Average Error: 14.6 → 8.7
Time: 26.5s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left(\left(\left|\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right| \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(\sqrt[3]{\left|\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right|} \cdot \sqrt[3]{\left|\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right|}\right) \cdot \left(\sqrt[3]{\left|\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right|} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \left(\left(\left|\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right| \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left(\sqrt[3]{\left|\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right|} \cdot \sqrt[3]{\left|\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right|}\right) \cdot \left(\sqrt[3]{\left|\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right|} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r225271 = w0;
        double r225272 = 1.0;
        double r225273 = M;
        double r225274 = D;
        double r225275 = r225273 * r225274;
        double r225276 = 2.0;
        double r225277 = d;
        double r225278 = r225276 * r225277;
        double r225279 = r225275 / r225278;
        double r225280 = pow(r225279, r225276);
        double r225281 = h;
        double r225282 = l;
        double r225283 = r225281 / r225282;
        double r225284 = r225280 * r225283;
        double r225285 = r225272 - r225284;
        double r225286 = sqrt(r225285);
        double r225287 = r225271 * r225286;
        return r225287;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r225288 = w0;
        double r225289 = 1.0;
        double r225290 = h;
        double r225291 = cbrt(r225290);
        double r225292 = l;
        double r225293 = cbrt(r225292);
        double r225294 = r225291 / r225293;
        double r225295 = fabs(r225294);
        double r225296 = M;
        double r225297 = D;
        double r225298 = r225296 * r225297;
        double r225299 = 2.0;
        double r225300 = d;
        double r225301 = r225299 * r225300;
        double r225302 = r225298 / r225301;
        double r225303 = 2.0;
        double r225304 = r225299 / r225303;
        double r225305 = pow(r225302, r225304);
        double r225306 = r225295 * r225305;
        double r225307 = cbrt(r225295);
        double r225308 = r225307 * r225307;
        double r225309 = r225307 * r225305;
        double r225310 = r225308 * r225309;
        double r225311 = r225306 * r225310;
        double r225312 = r225311 * r225294;
        double r225313 = r225289 - r225312;
        double r225314 = sqrt(r225313);
        double r225315 = r225288 * r225314;
        return r225315;
}

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 14.6

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

  3. Applied add-cube-cbrt14.6

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

  4. Applied add-cube-cbrt14.6

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

  5. Applied times-frac14.6

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

  6. Applied associate-*r*11.3

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

  8. Applied add-sqr-sqrt11.3

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

  9. Applied sqr-pow11.3

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

  10. Applied unswap-sqr9.6

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

  11. Simplified9.6

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

  12. Simplified8.7

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

  14. Applied add-cube-cbrt8.7

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

  15. Applied associate-*l*8.7

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

  16. Final simplification8.7

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

Reproduce

herbie shell --seed 2019347 
(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))))))