Average Error: 48.3 → 10.0
Time: 56.3s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -5.603375851643622422415683198718436485274 \cdot 10^{157} \lor \neg \left(k \le -3.459437993813782241454138708740781760984 \cdot 10^{-153}\right):\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \ell\right)}{\frac{\left|\sin k\right|}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \ell\right)\right)}{\frac{\left|\sin k\right|}{\ell}}\\ \end{array}\]
\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)}
↓
\begin{array}{l}
\mathbf{if}\;k \le -5.603375851643622422415683198718436485274 \cdot 10^{157} \lor \neg \left(k \le -3.459437993813782241454138708740781760984 \cdot 10^{-153}\right):\\
\;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \ell\right)}{\frac{\left|\sin k\right|}{\ell}}\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \ell\right)\right)}{\frac{\left|\sin k\right|}{\ell}}\\

\end{array}
double f(double t, double l, double k) {
        double r74930 = 2.0;
        double r74931 = t;
        double r74932 = 3.0;
        double r74933 = pow(r74931, r74932);
        double r74934 = l;
        double r74935 = r74934 * r74934;
        double r74936 = r74933 / r74935;
        double r74937 = k;
        double r74938 = sin(r74937);
        double r74939 = r74936 * r74938;
        double r74940 = tan(r74937);
        double r74941 = r74939 * r74940;
        double r74942 = 1.0;
        double r74943 = r74937 / r74931;
        double r74944 = pow(r74943, r74930);
        double r74945 = r74942 + r74944;
        double r74946 = r74945 - r74942;
        double r74947 = r74941 * r74946;
        double r74948 = r74930 / r74947;
        return r74948;
}
↓
double f(double t, double l, double k) {
        double r74949 = k;
        double r74950 = -5.603375851643622e+157;
        bool r74951 = r74949 <= r74950;
        double r74952 = -3.459437993813782e-153;
        bool r74953 = r74949 <= r74952;
        double r74954 = !r74953;
        bool r74955 = r74951 || r74954;
        double r74956 = 2.0;
        double r74957 = 1.0;
        double r74958 = 2.0;
        double r74959 = r74956 / r74958;
        double r74960 = pow(r74949, r74959);
        double r74961 = t;
        double r74962 = 1.0;
        double r74963 = pow(r74961, r74962);
        double r74964 = r74960 * r74963;
        double r74965 = r74960 * r74964;
        double r74966 = r74957 / r74965;
        double r74967 = pow(r74966, r74962);
        double r74968 = cos(r74949);
        double r74969 = sin(r74949);
        double r74970 = fabs(r74969);
        double r74971 = r74968 / r74970;
        double r74972 = l;
        double r74973 = r74971 * r74972;
        double r74974 = r74967 * r74973;
        double r74975 = r74970 / r74972;
        double r74976 = r74974 / r74975;
        double r74977 = r74956 * r74976;
        double r74978 = pow(r74949, r74956);
        double r74979 = r74957 / r74978;
        double r74980 = pow(r74979, r74962);
        double r74981 = r74957 / r74963;
        double r74982 = pow(r74981, r74962);
        double r74983 = r74982 * r74973;
        double r74984 = r74980 * r74983;
        double r74985 = r74984 / r74975;
        double r74986 = r74956 * r74985;
        double r74987 = r74955 ? r74977 : r74986;
        return r74987;
}

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. Split input into 2 regimes

  2. if k < -5.603375851643622e+157 or -3.459437993813782e-153 < k

    1. Initial program 46.2

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

    2. Simplified39.3

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

    3. Taylor expanded around inf 23.7

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

    5. Applied add-sqr-sqrt23.7

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

    6. Applied times-frac23.7

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

    7. Simplified23.7

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

    8. Simplified22.9

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

    10. Applied associate-*r/22.2

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

    11. Applied associate-*r/18.7

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

    13. Applied sqr-pow18.7

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

    14. Applied associate-*l*12.3

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

    if -5.603375851643622e+157 < k < -3.459437993813782e-153

    1. Initial program 53.5

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

    2. Simplified43.6

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

    3. Taylor expanded around inf 17.7

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

    5. Applied add-sqr-sqrt17.7

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

    6. Applied times-frac17.7

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

    7. Simplified17.7

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

    8. Simplified16.0

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

    10. Applied associate-*r/14.1

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

    11. Applied associate-*r/8.0

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

    13. Applied *-un-lft-identity8.0

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

    14. Applied times-frac7.7

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

    15. Applied unpow-prod-down7.7

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

    16. Applied associate-*l*4.1

      \[\leadsto 2 \cdot \frac{\color{blue}{{\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \ell\right)\right)}}{\frac{\left|\sin k\right|}{\ell}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -5.603375851643622422415683198718436485274 \cdot 10^{157} \lor \neg \left(k \le -3.459437993813782241454138708740781760984 \cdot 10^{-153}\right):\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \ell\right)}{\frac{\left|\sin k\right|}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{1}{{t}^{1}}\right)}^{1} \cdot \left(\frac{\cos k}{\left|\sin k\right|} \cdot \ell\right)\right)}{\frac{\left|\sin k\right|}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))