Henrywood and Agarwal, Equation (3)

Average Error: 18.9 → 1.1

Time: 16.6s

Precision: 64

Math

\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]

↓

\[\left(c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}\right)\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}\]

C

double f(double c0, double A, double V, double l) {
        double r2428114 = c0;
        double r2428115 = A;
        double r2428116 = V;
        double r2428117 = l;
        double r2428118 = r2428116 * r2428117;
        double r2428119 = r2428115 / r2428118;
        double r2428120 = sqrt(r2428119);
        double r2428121 = r2428114 * r2428120;
        return r2428121;
}

↓

double f(double c0, double A, double V, double l) {
        double r2428122 = c0;
        double r2428123 = A;
        double r2428124 = cbrt(r2428123);
        double r2428125 = V;
        double r2428126 = cbrt(r2428125);
        double r2428127 = r2428124 / r2428126;
        double r2428128 = fabs(r2428127);
        double r2428129 = 1.0;
        double r2428130 = l;
        double r2428131 = cbrt(r2428130);
        double r2428132 = r2428129 / r2428131;
        double r2428133 = r2428132 / r2428131;
        double r2428134 = sqrt(r2428133);
        double r2428135 = r2428128 * r2428134;
        double r2428136 = r2428122 * r2428135;
        double r2428137 = r2428127 / r2428131;
        double r2428138 = sqrt(r2428137);
        double r2428139 = r2428136 * r2428138;
        return r2428139;
}

Error

Bits error versus c0

Bits error versus A

Bits error versus V

Bits error versus l

Derivation

1. Initial program 18.9

   \[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
2. Using strategy rm

3. Applied associate-/r* 19.2

   \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{A}{V}}{\ell}}}\]
4. Using strategy rm

5. Applied add-cube-cbrt 19.5

   \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{V}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}\]

6. Applied add-cube-cbrt 19.6

   \[\leadsto c0 \cdot \sqrt{\frac{\frac{A}{\color{blue}{\left(\sqrt[3]{V} \cdot \sqrt[3]{V}\right) \cdot \sqrt[3]{V}}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\]

7. Applied add-cube-cbrt 19.7

   \[\leadsto c0 \cdot \sqrt{\frac{\frac{\color{blue}{\left(\sqrt[3]{A} \cdot \sqrt[3]{A}\right) \cdot \sqrt[3]{A}}}{\left(\sqrt[3]{V} \cdot \sqrt[3]{V}\right) \cdot \sqrt[3]{V}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\]

8. Applied times-frac 19.7

   \[\leadsto c0 \cdot \sqrt{\frac{\color{blue}{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{V}} \cdot \frac{\sqrt[3]{A}}{\sqrt[3]{V}}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\]

9. Applied times-frac 15.6

   \[\leadsto c0 \cdot \sqrt{\color{blue}{\frac{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{V}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}}\]

10. Applied sqrt-prod 7.2

    \[\leadsto c0 \cdot \color{blue}{\left(\sqrt{\frac{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{V}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}\right)}\]

11. Applied associate-*r* 7.2

    \[\leadsto \color{blue}{\left(c0 \cdot \sqrt{\frac{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\sqrt[3]{V} \cdot \sqrt[3]{V}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}}\]

12. Simplified 7.2

    \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}} \cdot \frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}} \cdot c0\right)} \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}\]
13. Using strategy rm

14. Applied *-un-lft-identity 7.2

    \[\leadsto \left(\sqrt{\frac{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}} \cdot \frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}{\color{blue}{1 \cdot \sqrt[3]{\ell}}}} \cdot c0\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}\]

15. Applied div-inv 7.2

    \[\leadsto \left(\sqrt{\frac{\color{blue}{\left(\frac{\sqrt[3]{A}}{\sqrt[3]{V}} \cdot \frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right) \cdot \frac{1}{\sqrt[3]{\ell}}}}{1 \cdot \sqrt[3]{\ell}}} \cdot c0\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}\]

16. Applied times-frac 7.2

    \[\leadsto \left(\sqrt{\color{blue}{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}} \cdot \frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{1} \cdot \frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}} \cdot c0\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}\]

17. Applied sqrt-prod 3.4

    \[\leadsto \left(\color{blue}{\left(\sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}} \cdot \frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{1}} \cdot \sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}\right)} \cdot c0\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}\]

18. Simplified 3.4

    \[\leadsto \left(\left(\color{blue}{\sqrt{\frac{\sqrt[3]{A}}{\sqrt[3]{V}} \cdot \frac{\sqrt[3]{A}}{\sqrt[3]{V}}}} \cdot \sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}\right) \cdot c0\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}\]
19. Using strategy rm

20. Applied rem-sqrt-square 1.1

    \[\leadsto \left(\left(\color{blue}{\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right|} \cdot \sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}\right) \cdot c0\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}\]

21. Final simplification 1.1

    \[\leadsto \left(c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}\right)\right) \cdot \sqrt{\frac{\frac{\sqrt[3]{A}}{\sqrt[3]{V}}}{\sqrt[3]{\ell}}}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  (* c0 (sqrt (/ A (* V l)))))