Average Error: 13.7 → 7.8
Time: 10.0s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\frac{\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\frac{\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)}
double code(double w0, double M, double D, double h, double l, double d) {
	return (w0 * sqrt((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l)))));
}
↓
double code(double w0, double M, double D, double h, double l, double d) {
	return (w0 * sqrt((1.0 - ((cbrt(h) * pow(((M * D) / (2.0 * d)), (2.0 / 2.0))) * (((cbrt(h) * pow(((M * D) / (2.0 * d)), (2.0 / 2.0))) / (cbrt(l) * cbrt(l))) * (cbrt(h) / cbrt(l)))))));
}

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.7

    \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
  2. Using strategy rm

  3. Applied *-un-lft-identity13.7

    \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\color{blue}{1 \cdot \ell}}}\]

  4. Applied add-cube-cbrt13.8

    \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}{1 \cdot \ell}}\]

  5. Applied times-frac13.8

    \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(\frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{1} \cdot \frac{\sqrt[3]{h}}{\ell}\right)}}\]

  6. Applied associate-*r*11.4

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{1}\right) \cdot \frac{\sqrt[3]{h}}{\ell}}}\]

  7. Simplified11.4

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right)} \cdot \frac{\sqrt[3]{h}}{\ell}}\]
  8. Using strategy rm

  9. Applied sqr-pow11.4

    \[\leadsto w0 \cdot \sqrt{1 - \left(\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \frac{\sqrt[3]{h}}{\ell}}\]

  10. Applied unswap-sqr10.3

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)\right)} \cdot \frac{\sqrt[3]{h}}{\ell}}\]
  11. Using strategy rm

  12. Applied associate-*l*9.1

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)}}\]
  13. Using strategy rm

  14. Applied add-cube-cbrt9.1

    \[\leadsto w0 \cdot \sqrt{1 - \left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{\sqrt[3]{h}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}\right)}\]

  15. Applied *-un-lft-identity9.1

    \[\leadsto w0 \cdot \sqrt{1 - \left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{\sqrt[3]{\color{blue}{1 \cdot h}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}\right)}\]

  16. Applied cbrt-prod9.1

    \[\leadsto w0 \cdot \sqrt{1 - \left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{\color{blue}{\sqrt[3]{1} \cdot \sqrt[3]{h}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}\right)}\]

  17. Applied times-frac9.1

    \[\leadsto w0 \cdot \sqrt{1 - \left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{1}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)}\right)}\]

  18. Applied associate-*r*7.8

    \[\leadsto w0 \cdot \sqrt{1 - \left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \color{blue}{\left(\left(\left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)}}\]

  19. Simplified7.8

    \[\leadsto w0 \cdot \sqrt{1 - \left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\color{blue}{\frac{\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)}\]

  20. Final simplification7.8

    \[\leadsto w0 \cdot \sqrt{1 - \left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\frac{\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)}\]

Reproduce

herbie shell --seed 2020057 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))