Average Error: 13.7 → 8.2
Time: 24.5s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left(\left(\left|\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right| \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left(\left|\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right| \cdot {\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\]
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|\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right| \cdot {\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\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 r123142 = w0;
        double r123143 = 1.0;
        double r123144 = M;
        double r123145 = D;
        double r123146 = r123144 * r123145;
        double r123147 = 2.0;
        double r123148 = d;
        double r123149 = r123147 * r123148;
        double r123150 = r123146 / r123149;
        double r123151 = pow(r123150, r123147);
        double r123152 = h;
        double r123153 = l;
        double r123154 = r123152 / r123153;
        double r123155 = r123151 * r123154;
        double r123156 = r123143 - r123155;
        double r123157 = sqrt(r123156);
        double r123158 = r123142 * r123157;
        return r123158;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r123159 = w0;
        double r123160 = 1.0;
        double r123161 = h;
        double r123162 = cbrt(r123161);
        double r123163 = l;
        double r123164 = cbrt(r123163);
        double r123165 = r123162 / r123164;
        double r123166 = fabs(r123165);
        double r123167 = M;
        double r123168 = D;
        double r123169 = r123167 * r123168;
        double r123170 = 2.0;
        double r123171 = d;
        double r123172 = r123170 * r123171;
        double r123173 = r123169 / r123172;
        double r123174 = 2.0;
        double r123175 = r123170 / r123174;
        double r123176 = pow(r123173, r123175);
        double r123177 = r123166 * r123176;
        double r123178 = 1.0;
        double r123179 = r123178 / r123172;
        double r123180 = r123169 * r123179;
        double r123181 = pow(r123180, r123175);
        double r123182 = r123166 * r123181;
        double r123183 = r123177 * r123182;
        double r123184 = r123183 * r123165;
        double r123185 = r123160 - r123184;
        double r123186 = sqrt(r123185);
        double r123187 = r123159 * r123186;
        return r123187;
}

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 add-cube-cbrt13.7

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

  4. Applied add-cube-cbrt13.7

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

  5. Applied times-frac13.7

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

  6. Applied associate-*r*10.7

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

  8. Applied add-sqr-sqrt10.7

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

  9. Applied sqr-pow10.7

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

  10. Applied unswap-sqr9.1

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

  11. Simplified9.0

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

  12. Simplified8.2

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

  14. Applied div-inv8.2

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

  15. Final simplification8.2

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

Reproduce

herbie shell --seed 2019212 
(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))))))