Average Error: 13.9 → 9.2
Time: 24.9s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{{\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \frac{{\left(\frac{1}{\frac{2 \cdot d}{M \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}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r137235 = w0;
        double r137236 = 1.0;
        double r137237 = M;
        double r137238 = D;
        double r137239 = r137237 * r137238;
        double r137240 = 2.0;
        double r137241 = d;
        double r137242 = r137240 * r137241;
        double r137243 = r137239 / r137242;
        double r137244 = pow(r137243, r137240);
        double r137245 = h;
        double r137246 = l;
        double r137247 = r137245 / r137246;
        double r137248 = r137244 * r137247;
        double r137249 = r137236 - r137248;
        double r137250 = sqrt(r137249);
        double r137251 = r137235 * r137250;
        return r137251;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r137252 = w0;
        double r137253 = 1.0;
        double r137254 = 1.0;
        double r137255 = 2.0;
        double r137256 = d;
        double r137257 = r137255 * r137256;
        double r137258 = M;
        double r137259 = D;
        double r137260 = r137258 * r137259;
        double r137261 = r137257 / r137260;
        double r137262 = r137254 / r137261;
        double r137263 = 2.0;
        double r137264 = r137255 / r137263;
        double r137265 = pow(r137262, r137264);
        double r137266 = r137260 / r137257;
        double r137267 = pow(r137266, r137264);
        double r137268 = h;
        double r137269 = r137267 * r137268;
        double r137270 = r137265 * r137269;
        double r137271 = l;
        double r137272 = r137270 / r137271;
        double r137273 = r137253 - r137272;
        double r137274 = sqrt(r137273);
        double r137275 = r137252 * r137274;
        return r137275;
}

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 13.9

    \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
  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 clear-num9.2

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

  9. Final simplification9.2

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

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(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))))))