Average Error: 31.9 → 11.9
Time: 1.2m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\left(\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}} \cdot \left(\frac{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}{\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}} \cdot \left(\left(\frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}} \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}\right) \cdot \frac{\sqrt[3]{\frac{\ell}{t}}}{\sqrt[3]{\sqrt[3]{\sin k}}}\right)\right)\right) \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}\]
\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(\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}} \cdot \left(\frac{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}{\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}} \cdot \left(\left(\frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}} \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}\right) \cdot \frac{\sqrt[3]{\frac{\ell}{t}}}{\sqrt[3]{\sqrt[3]{\sin k}}}\right)\right)\right) \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}
double f(double t, double l, double k) {
        double r3705932 = 2.0;
        double r3705933 = t;
        double r3705934 = 3.0;
        double r3705935 = pow(r3705933, r3705934);
        double r3705936 = l;
        double r3705937 = r3705936 * r3705936;
        double r3705938 = r3705935 / r3705937;
        double r3705939 = k;
        double r3705940 = sin(r3705939);
        double r3705941 = r3705938 * r3705940;
        double r3705942 = tan(r3705939);
        double r3705943 = r3705941 * r3705942;
        double r3705944 = 1.0;
        double r3705945 = r3705939 / r3705933;
        double r3705946 = pow(r3705945, r3705932);
        double r3705947 = r3705944 + r3705946;
        double r3705948 = r3705947 + r3705944;
        double r3705949 = r3705943 * r3705948;
        double r3705950 = r3705932 / r3705949;
        return r3705950;
}
↓
double f(double t, double l, double k) {
        double r3705951 = l;
        double r3705952 = t;
        double r3705953 = r3705951 / r3705952;
        double r3705954 = k;
        double r3705955 = sin(r3705954);
        double r3705956 = cbrt(r3705955);
        double r3705957 = r3705953 / r3705956;
        double r3705958 = cbrt(r3705953);
        double r3705959 = r3705958 * r3705958;
        double r3705960 = cbrt(r3705956);
        double r3705961 = r3705960 * r3705960;
        double r3705962 = r3705959 / r3705961;
        double r3705963 = 2.0;
        double r3705964 = r3705963 / r3705952;
        double r3705965 = r3705964 / r3705956;
        double r3705966 = cbrt(r3705965);
        double r3705967 = tan(r3705954);
        double r3705968 = r3705954 / r3705952;
        double r3705969 = 1.0;
        double r3705970 = fma(r3705968, r3705968, r3705969);
        double r3705971 = fma(r3705967, r3705970, r3705967);
        double r3705972 = cbrt(r3705971);
        double r3705973 = r3705966 / r3705972;
        double r3705974 = r3705973 * r3705973;
        double r3705975 = r3705958 / r3705960;
        double r3705976 = r3705974 * r3705975;
        double r3705977 = r3705962 * r3705976;
        double r3705978 = r3705957 * r3705977;
        double r3705979 = r3705978 * r3705973;
        return r3705979;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Initial program 31.9

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

  2. Simplified20.0

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

  4. Applied *-un-lft-identity20.0

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

  5. Applied add-cube-cbrt20.1

    \[\leadsto \frac{\frac{\left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right) \cdot \frac{2}{t}}{\color{blue}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\sin k}}}}{1 \cdot \mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}\]

  6. Applied times-frac18.4

    \[\leadsto \frac{\color{blue}{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}} \cdot \frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{1 \cdot \mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}\]

  7. Applied times-frac17.1

    \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{\sqrt[3]{\sin k} \cdot \sqrt[3]{\sin k}}}{1} \cdot \frac{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}\]

  8. Simplified15.4

    \[\leadsto \color{blue}{\left(\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}} \cdot \frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}}\right)} \cdot \frac{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt15.5

    \[\leadsto \left(\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}} \cdot \frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}}\right) \cdot \frac{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}}\]

  11. Applied add-cube-cbrt15.5

    \[\leadsto \left(\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}} \cdot \frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}}\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}} \cdot \sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}\right) \cdot \sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}}{\left(\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}\]

  12. Applied times-frac15.5

    \[\leadsto \left(\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}} \cdot \frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}}\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}} \cdot \sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}} \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}\right)}\]

  13. Applied associate-*r*14.4

    \[\leadsto \color{blue}{\left(\left(\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}} \cdot \frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}}\right) \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}} \cdot \sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}\right) \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}}\]

  14. Simplified11.9

    \[\leadsto \color{blue}{\left(\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}} \cdot \left(\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}} \cdot \left(\frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}} \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}\right)\right)\right)} \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}\]
  15. Using strategy rm

  16. Applied add-cube-cbrt11.9

    \[\leadsto \left(\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}} \cdot \left(\frac{\frac{\ell}{t}}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right) \cdot \sqrt[3]{\sqrt[3]{\sin k}}}} \cdot \left(\frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}} \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}\right)\right)\right) \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}\]

  17. Applied add-cube-cbrt11.9

    \[\leadsto \left(\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}} \cdot \left(\frac{\color{blue}{\left(\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}\right) \cdot \sqrt[3]{\frac{\ell}{t}}}}{\left(\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}\right) \cdot \sqrt[3]{\sqrt[3]{\sin k}}} \cdot \left(\frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}} \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}\right)\right)\right) \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}\]

  18. Applied times-frac11.9

    \[\leadsto \left(\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}} \cdot \left(\color{blue}{\left(\frac{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}{\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}} \cdot \frac{\sqrt[3]{\frac{\ell}{t}}}{\sqrt[3]{\sqrt[3]{\sin k}}}\right)} \cdot \left(\frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}} \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}\right)\right)\right) \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}\]

  19. Applied associate-*l*11.9

    \[\leadsto \left(\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}} \cdot \color{blue}{\left(\frac{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}{\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}} \cdot \left(\frac{\sqrt[3]{\frac{\ell}{t}}}{\sqrt[3]{\sqrt[3]{\sin k}}} \cdot \left(\frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}} \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}\right)\right)\right)}\right) \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}\]

  20. Final simplification11.9

    \[\leadsto \left(\frac{\frac{\ell}{t}}{\sqrt[3]{\sin k}} \cdot \left(\frac{\sqrt[3]{\frac{\ell}{t}} \cdot \sqrt[3]{\frac{\ell}{t}}}{\sqrt[3]{\sqrt[3]{\sin k}} \cdot \sqrt[3]{\sqrt[3]{\sin k}}} \cdot \left(\left(\frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}} \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}\right) \cdot \frac{\sqrt[3]{\frac{\ell}{t}}}{\sqrt[3]{\sqrt[3]{\sin k}}}\right)\right)\right) \cdot \frac{\sqrt[3]{\frac{\frac{2}{t}}{\sqrt[3]{\sin k}}}}{\sqrt[3]{\mathsf{fma}\left(\tan k, \mathsf{fma}\left(\frac{k}{t}, \frac{k}{t}, 1\right), \tan k\right)}}\]

Reproduce

herbie shell --seed 2019134 +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))))