Average Error: 46.7 → 1.6
Time: 1.3m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\frac{\frac{\frac{-2}{\frac{k}{\ell}}}{\frac{k}{\ell} \cdot \left(\sin k \cdot t\right)}}{-\tan k}\]
\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)}
↓
\frac{\frac{\frac{-2}{\frac{k}{\ell}}}{\frac{k}{\ell} \cdot \left(\sin k \cdot t\right)}}{-\tan k}
double f(double t, double l, double k) {
        double r2393950 = 2.0;
        double r2393951 = t;
        double r2393952 = 3.0;
        double r2393953 = pow(r2393951, r2393952);
        double r2393954 = l;
        double r2393955 = r2393954 * r2393954;
        double r2393956 = r2393953 / r2393955;
        double r2393957 = k;
        double r2393958 = sin(r2393957);
        double r2393959 = r2393956 * r2393958;
        double r2393960 = tan(r2393957);
        double r2393961 = r2393959 * r2393960;
        double r2393962 = 1.0;
        double r2393963 = r2393957 / r2393951;
        double r2393964 = pow(r2393963, r2393950);
        double r2393965 = r2393962 + r2393964;
        double r2393966 = r2393965 - r2393962;
        double r2393967 = r2393961 * r2393966;
        double r2393968 = r2393950 / r2393967;
        return r2393968;
}
↓
double f(double t, double l, double k) {
        double r2393969 = -2.0;
        double r2393970 = k;
        double r2393971 = l;
        double r2393972 = r2393970 / r2393971;
        double r2393973 = r2393969 / r2393972;
        double r2393974 = sin(r2393970);
        double r2393975 = t;
        double r2393976 = r2393974 * r2393975;
        double r2393977 = r2393972 * r2393976;
        double r2393978 = r2393973 / r2393977;
        double r2393979 = tan(r2393970);
        double r2393980 = -r2393979;
        double r2393981 = r2393978 / r2393980;
        return r2393981;
}

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 46.7

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

  2. Simplified29.3

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

  4. Applied *-un-lft-identity29.3

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

  5. Applied times-frac28.9

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

  6. Applied times-frac17.9

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

  7. Simplified17.9

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

  8. Simplified11.6

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

  9. Taylor expanded around -inf 11.5

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

  10. Taylor expanded around -inf 2.9

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

  12. Applied frac-2neg2.9

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

  13. Simplified1.6

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

  14. Final simplification1.6

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

Reproduce

herbie shell --seed 2019130 +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))))