Average Error: 13.5 → 8.3
Time: 30.3s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \left(\sqrt[3]{1 - \frac{\frac{D \cdot M}{2 \cdot d} \cdot h}{\ell} \cdot \frac{D \cdot M}{2 \cdot d}} \cdot \sqrt[3]{\sqrt{1 - \frac{\frac{D \cdot M}{2 \cdot d} \cdot h}{\ell} \cdot \frac{D \cdot M}{2 \cdot d}}}\right)\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \left(\sqrt[3]{1 - \frac{\frac{D \cdot M}{2 \cdot d} \cdot h}{\ell} \cdot \frac{D \cdot M}{2 \cdot d}} \cdot \sqrt[3]{\sqrt{1 - \frac{\frac{D \cdot M}{2 \cdot d} \cdot h}{\ell} \cdot \frac{D \cdot M}{2 \cdot d}}}\right)
double f(double w0, double M, double D, double h, double l, double d) {
        double r4516020 = w0;
        double r4516021 = 1.0;
        double r4516022 = M;
        double r4516023 = D;
        double r4516024 = r4516022 * r4516023;
        double r4516025 = 2.0;
        double r4516026 = d;
        double r4516027 = r4516025 * r4516026;
        double r4516028 = r4516024 / r4516027;
        double r4516029 = pow(r4516028, r4516025);
        double r4516030 = h;
        double r4516031 = l;
        double r4516032 = r4516030 / r4516031;
        double r4516033 = r4516029 * r4516032;
        double r4516034 = r4516021 - r4516033;
        double r4516035 = sqrt(r4516034);
        double r4516036 = r4516020 * r4516035;
        return r4516036;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r4516037 = w0;
        double r4516038 = 1.0;
        double r4516039 = D;
        double r4516040 = M;
        double r4516041 = r4516039 * r4516040;
        double r4516042 = 2.0;
        double r4516043 = d;
        double r4516044 = r4516042 * r4516043;
        double r4516045 = r4516041 / r4516044;
        double r4516046 = h;
        double r4516047 = r4516045 * r4516046;
        double r4516048 = l;
        double r4516049 = r4516047 / r4516048;
        double r4516050 = r4516049 * r4516045;
        double r4516051 = r4516038 - r4516050;
        double r4516052 = cbrt(r4516051);
        double r4516053 = sqrt(r4516051);
        double r4516054 = cbrt(r4516053);
        double r4516055 = r4516052 * r4516054;
        double r4516056 = r4516037 * r4516055;
        return r4516056;
}

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 13.5

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

  2. Simplified13.5

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

  4. Applied add-cbrt-cube14.5

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

  5. Simplified9.6

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

  7. Applied cbrt-prod8.3

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

  8. Final simplification8.3

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

Reproduce

herbie shell --seed 2019135 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))