Average Error: 13.0 → 7.8
Time: 1.1m
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{1}{\sqrt[3]{\frac{2}{\frac{D}{d}}} \cdot \sqrt[3]{\frac{2}{\frac{D}{d}}}}\right) \cdot \frac{M}{\sqrt[3]{\frac{2}{\frac{D}{d}}}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right)\right)}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{1}{\sqrt[3]{\frac{2}{\frac{D}{d}}} \cdot \sqrt[3]{\frac{2}{\frac{D}{d}}}}\right) \cdot \frac{M}{\sqrt[3]{\frac{2}{\frac{D}{d}}}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{M}{\frac{2}{\frac{D}{d}}}\right)\right)}
double f(double w0, double M, double D, double h, double l, double d) {
        double r2565330 = w0;
        double r2565331 = 1.0;
        double r2565332 = M;
        double r2565333 = D;
        double r2565334 = r2565332 * r2565333;
        double r2565335 = 2.0;
        double r2565336 = d;
        double r2565337 = r2565335 * r2565336;
        double r2565338 = r2565334 / r2565337;
        double r2565339 = pow(r2565338, r2565335);
        double r2565340 = h;
        double r2565341 = l;
        double r2565342 = r2565340 / r2565341;
        double r2565343 = r2565339 * r2565342;
        double r2565344 = r2565331 - r2565343;
        double r2565345 = sqrt(r2565344);
        double r2565346 = r2565330 * r2565345;
        return r2565346;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r2565347 = w0;
        double r2565348 = 1.0;
        double r2565349 = h;
        double r2565350 = cbrt(r2565349);
        double r2565351 = l;
        double r2565352 = cbrt(r2565351);
        double r2565353 = r2565350 / r2565352;
        double r2565354 = 2.0;
        double r2565355 = D;
        double r2565356 = d;
        double r2565357 = r2565355 / r2565356;
        double r2565358 = r2565354 / r2565357;
        double r2565359 = cbrt(r2565358);
        double r2565360 = r2565359 * r2565359;
        double r2565361 = r2565348 / r2565360;
        double r2565362 = r2565353 * r2565361;
        double r2565363 = M;
        double r2565364 = r2565363 / r2565359;
        double r2565365 = r2565362 * r2565364;
        double r2565366 = r2565363 / r2565358;
        double r2565367 = r2565353 * r2565366;
        double r2565368 = r2565365 * r2565367;
        double r2565369 = r2565353 * r2565368;
        double r2565370 = r2565348 - r2565369;
        double r2565371 = sqrt(r2565370);
        double r2565372 = r2565347 * r2565371;
        return r2565372;
}

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 13.0

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

  2. Simplified13.0

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

  4. Applied add-cube-cbrt13.0

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

  5. Applied add-cube-cbrt13.1

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

  6. Applied times-frac13.1

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

  7. Applied associate-*r*10.1

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

  8. Simplified7.8

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

  10. Applied add-cube-cbrt7.8

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

  11. Applied *-un-lft-identity7.8

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

  12. Applied times-frac7.8

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

  13. Applied associate-*r*7.8

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

  14. Final simplification7.8

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

Reproduce

herbie shell --seed 2019152 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))