Average Error: 48.0 → 14.6
Time: 33.4s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 4.234159102496386151558578367249599059363 \cdot 10^{298}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k \cdot \ell}{\left|\sin k\right|} \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\left|\sin k\right|}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]
\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}\;\ell \cdot \ell \le 4.234159102496386151558578367249599059363 \cdot 10^{298}:\\
\;\;\;\;2 \cdot \left({\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 \cdot \ell}{\left|\sin k\right|} \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\left|\sin k\right|}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\

\end{array}
double f(double t, double l, double k) {
        double r89041 = 2.0;
        double r89042 = t;
        double r89043 = 3.0;
        double r89044 = pow(r89042, r89043);
        double r89045 = l;
        double r89046 = r89045 * r89045;
        double r89047 = r89044 / r89046;
        double r89048 = k;
        double r89049 = sin(r89048);
        double r89050 = r89047 * r89049;
        double r89051 = tan(r89048);
        double r89052 = r89050 * r89051;
        double r89053 = 1.0;
        double r89054 = r89048 / r89042;
        double r89055 = pow(r89054, r89041);
        double r89056 = r89053 + r89055;
        double r89057 = r89056 - r89053;
        double r89058 = r89052 * r89057;
        double r89059 = r89041 / r89058;
        return r89059;
}
↓
double f(double t, double l, double k) {
        double r89060 = l;
        double r89061 = r89060 * r89060;
        double r89062 = 4.234159102496386e+298;
        bool r89063 = r89061 <= r89062;
        double r89064 = 2.0;
        double r89065 = 1.0;
        double r89066 = k;
        double r89067 = 2.0;
        double r89068 = r89064 / r89067;
        double r89069 = pow(r89066, r89068);
        double r89070 = t;
        double r89071 = 1.0;
        double r89072 = pow(r89070, r89071);
        double r89073 = r89069 * r89072;
        double r89074 = r89069 * r89073;
        double r89075 = r89065 / r89074;
        double r89076 = pow(r89075, r89071);
        double r89077 = cos(r89066);
        double r89078 = r89077 * r89060;
        double r89079 = sin(r89066);
        double r89080 = fabs(r89079);
        double r89081 = r89078 / r89080;
        double r89082 = r89067 / r89067;
        double r89083 = pow(r89060, r89082);
        double r89084 = r89083 / r89080;
        double r89085 = r89081 * r89084;
        double r89086 = r89076 * r89085;
        double r89087 = r89064 * r89086;
        double r89088 = 3.0;
        double r89089 = r89088 / r89067;
        double r89090 = pow(r89070, r89089);
        double r89091 = r89090 / r89060;
        double r89092 = r89091 * r89079;
        double r89093 = r89091 * r89092;
        double r89094 = tan(r89066);
        double r89095 = r89093 * r89094;
        double r89096 = r89064 / r89095;
        double r89097 = r89066 / r89070;
        double r89098 = pow(r89097, r89064);
        double r89099 = r89096 / r89098;
        double r89100 = r89063 ? r89087 : r89099;
        return r89100;
}

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 (* l l) < 4.234159102496386e+298

    1. Initial program 45.1

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

    2. Simplified36.1

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

    3. Taylor expanded around inf 14.3

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

    5. Applied sqr-pow14.3

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

    6. Applied associate-*l*11.8

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

    8. Applied add-sqr-sqrt11.8

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

    9. Applied times-frac11.8

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

    10. Simplified11.8

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

    11. Simplified11.4

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

    13. Applied *-un-lft-identity11.4

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

    14. Applied sqr-pow11.4

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

    15. Applied times-frac9.3

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

    16. Applied associate-*r*7.6

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

    17. Simplified7.6

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

    if 4.234159102496386e+298 < (* l l)

    1. Initial program 63.4

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

    2. Simplified63.0

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

    4. Applied sqr-pow63.7

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

    5. Applied times-frac52.0

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

    6. Applied associate-*l*52.0

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

  4. Final simplification14.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \le 4.234159102496386151558578367249599059363 \cdot 10^{298}:\\ \;\;\;\;2 \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)} \cdot \left({k}^{\left(\frac{2}{2}\right)} \cdot {t}^{1}\right)}\right)}^{1} \cdot \left(\frac{\cos k \cdot \ell}{\left|\sin k\right|} \cdot \frac{{\ell}^{\left(\frac{2}{2}\right)}}{\left|\sin k\right|}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \left(\frac{{t}^{\left(\frac{3}{2}\right)}}{\ell} \cdot \sin k\right)\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(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))))