Average Error: 14.5 → 9.6
Time: 10.1s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)\right) \cdot \sqrt[3]{h}\right)}{\ell}}\]
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{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)\right) \cdot \sqrt[3]{h}\right)}{\ell}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r192300 = w0;
        double r192301 = 1.0;
        double r192302 = M;
        double r192303 = D;
        double r192304 = r192302 * r192303;
        double r192305 = 2.0;
        double r192306 = d;
        double r192307 = r192305 * r192306;
        double r192308 = r192304 / r192307;
        double r192309 = pow(r192308, r192305);
        double r192310 = h;
        double r192311 = l;
        double r192312 = r192310 / r192311;
        double r192313 = r192309 * r192312;
        double r192314 = r192301 - r192313;
        double r192315 = sqrt(r192314);
        double r192316 = r192300 * r192315;
        return r192316;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r192317 = w0;
        double r192318 = 1.0;
        double r192319 = M;
        double r192320 = D;
        double r192321 = r192319 * r192320;
        double r192322 = 2.0;
        double r192323 = d;
        double r192324 = r192322 * r192323;
        double r192325 = r192321 / r192324;
        double r192326 = 2.0;
        double r192327 = r192322 / r192326;
        double r192328 = pow(r192325, r192327);
        double r192329 = h;
        double r192330 = cbrt(r192329);
        double r192331 = r192330 * r192330;
        double r192332 = r192328 * r192331;
        double r192333 = r192332 * r192330;
        double r192334 = r192328 * r192333;
        double r192335 = l;
        double r192336 = r192334 / r192335;
        double r192337 = r192318 - r192336;
        double r192338 = sqrt(r192337);
        double r192339 = r192317 * r192338;
        return r192339;
}

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 14.5

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

  3. Applied associate-*r/11.0

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

  5. Applied sqr-pow11.0

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

  6. Applied associate-*l*9.6

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

  8. Applied add-cube-cbrt9.6

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

  9. Applied associate-*r*9.6

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

  10. Final simplification9.6

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

Reproduce

herbie shell --seed 2020060 +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))))))