Average Error: 14.4 → 9.1
Time: 17.1s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\]
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{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}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r205043 = w0;
        double r205044 = 1.0;
        double r205045 = M;
        double r205046 = D;
        double r205047 = r205045 * r205046;
        double r205048 = 2.0;
        double r205049 = d;
        double r205050 = r205048 * r205049;
        double r205051 = r205047 / r205050;
        double r205052 = pow(r205051, r205048);
        double r205053 = h;
        double r205054 = l;
        double r205055 = r205053 / r205054;
        double r205056 = r205052 * r205055;
        double r205057 = r205044 - r205056;
        double r205058 = sqrt(r205057);
        double r205059 = r205043 * r205058;
        return r205059;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r205060 = w0;
        double r205061 = 1.0;
        double r205062 = M;
        double r205063 = D;
        double r205064 = r205062 * r205063;
        double r205065 = 2.0;
        double r205066 = d;
        double r205067 = r205065 * r205066;
        double r205068 = r205064 / r205067;
        double r205069 = 2.0;
        double r205070 = r205065 / r205069;
        double r205071 = pow(r205068, r205070);
        double r205072 = h;
        double r205073 = r205071 * r205072;
        double r205074 = r205071 * r205073;
        double r205075 = l;
        double r205076 = r205074 / r205075;
        double r205077 = r205061 - r205076;
        double r205078 = sqrt(r205077);
        double r205079 = r205060 * r205078;
        return r205079;
}

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 14.4

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

  3. Applied associate-*r/10.8

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

  5. Applied sqr-pow10.8

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

  6. Applied associate-*l*9.1

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

  7. Final simplification9.1

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

Reproduce

herbie shell --seed 2020046 
(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))))))