8416155;
        double r8416157 = tan(r8416147);
        double r8416158 = r8416149 * r8416157;
        double r8416159 = r8416156 / r8416158;
        return r8416159;
}

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 46.9

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

  2. Simplified26.5

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

  3. Taylor expanded around inf 14.5

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

  4. Simplified14.5

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

  6. Applied *-un-lft-identity14.5

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

  7. Applied div-inv14.6

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

  8. Applied *-un-lft-identity14.6

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

  9. Applied times-frac10.6

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

  10. Applied times-frac3.0

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

  11. Applied *-un-lft-identity3.0

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

  12. Applied times-frac2.6

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

  13. Applied times-frac1.6

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

  14. Simplified1.6

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

  16. Applied associate-*l/1.6

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

  17. Applied associate-/l/1.3

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

  18. Final simplification1.3

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

Reproduce

herbie shell --seed 2019132 
(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))))