Average Error: 46.8 → 10.8
Time: 5.9m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -5.013219891376737 \cdot 10^{-240}:\\ \;\;\;\;\left(\frac{\frac{2}{t}}{\frac{k}{t}} \cdot \frac{1}{\frac{\tan k \cdot t}{\ell}}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\sin k}}{\frac{k}{t}}\\ \mathbf{elif}\;t \le 1.625274404785811 \cdot 10^{-294}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t \cdot \left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2}{t}}{\frac{k}{t}} \cdot \frac{1}{\frac{t}{\ell} \cdot \tan k}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\sin k}}{\frac{k}{t}}\\ \end{array}\]
\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}\;t \le -5.013219891376737 \cdot 10^{-240}:\\
\;\;\;\;\left(\frac{\frac{2}{t}}{\frac{k}{t}} \cdot \frac{1}{\frac{\tan k \cdot t}{\ell}}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\sin k}}{\frac{k}{t}}\\

\mathbf{elif}\;t \le 1.625274404785811 \cdot 10^{-294}:\\
\;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t \cdot \left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right)}\\

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

\end{array}
double f(double t, double l, double k) {
        double r12023969 = 2.0;
        double r12023970 = t;
        double r12023971 = 3.0;
        double r12023972 = pow(r12023970, r12023971);
        double r12023973 = l;
        double r12023974 = r12023973 * r12023973;
        double r12023975 = r12023972 / r12023974;
        double r12023976 = k;
        double r12023977 = sin(r12023976);
        double r12023978 = r12023975 * r12023977;
        double r12023979 = tan(r12023976);
        double r12023980 = r12023978 * r12023979;
        double r12023981 = 1.0;
        double r12023982 = r12023976 / r12023970;
        double r12023983 = pow(r12023982, r12023969);
        double r12023984 = r12023981 + r12023983;
        double r12023985 = r12023984 - r12023981;
        double r12023986 = r12023980 * r12023985;
        double r12023987 = r12023969 / r12023986;
        return r12023987;
}
↓
double f(double t, double l, double k) {
        double r12023988 = t;
        double r12023989 = -5.013219891376737e-240;
        bool r12023990 = r12023988 <= r12023989;
        double r12023991 = 2.0;
        double r12023992 = r12023991 / r12023988;
        double r12023993 = k;
        double r12023994 = r12023993 / r12023988;
        double r12023995 = r12023992 / r12023994;
        double r12023996 = 1.0;
        double r12023997 = tan(r12023993);
        double r12023998 = r12023997 * r12023988;
        double r12023999 = l;
        double r12024000 = r12023998 / r12023999;
        double r12024001 = r12023996 / r12024000;
        double r12024002 = r12023995 * r12024001;
        double r12024003 = r12023999 / r12023988;
        double r12024004 = sin(r12023993);
        double r12024005 = r12024003 / r12024004;
        double r12024006 = r12024005 / r12023994;
        double r12024007 = r12024002 * r12024006;
        double r12024008 = 1.625274404785811e-294;
        bool r12024009 = r12023988 <= r12024008;
        double r12024010 = cos(r12023993);
        double r12024011 = pow(r12023999, r12023991);
        double r12024012 = r12024010 * r12024011;
        double r12024013 = pow(r12024004, r12023991);
        double r12024014 = pow(r12023993, r12023991);
        double r12024015 = r12024013 * r12024014;
        double r12024016 = r12023988 * r12024015;
        double r12024017 = r12024012 / r12024016;
        double r12024018 = r12023991 * r12024017;
        double r12024019 = r12023988 / r12023999;
        double r12024020 = r12024019 * r12023997;
        double r12024021 = r12023996 / r12024020;
        double r12024022 = r12023995 * r12024021;
        double r12024023 = r12024022 * r12024006;
        double r12024024 = r12024009 ? r12024018 : r12024023;
        double r12024025 = r12023990 ? r12024007 : r12024024;
        return r12024025;
}

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 3 regimes

  2. if t < -5.013219891376737e-240

    1. Initial program 45.9

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

    2. Simplified27.9

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

    4. Applied *-un-lft-identity27.9

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

    5. Applied times-frac27.1

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

    6. Applied associate-*r*26.4

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

    7. Applied times-frac14.3

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

    9. Applied clear-num14.3

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

    10. Applied associate-/l/14.3

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

    11. Applied frac-times14.2

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

    12. Applied associate-/l/9.2

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

    14. Applied *-commutative9.2

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

    15. Applied *-commutative9.2

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

    16. Applied times-frac9.2

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

    18. Applied associate-*r/9.2

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

    19. Applied associate-*r/9.2

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

    if -5.013219891376737e-240 < t < 1.625274404785811e-294

    1. Initial program 62.8

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

    2. Simplified57.7

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

    3. Taylor expanded around inf 28.8

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

    if 1.625274404785811e-294 < t

    1. Initial program 45.8

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

    2. Simplified29.1

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

    4. Applied *-un-lft-identity29.1

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

    5. Applied times-frac28.5

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

    6. Applied associate-*r*28.0

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

    7. Applied times-frac16.3

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

    9. Applied clear-num16.3

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

    10. Applied associate-/l/16.3

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

    11. Applied frac-times16.1

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

    12. Applied associate-/l/10.3

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

    14. Applied *-commutative10.3

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

    15. Applied *-commutative10.3

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

    16. Applied times-frac10.3

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

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.013219891376737 \cdot 10^{-240}:\\ \;\;\;\;\left(\frac{\frac{2}{t}}{\frac{k}{t}} \cdot \frac{1}{\frac{\tan k \cdot t}{\ell}}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\sin k}}{\frac{k}{t}}\\ \mathbf{elif}\;t \le 1.625274404785811 \cdot 10^{-294}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t \cdot \left({\left(\sin k\right)}^{2} \cdot {k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{2}{t}}{\frac{k}{t}} \cdot \frac{1}{\frac{t}{\ell} \cdot \tan k}\right) \cdot \frac{\frac{\frac{\ell}{t}}{\sin k}}{\frac{k}{t}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +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))))