Average Error: 47.2 → 1.0
Time: 4.9m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}} \cdot \left(\frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\ell}} \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}\right)}{\frac{k}{\ell} \cdot \sin k}\]
\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{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}} \cdot \left(\frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\ell}} \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}\right)}{\frac{k}{\ell} \cdot \sin k}
double f(double t, double l, double k) {
        double r11454003 = 2.0;
        double r11454004 = t;
        double r11454005 = 3.0;
        double r11454006 = pow(r11454004, r11454005);
        double r11454007 = l;
        double r11454008 = r11454007 * r11454007;
        double r11454009 = r11454006 / r11454008;
        double r11454010 = k;
        double r11454011 = sin(r11454010);
        double r11454012 = r11454009 * r11454011;
        double r11454013 = tan(r11454010);
        double r11454014 = r11454012 * r11454013;
        double r11454015 = 1.0;
        double r11454016 = r11454010 / r11454004;
        double r11454017 = pow(r11454016, r11454003);
        double r11454018 = r11454015 + r11454017;
        double r11454019 = r11454018 - r11454015;
        double r11454020 = r11454014 * r11454019;
        double r11454021 = r11454003 / r11454020;
        return r11454021;
}
↓
double f(double t, double l, double k) {
        double r11454022 = 2.0;
        double r11454023 = t;
        double r11454024 = r11454022 / r11454023;
        double r11454025 = cbrt(r11454024);
        double r11454026 = k;
        double r11454027 = tan(r11454026);
        double r11454028 = cbrt(r11454027);
        double r11454029 = r11454025 / r11454028;
        double r11454030 = l;
        double r11454031 = r11454026 / r11454030;
        double r11454032 = r11454029 / r11454031;
        double r11454033 = r11454032 * r11454029;
        double r11454034 = r11454029 * r11454033;
        double r11454035 = sin(r11454026);
        double r11454036 = r11454031 * r11454035;
        double r11454037 = r11454034 / r11454036;
        return r11454037;
}

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 47.2

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

  2. Simplified30.6

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

  4. Applied associate-/l*28.0

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

  5. Simplified15.0

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

  7. Applied add-cube-cbrt15.2

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

  8. Applied add-cube-cbrt15.3

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

  9. Applied times-frac15.3

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

  10. Applied times-frac14.7

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{t}} \cdot \sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{\tan k}}}{\frac{\frac{k}{t}}{\frac{\ell}{t}} \cdot \frac{\frac{k}{t}}{\frac{\ell}{t}}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\sin k}}\]

  11. Simplified2.7

    \[\leadsto \color{blue}{\left(\frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\frac{\ell}{1}}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\frac{\ell}{1}}}\right)} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\sin k}\]
  12. Using strategy rm

  13. Applied associate-*l/2.7

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\frac{\ell}{1}}}}{\frac{k}{\frac{\ell}{1}}}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\sin k}\]

  14. Applied frac-times1.0

    \[\leadsto \color{blue}{\frac{\left(\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}} \cdot \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\frac{\ell}{1}}}\right) \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\frac{\ell}{1}} \cdot \sin k}}\]

  15. Final simplification1.0

    \[\leadsto \frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}} \cdot \left(\frac{\frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}}{\frac{k}{\ell}} \cdot \frac{\sqrt[3]{\frac{2}{t}}}{\sqrt[3]{\tan k}}\right)}{\frac{k}{\ell} \cdot \sin k}\]

Reproduce

herbie shell --seed 2019164 +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))))