Average Error: 13.4 → 8.9
Time: 10.9s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{{\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \frac{{\left(\left(M \cdot D\right) \cdot \frac{1}{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}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r171910 = w0;
        double r171911 = 1.0;
        double r171912 = M;
        double r171913 = D;
        double r171914 = r171912 * r171913;
        double r171915 = 2.0;
        double r171916 = d;
        double r171917 = r171915 * r171916;
        double r171918 = r171914 / r171917;
        double r171919 = pow(r171918, r171915);
        double r171920 = h;
        double r171921 = l;
        double r171922 = r171920 / r171921;
        double r171923 = r171919 * r171922;
        double r171924 = r171911 - r171923;
        double r171925 = sqrt(r171924);
        double r171926 = r171910 * r171925;
        return r171926;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r171927 = w0;
        double r171928 = 1.0;
        double r171929 = M;
        double r171930 = D;
        double r171931 = r171929 * r171930;
        double r171932 = 1.0;
        double r171933 = 2.0;
        double r171934 = d;
        double r171935 = r171933 * r171934;
        double r171936 = r171932 / r171935;
        double r171937 = r171931 * r171936;
        double r171938 = 2.0;
        double r171939 = r171933 / r171938;
        double r171940 = pow(r171937, r171939);
        double r171941 = r171931 / r171935;
        double r171942 = pow(r171941, r171939);
        double r171943 = h;
        double r171944 = r171942 * r171943;
        double r171945 = r171940 * r171944;
        double r171946 = l;
        double r171947 = r171945 / r171946;
        double r171948 = r171928 - r171947;
        double r171949 = sqrt(r171948);
        double r171950 = r171927 * r171949;
        return r171950;
}

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 13.4

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

  3. Applied associate-*r/10.4

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

  5. Applied sqr-pow10.4

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

  6. Applied associate-*l*8.9

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

  8. Applied div-inv8.9

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

  9. Final simplification8.9

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

Reproduce

herbie shell --seed 2020018 
(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))))))