Average Error: 13.4 → 8.0
Time: 20.8s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\frac{\frac{D \cdot M}{2}}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\frac{\frac{D \cdot M}{2}}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{\frac{D \cdot M}{2}}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\frac{D \cdot M}{2}}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}\right)\right) \cdot \left(\sqrt[3]{\frac{\frac{D \cdot M}{2}}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{\frac{\frac{D \cdot M}{2}}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right)\right)\right)}\]
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(\frac{\frac{D \cdot M}{2}}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{\frac{\frac{D \cdot M}{2}}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{\frac{D \cdot M}{2}}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\frac{D \cdot M}{2}}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}\right)\right) \cdot \left(\sqrt[3]{\frac{\frac{D \cdot M}{2}}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{\frac{\frac{D \cdot M}{2}}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right)\right)\right)}
double f(double w0, double M, double D, double h, double l, double d) {
        double r2374207 = w0;
        double r2374208 = 1.0;
        double r2374209 = M;
        double r2374210 = D;
        double r2374211 = r2374209 * r2374210;
        double r2374212 = 2.0;
        double r2374213 = d;
        double r2374214 = r2374212 * r2374213;
        double r2374215 = r2374211 / r2374214;
        double r2374216 = pow(r2374215, r2374212);
        double r2374217 = h;
        double r2374218 = l;
        double r2374219 = r2374217 / r2374218;
        double r2374220 = r2374216 * r2374219;
        double r2374221 = r2374208 - r2374220;
        double r2374222 = sqrt(r2374221);
        double r2374223 = r2374207 * r2374222;
        return r2374223;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r2374224 = w0;
        double r2374225 = 1.0;
        double r2374226 = h;
        double r2374227 = cbrt(r2374226);
        double r2374228 = l;
        double r2374229 = cbrt(r2374228);
        double r2374230 = r2374227 / r2374229;
        double r2374231 = D;
        double r2374232 = M;
        double r2374233 = r2374231 * r2374232;
        double r2374234 = 2.0;
        double r2374235 = r2374233 / r2374234;
        double r2374236 = d;
        double r2374237 = r2374235 / r2374236;
        double r2374238 = r2374237 * r2374230;
        double r2374239 = cbrt(r2374238);
        double r2374240 = cbrt(r2374239);
        double r2374241 = r2374240 * r2374240;
        double r2374242 = r2374240 * r2374241;
        double r2374243 = r2374239 * r2374239;
        double r2374244 = r2374242 * r2374243;
        double r2374245 = r2374238 * r2374244;
        double r2374246 = r2374230 * r2374245;
        double r2374247 = r2374225 - r2374246;
        double r2374248 = sqrt(r2374247);
        double r2374249 = r2374224 * r2374248;
        return r2374249;
}

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 13.4

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

  2. Simplified13.4

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

  4. Applied add-cube-cbrt13.4

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

  5. Applied add-cube-cbrt13.4

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

  6. Applied times-frac13.4

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

  7. Applied associate-*r*10.5

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

  8. Simplified8.0

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

  10. Applied add-cube-cbrt8.0

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

  12. Applied add-cube-cbrt8.0

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

  13. Final simplification8.0

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

Reproduce

herbie shell --seed 2019156 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))