Average Error: 48.1 → 9.4
Time: 3.0m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\left({\left(\frac{1}{{t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\frac{\ell}{\sin k} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{\ell}{\frac{\sin k}{\cos k}}\right)\right)\right) \cdot 2\]
\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)}
↓
\left({\left(\frac{1}{{t}^{1} \cdot {k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \left(\frac{\ell}{\sin k} \cdot \left({\left(\frac{1}{{k}^{\left(\frac{2}{2}\right)}}\right)}^{1} \cdot \frac{\ell}{\frac{\sin k}{\cos k}}\right)\right)\right) \cdot 2
double f(double t, double l, double k) {
        double r7620067 = 2.0;
        double r7620068 = t;
        double r7620069 = 3.0;
        double r7620070 = pow(r7620068, r7620069);
        double r7620071 = l;
        double r7620072 = r7620071 * r7620071;
        double r7620073 = r7620070 / r7620072;
        double r7620074 = k;
        double r7620075 = sin(r7620074);
        double r7620076 = r7620073 * r7620075;
        double r7620077 = tan(r7620074);
        double r7620078 = r7620076 * r7620077;
        double r7620079 = 1.0;
        double r7620080 = r7620074 / r7620068;
        double r7620081 = pow(r7620080, r7620067);
        double r7620082 = r7620079 + r7620081;
        double r7620083 = r7620082 - r7620079;
        double r7620084 = r7620078 * r7620083;
        double r7620085 = r7620067 / r7620084;
        return r7620085;
}
↓
double f(double t, double l, double k) {
        double r7620086 = 1.0;
        double r7620087 = t;
        double r7620088 = 1.0;
        double r7620089 = pow(r7620087, r7620088);
        double r7620090 = k;
        double r7620091 = 2.0;
        double r7620092 = 2.0;
        double r7620093 = r7620091 / r7620092;
        double r7620094 = pow(r7620090, r7620093);
        double r7620095 = r7620089 * r7620094;
        double r7620096 = r7620086 / r7620095;
        double r7620097 = pow(r7620096, r7620088);
        double r7620098 = l;
        double r7620099 = sin(r7620090);
        double r7620100 = r7620098 / r7620099;
        double r7620101 = r7620086 / r7620094;
        double r7620102 = pow(r7620101, r7620088);
        double r7620103 = cos(r7620090);
        double r7620104 = r7620099 / r7620103;
        double r7620105 = r7620098 / r7620104;
        double r7620106 = r7620102 * r7620105;
        double r7620107 = r7620100 * r7620106;
        double r7620108 = r7620097 * r7620107;
        double r7620109 = r7620108 * r7620091;
        return r7620109;
}

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 48.1

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

  2. Simplified40.2

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

  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. Using strategy rm

  5. Applied sqr-pow21.8

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

  6. Applied associate-*r*19.7

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

  8. Applied *-un-lft-identity19.7

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

  9. Applied times-frac19.6

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

  10. Applied unpow-prod-down19.6

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

  11. Applied associate-*l*18.6

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

  12. Simplified18.6

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

  14. Applied *-un-lft-identity18.6

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

  15. Applied times-frac18.6

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

  16. Applied times-frac14.6

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

  17. Applied associate-*l*9.4

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

  18. Final simplification9.4

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

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(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))))