Average Error: 48.4 → 8.0
Time: 1.9m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left(\left({\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}\right)}^{1} \cdot \frac{2}{\frac{\sin k}{\ell}}\right) \cdot \frac{\cos k}{\frac{\sin k}{\ell}}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}\]
\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)}
↓
\left(\left({\left(\frac{\frac{1}{{k}^{\left(\frac{2}{2}\right)}}}{{t}^{1}}\right)}^{1} \cdot \frac{2}{\frac{\sin k}{\ell}}\right) \cdot \frac{\cos k}{\frac{\sin k}{\ell}}\right) \cdot {\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1}
double f(double t, double l, double k) {
        double r10631122 = 2.0;
        double r10631123 = t;
        double r10631124 = 3.0;
        double r10631125 = pow(r10631123, r10631124);
        double r10631126 = l;
        double r10631127 = r10631126 * r10631126;
        double r10631128 = r10631125 / r10631127;
        double r10631129 = k;
        double r10631130 = sin(r10631129);
        double r10631131 = r10631128 * r10631130;
        double r10631132 = tan(r10631129);
        double r10631133 = r10631131 * r10631132;
        double r10631134 = 1.0;
        double r10631135 = r10631129 / r10631123;
        double r10631136 = pow(r10631135, r10631122);
        double r10631137 = r10631134 + r10631136;
        double r10631138 = r10631137 - r10631134;
        double r10631139 = r10631133 * r10631138;
        double r10631140 = r10631122 / r10631139;
        return r10631140;
}
↓
double f(double t, double l, double k) {
        double r10631141 = 1.0;
        double r10631142 = k;
        double r10631143 = 2.0;
        double r10631144 = 2.0;
        double r10631145 = r10631143 / r10631144;
        double r10631146 = pow(r10631142, r10631145);
        double r10631147 = r10631141 / r10631146;
        double r10631148 = t;
        double r10631149 = 1.0;
        double r10631150 = pow(r10631148, r10631149);
        double r10631151 = r10631147 / r10631150;
        double r10631152 = pow(r10631151, r10631149);
        double r10631153 = sin(r10631142);
        double r10631154 = l;
        double r10631155 = r10631153 / r10631154;
        double r10631156 = r10631143 / r10631155;
        double r10631157 = r10631152 * r10631156;
        double r10631158 = cos(r10631142);
        double r10631159 = r10631158 / r10631155;
        double r10631160 = r10631157 * r10631159;
        double r10631161 = pow(r10631147, r10631149);
        double r10631162 = r10631160 * r10631161;
        return r10631162;
}

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 48.4

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

  2. Simplified40.1

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

  3. Taylor expanded around inf 21.8

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

  4. Simplified20.2

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

  6. Applied sqr-pow20.2

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

  7. Applied associate-*l*16.7

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

  9. Applied *-un-lft-identity16.7

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

  10. Applied times-frac16.5

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

  11. Applied unpow-prod-down16.5

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

  12. Applied associate-*l*14.2

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

  13. Simplified14.2

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

  15. Applied times-frac14.0

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

  16. Applied associate-*r*8.0

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

  17. Final simplification8.0

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

Reproduce

herbie shell --seed 2019170 
(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))))