Average Error: 32.1 → 14.3
Time: 26.1s
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\left(\frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\sqrt{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}} \cdot \frac{\frac{\sqrt[3]{2}}{\tan k}}{\sqrt{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}\right) \cdot \frac{\frac{\ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}\]
\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{\sqrt[3]{2} \cdot \sqrt[3]{2}}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\sqrt{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}} \cdot \frac{\frac{\sqrt[3]{2}}{\tan k}}{\sqrt{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}\right) \cdot \frac{\frac{\ell}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}
double f(double t, double l, double k) {
        double r125117 = 2.0;
        double r125118 = t;
        double r125119 = 3.0;
        double r125120 = pow(r125118, r125119);
        double r125121 = l;
        double r125122 = r125121 * r125121;
        double r125123 = r125120 / r125122;
        double r125124 = k;
        double r125125 = sin(r125124);
        double r125126 = r125123 * r125125;
        double r125127 = tan(r125124);
        double r125128 = r125126 * r125127;
        double r125129 = 1.0;
        double r125130 = r125124 / r125118;
        double r125131 = pow(r125130, r125117);
        double r125132 = r125129 + r125131;
        double r125133 = r125132 + r125129;
        double r125134 = r125128 * r125133;
        double r125135 = r125117 / r125134;
        return r125135;
}
↓
double f(double t, double l, double k) {
        double r125136 = 2.0;
        double r125137 = cbrt(r125136);
        double r125138 = r125137 * r125137;
        double r125139 = t;
        double r125140 = cbrt(r125139);
        double r125141 = r125140 * r125140;
        double r125142 = 3.0;
        double r125143 = 2.0;
        double r125144 = r125142 / r125143;
        double r125145 = pow(r125141, r125144);
        double r125146 = pow(r125140, r125142);
        double r125147 = l;
        double r125148 = r125146 / r125147;
        double r125149 = k;
        double r125150 = sin(r125149);
        double r125151 = r125148 * r125150;
        double r125152 = r125145 * r125151;
        double r125153 = r125138 / r125152;
        double r125154 = 1.0;
        double r125155 = r125149 / r125139;
        double r125156 = pow(r125155, r125136);
        double r125157 = fma(r125143, r125154, r125156);
        double r125158 = sqrt(r125157);
        double r125159 = sqrt(r125158);
        double r125160 = r125153 / r125159;
        double r125161 = tan(r125149);
        double r125162 = r125137 / r125161;
        double r125163 = r125162 / r125159;
        double r125164 = r125160 * r125163;
        double r125165 = r125147 / r125145;
        double r125166 = r125165 / r125158;
        double r125167 = r125164 * r125166;
        return r125167;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Initial program 32.1

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

  2. Simplified32.1

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

  4. Applied add-cube-cbrt32.3

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

  5. Applied unpow-prod-down32.3

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

  6. Applied times-frac25.9

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

  7. Applied associate-*l*24.0

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

  9. Applied sqr-pow24.0

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

  10. Applied associate-/l*18.3

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

  12. Applied add-sqr-sqrt18.3

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

  13. Applied associate-*l/17.5

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

  14. Applied associate-*l/16.2

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

  15. Applied associate-/r/16.2

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

  16. Applied times-frac14.6

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

  18. Applied add-sqr-sqrt14.6

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

  19. Applied sqrt-prod14.6

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

  20. Applied add-cube-cbrt14.6

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

  21. Applied times-frac14.6

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

  22. Applied times-frac14.3

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

  23. Final simplification14.3

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

Reproduce

herbie shell --seed 2020045 +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))))