Average Error: 13.0 → 7.5
Time: 20.7s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\left(\frac{D \cdot M}{2 \cdot d} \cdot \sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right)} \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\left(\frac{D \cdot M}{2 \cdot d} \cdot \sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right) \cdot \left(\sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right)} \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r3403174 = w0;
        double r3403175 = 1.0;
        double r3403176 = M;
        double r3403177 = D;
        double r3403178 = r3403176 * r3403177;
        double r3403179 = 2.0;
        double r3403180 = d;
        double r3403181 = r3403179 * r3403180;
        double r3403182 = r3403178 / r3403181;
        double r3403183 = pow(r3403182, r3403179);
        double r3403184 = h;
        double r3403185 = l;
        double r3403186 = r3403184 / r3403185;
        double r3403187 = r3403183 * r3403186;
        double r3403188 = r3403175 - r3403187;
        double r3403189 = sqrt(r3403188);
        double r3403190 = r3403174 * r3403189;
        return r3403190;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r3403191 = 1.0;
        double r3403192 = h;
        double r3403193 = cbrt(r3403192);
        double r3403194 = l;
        double r3403195 = cbrt(r3403194);
        double r3403196 = r3403193 / r3403195;
        double r3403197 = D;
        double r3403198 = M;
        double r3403199 = r3403197 * r3403198;
        double r3403200 = 2.0;
        double r3403201 = d;
        double r3403202 = r3403200 * r3403201;
        double r3403203 = r3403199 / r3403202;
        double r3403204 = cbrt(r3403196);
        double r3403205 = r3403203 * r3403204;
        double r3403206 = r3403204 * r3403204;
        double r3403207 = r3403205 * r3403206;
        double r3403208 = r3403203 * r3403196;
        double r3403209 = r3403207 * r3403208;
        double r3403210 = r3403196 * r3403209;
        double r3403211 = r3403191 - r3403210;
        double r3403212 = sqrt(r3403211);
        double r3403213 = w0;
        double r3403214 = r3403212 * r3403213;
        return r3403214;
}

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. Simplified7.5

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

  10. Applied add-cube-cbrt7.5

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

  11. Applied associate-*l*7.5

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

  12. Final simplification7.5

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

Reproduce

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