Average Error: 13.6 → 8.3
Time: 9.6s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right) \cdot \frac{1}{\ell}\right)}\]
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 \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right) \cdot \frac{1}{\ell}\right)}
double f(double w0, double M, double D, double h, double l, double d) {
        double r267188 = w0;
        double r267189 = 1.0;
        double r267190 = M;
        double r267191 = D;
        double r267192 = r267190 * r267191;
        double r267193 = 2.0;
        double r267194 = d;
        double r267195 = r267193 * r267194;
        double r267196 = r267192 / r267195;
        double r267197 = pow(r267196, r267193);
        double r267198 = h;
        double r267199 = l;
        double r267200 = r267198 / r267199;
        double r267201 = r267197 * r267200;
        double r267202 = r267189 - r267201;
        double r267203 = sqrt(r267202);
        double r267204 = r267188 * r267203;
        return r267204;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r267205 = w0;
        double r267206 = 1.0;
        double r267207 = M;
        double r267208 = D;
        double r267209 = r267207 * r267208;
        double r267210 = 2.0;
        double r267211 = d;
        double r267212 = r267210 * r267211;
        double r267213 = r267209 / r267212;
        double r267214 = 2.0;
        double r267215 = r267210 / r267214;
        double r267216 = pow(r267213, r267215);
        double r267217 = h;
        double r267218 = r267216 * r267217;
        double r267219 = 1.0;
        double r267220 = l;
        double r267221 = r267219 / r267220;
        double r267222 = r267218 * r267221;
        double r267223 = r267216 * r267222;
        double r267224 = r267206 - r267223;
        double r267225 = sqrt(r267224);
        double r267226 = r267205 * r267225;
        return r267226;
}

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 13.6

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

  3. Applied div-inv13.6

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

  4. Applied associate-*r*10.3

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

  6. Applied sqr-pow10.3

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

  7. Applied associate-*l*8.9

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

  9. Applied associate-*l*8.3

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

  10. Final simplification8.3

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

Reproduce

herbie shell --seed 2020049 
(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))))))