Average Error: 48.1 → 5.8
Time: 1.3m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\ell}{\tan k} \cdot \frac{2 \cdot \left(\left(\ell \cdot {\left(\frac{\frac{1}{{t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)}{\sin k}\]
\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{\ell}{\tan k} \cdot \frac{2 \cdot \left(\left(\ell \cdot {\left(\frac{\frac{1}{{t}^{1}}}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\right)}{\sin k}
double f(double t, double l, double k) {
        double r136937 = 2.0;
        double r136938 = t;
        double r136939 = 3.0;
        double r136940 = pow(r136938, r136939);
        double r136941 = l;
        double r136942 = r136941 * r136941;
        double r136943 = r136940 / r136942;
        double r136944 = k;
        double r136945 = sin(r136944);
        double r136946 = r136943 * r136945;
        double r136947 = tan(r136944);
        double r136948 = r136946 * r136947;
        double r136949 = 1.0;
        double r136950 = r136944 / r136938;
        double r136951 = pow(r136950, r136937);
        double r136952 = r136949 + r136951;
        double r136953 = r136952 - r136949;
        double r136954 = r136948 * r136953;
        double r136955 = r136937 / r136954;
        return r136955;
}
↓
double f(double t, double l, double k) {
        double r136956 = l;
        double r136957 = k;
        double r136958 = tan(r136957);
        double r136959 = r136956 / r136958;
        double r136960 = 2.0;
        double r136961 = 1.0;
        double r136962 = t;
        double r136963 = 1.0;
        double r136964 = pow(r136962, r136963);
        double r136965 = r136961 / r136964;
        double r136966 = 2.0;
        double r136967 = r136960 / r136966;
        double r136968 = pow(r136957, r136967);
        double r136969 = r136965 / r136968;
        double r136970 = pow(r136969, r136963);
        double r136971 = r136956 * r136970;
        double r136972 = r136961 / r136968;
        double r136973 = pow(r136972, r136963);
        double r136974 = r136971 * r136973;
        double r136975 = r136960 * r136974;
        double r136976 = sin(r136957);
        double r136977 = r136975 / r136976;
        double r136978 = r136959 * r136977;
        return r136978;
}

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 48.1

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

  2. Simplified36.9

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

  3. Taylor expanded around inf 15.8

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

  4. Simplified15.5

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

  6. Applied sqr-pow15.5

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

  7. Applied *-un-lft-identity15.5

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

  8. Applied unpow-prod-down15.5

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

  9. Applied add-cube-cbrt15.5

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

  10. Applied times-frac15.5

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

  11. Applied times-frac10.5

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

  12. Simplified10.5

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

  13. Simplified10.5

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

  15. Applied associate-*r/10.5

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

  16. Simplified10.5

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

  18. Applied unpow-prod-down10.5

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

  19. Applied associate-*l*5.8

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

  20. Final simplification5.8

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2.0 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) (- (+ 1.0 (pow (/ k t) 2.0)) 1.0))))