Average Error: 13.8 → 8.4
Time: 12.0s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{{\left(\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\ell}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{{\left(\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right) \cdot \sqrt[3]{\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 r209286 = w0;
        double r209287 = 1.0;
        double r209288 = M;
        double r209289 = D;
        double r209290 = r209288 * r209289;
        double r209291 = 2.0;
        double r209292 = d;
        double r209293 = r209291 * r209292;
        double r209294 = r209290 / r209293;
        double r209295 = pow(r209294, r209291);
        double r209296 = h;
        double r209297 = l;
        double r209298 = r209296 / r209297;
        double r209299 = r209295 * r209298;
        double r209300 = r209287 - r209299;
        double r209301 = sqrt(r209300);
        double r209302 = r209286 * r209301;
        return r209302;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r209303 = w0;
        double r209304 = 1.0;
        double r209305 = M;
        double r209306 = D;
        double r209307 = r209305 * r209306;
        double r209308 = 2.0;
        double r209309 = d;
        double r209310 = r209308 * r209309;
        double r209311 = r209307 / r209310;
        double r209312 = 2.0;
        double r209313 = r209308 / r209312;
        double r209314 = pow(r209311, r209313);
        double r209315 = cbrt(r209311);
        double r209316 = r209315 * r209315;
        double r209317 = r209316 * r209315;
        double r209318 = pow(r209317, r209313);
        double r209319 = h;
        double r209320 = r209318 * r209319;
        double r209321 = l;
        double r209322 = r209320 / r209321;
        double r209323 = r209314 * r209322;
        double r209324 = r209304 - r209323;
        double r209325 = sqrt(r209324);
        double r209326 = r209303 * r209325;
        return r209326;
}

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 13.8

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

  3. Applied sqr-pow13.8

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

  4. Applied associate-*l*12.4

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

  6. Applied associate-*r/8.4

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

  8. Applied add-cube-cbrt8.4

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

  9. Final simplification8.4

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

Reproduce

herbie shell --seed 2019318 
(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))))))