Average Error: 48.1 → 10.3
Time: 1.0m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;k \le -2.520331926479214337700732873487972896391 \cdot 10^{161} \lor \neg \left(k \le \frac{-5063473905106917}{1.179363257756731672575485806558834028412 \cdot 10^{167}}\right):\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\sqrt{1}}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)\right)}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\ \end{array}\]
\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 -2.520331926479214337700732873487972896391 \cdot 10^{161} \lor \neg \left(k \le \frac{-5063473905106917}{1.179363257756731672575485806558834028412 \cdot 10^{167}}\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(\cos k \cdot \ell\right)}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\

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

\end{array}
double f(double t, double l, double k) {
        double r92952 = 2.0;
        double r92953 = t;
        double r92954 = 3.0;
        double r92955 = pow(r92953, r92954);
        double r92956 = l;
        double r92957 = r92956 * r92956;
        double r92958 = r92955 / r92957;
        double r92959 = k;
        double r92960 = sin(r92959);
        double r92961 = r92958 * r92960;
        double r92962 = tan(r92959);
        double r92963 = r92961 * r92962;
        double r92964 = 1.0;
        double r92965 = r92959 / r92953;
        double r92966 = pow(r92965, r92952);
        double r92967 = r92964 + r92966;
        double r92968 = r92967 - r92964;
        double r92969 = r92963 * r92968;
        double r92970 = r92952 / r92969;
        return r92970;
}
↓
double f(double t, double l, double k) {
        double r92971 = k;
        double r92972 = -2.5203319264792143e+161;
        bool r92973 = r92971 <= r92972;
        double r92974 = -5063473905106917.0;
        double r92975 = 1.1793632577567317e+167;
        double r92976 = r92974 / r92975;
        bool r92977 = r92971 <= r92976;
        double r92978 = !r92977;
        bool r92979 = r92973 || r92978;
        double r92980 = 2.0;
        double r92981 = 1.0;
        double r92982 = 2.0;
        double r92983 = r92980 / r92982;
        double r92984 = pow(r92971, r92983);
        double r92985 = t;
        double r92986 = 1.0;
        double r92987 = pow(r92985, r92986);
        double r92988 = r92984 * r92987;
        double r92989 = r92984 * r92988;
        double r92990 = r92981 / r92989;
        double r92991 = pow(r92990, r92986);
        double r92992 = cos(r92971);
        double r92993 = l;
        double r92994 = r92992 * r92993;
        double r92995 = r92991 * r92994;
        double r92996 = sin(r92971);
        double r92997 = fabs(r92996);
        double r92998 = r92997 / r92993;
        double r92999 = r92997 * r92998;
        double r93000 = r92995 / r92999;
        double r93001 = r92980 * r93000;
        double r93002 = sqrt(r92981);
        double r93003 = pow(r92971, r92980);
        double r93004 = r93002 / r93003;
        double r93005 = pow(r93004, r92986);
        double r93006 = r93002 / r92987;
        double r93007 = pow(r93006, r92986);
        double r93008 = r93007 * r92994;
        double r93009 = r93005 * r93008;
        double r93010 = r93009 / r92999;
        double r93011 = r92980 * r93010;
        double r93012 = r92979 ? r93001 : r93011;
        return r93012;
}

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 < -2.5203319264792143e+161 or -4.293396349092778e-152 < k

    1. Initial program 46.0

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

    2. Simplified39.1

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

    3. Taylor expanded around inf 23.8

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

    5. Applied add-sqr-sqrt23.8

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

    6. Applied times-frac23.8

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

    7. Simplified23.8

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

    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 frac-times22.1

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

    11. Applied associate-*r/18.6

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

    13. Applied sqr-pow18.6

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

    14. Applied associate-*l*12.3

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

    if -2.5203319264792143e+161 < k < -4.293396349092778e-152

    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 17.8

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

    5. Applied add-sqr-sqrt17.8

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

    6. Applied times-frac17.8

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

    7. Simplified17.8

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

    8. Simplified16.2

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

    10. Applied frac-times15.0

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

    11. Applied associate-*r/8.6

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

    13. Applied add-sqr-sqrt8.6

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

    14. Applied times-frac8.3

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

    15. Applied unpow-prod-down8.3

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

    16. Applied associate-*l*5.0

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \le -2.520331926479214337700732873487972896391 \cdot 10^{161} \lor \neg \left(k \le \frac{-5063473905106917}{1.179363257756731672575485806558834028412 \cdot 10^{167}}\right):\\ \;\;\;\;2 \cdot \frac{{\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\sqrt{1}}{{k}^{2}}\right)}^{1} \cdot \left({\left(\frac{\sqrt{1}}{{t}^{1}}\right)}^{1} \cdot \left(\cos k \cdot \ell\right)\right)}{\left|\sin k\right| \cdot \frac{\left|\sin k\right|}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))))