Average Error: 19.3 → 2.4
Time: 26.1s
Precision: binary64
\[c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}\]
↓
\[c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\sqrt{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}{\frac{\sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}}}{\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}}}} \cdot \sqrt{\frac{\sqrt[3]{\sqrt[3]{A}}}{\frac{\sqrt[3]{\sqrt[3]{V}}}{\sqrt[3]{\frac{1}{\ell}}}}}\right)\right)\]
c0 \cdot \sqrt{\frac{A}{V \cdot \ell}}
↓
c0 \cdot \left(\left|\frac{\sqrt[3]{A}}{\sqrt[3]{V}}\right| \cdot \left(\sqrt{\frac{\sqrt[3]{\sqrt[3]{A} \cdot \sqrt[3]{A}}}{\frac{\sqrt[3]{\sqrt[3]{V} \cdot \sqrt[3]{V}}}{\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}}}} \cdot \sqrt{\frac{\sqrt[3]{\sqrt[3]{A}}}{\frac{\sqrt[3]{\sqrt[3]{V}}}{\sqrt[3]{\frac{1}{\ell}}}}}\right)\right)
(FPCore (c0 A V l) :precision binary64 (* c0 (sqrt (/ A (* V l)))))
↓
(FPCore (c0 A V l)
 :precision binary64
 (*
  c0
  (*
   (fabs (/ (cbrt A) (cbrt V)))
   (*
    (sqrt
     (/
      (cbrt (* (cbrt A) (cbrt A)))
      (/ (cbrt (* (cbrt V) (cbrt V))) (* (cbrt (/ 1.0 l)) (cbrt (/ 1.0 l))))))
    (sqrt (/ (cbrt (cbrt A)) (/ (cbrt (cbrt V)) (cbrt (/ 1.0 l)))))))))
double code(double c0, double A, double V, double l) {
	return c0 * sqrt(A / (V * l));
}
↓
double code(double c0, double A, double V, double l) {
	return c0 * (fabs(cbrt(A) / cbrt(V)) * (sqrt(cbrt(cbrt(A) * cbrt(A)) / (cbrt(cbrt(V) * cbrt(V)) / (cbrt(1.0 / l) * cbrt(1.0 / l)))) * sqrt(cbrt(cbrt(A)) / (cbrt(cbrt(V)) / cbrt(1.0 / l)))));
}

Error

Bits error versus c0

Bits error versus A

Bits error versus V

Bits error versus l

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 19.3

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

  3. Applied add-cube-cbrt_binary64_113619.6

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

  4. Applied associate-/l*_binary64_104619.6

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

  5. Simplified18.3

    \[\leadsto c0 \cdot \sqrt{\frac{\sqrt[3]{A} \cdot \sqrt[3]{A}}{\color{blue}{\frac{V}{\frac{\sqrt[3]{A}}{\ell}}}}}\]
  6. Using strategy rm

  7. Applied div-inv_binary64_109818.3

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

  8. Applied add-cube-cbrt_binary64_113618.4

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

  9. Applied times-frac_binary64_110717.7

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

  10. Applied times-frac_binary64_110717.0

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

  11. Applied sqrt-prod_binary64_11178.5

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

  12. Applied associate-*r*_binary64_10419.7

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

  13. Simplified7.8

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

  15. Applied add-cube-cbrt_binary64_11367.9

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

  16. Applied add-cube-cbrt_binary64_11367.9

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

  17. Applied cbrt-prod_binary64_11328.0

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

  18. Applied times-frac_binary64_11078.0

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

  19. Applied add-cube-cbrt_binary64_11368.0

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

  20. Applied times-frac_binary64_11076.8

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

  21. Applied sqrt-prod_binary64_11172.6

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

  23. Applied associate-*l*_binary64_10422.5

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

  24. Simplified2.4

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

  25. Final simplification2.4

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

Reproduce

herbie shell --seed 2021024 
(FPCore (c0 A V l)
  :name "Henrywood and Agarwal, Equation (3)"
  :precision binary64
  (* c0 (sqrt (/ A (* V l)))))