Average Error: 14.0 → 8.7
Time: 10.8s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - {\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\ell}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - {\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\ell}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r215203 = w0;
        double r215204 = 1.0;
        double r215205 = M;
        double r215206 = D;
        double r215207 = r215205 * r215206;
        double r215208 = 2.0;
        double r215209 = d;
        double r215210 = r215208 * r215209;
        double r215211 = r215207 / r215210;
        double r215212 = pow(r215211, r215208);
        double r215213 = h;
        double r215214 = l;
        double r215215 = r215213 / r215214;
        double r215216 = r215212 * r215215;
        double r215217 = r215204 - r215216;
        double r215218 = sqrt(r215217);
        double r215219 = r215203 * r215218;
        return r215219;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r215220 = w0;
        double r215221 = 1.0;
        double r215222 = 1.0;
        double r215223 = 2.0;
        double r215224 = d;
        double r215225 = r215223 * r215224;
        double r215226 = M;
        double r215227 = D;
        double r215228 = r215226 * r215227;
        double r215229 = r215225 / r215228;
        double r215230 = r215222 / r215229;
        double r215231 = 2.0;
        double r215232 = r215223 / r215231;
        double r215233 = pow(r215230, r215232);
        double r215234 = r215228 / r215225;
        double r215235 = pow(r215234, r215232);
        double r215236 = h;
        double r215237 = r215235 * r215236;
        double r215238 = l;
        double r215239 = r215237 / r215238;
        double r215240 = r215233 * r215239;
        double r215241 = r215221 - r215240;
        double r215242 = sqrt(r215241);
        double r215243 = r215220 * r215242;
        return r215243;
}

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 14.0

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

  3. Applied associate-*r/10.7

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

  5. Applied sqr-pow10.7

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

  6. Applied associate-*l*9.2

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

  8. Applied *-un-lft-identity9.2

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

  9. Applied times-frac8.7

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

  10. Simplified8.7

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

  12. Applied clear-num8.7

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

  13. Final simplification8.7

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

Reproduce

herbie shell --seed 2019346 
(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))))))