Average Error: 13.3 → 7.9
Time: 2.9m
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{M}{2 \cdot d} \cdot D\right)\right) \cdot \frac{\sqrt[3]{h} \cdot \left(\frac{M}{2 \cdot d} \cdot D\right)}{\sqrt[3]{\ell}}\right)} \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{M}{2 \cdot d} \cdot D\right)\right) \cdot \frac{\sqrt[3]{h} \cdot \left(\frac{M}{2 \cdot d} \cdot D\right)}{\sqrt[3]{\ell}}\right)} \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r52913091 = w0;
        double r52913092 = 1.0;
        double r52913093 = M;
        double r52913094 = D;
        double r52913095 = r52913093 * r52913094;
        double r52913096 = 2.0;
        double r52913097 = d;
        double r52913098 = r52913096 * r52913097;
        double r52913099 = r52913095 / r52913098;
        double r52913100 = pow(r52913099, r52913096);
        double r52913101 = h;
        double r52913102 = l;
        double r52913103 = r52913101 / r52913102;
        double r52913104 = r52913100 * r52913103;
        double r52913105 = r52913092 - r52913104;
        double r52913106 = sqrt(r52913105);
        double r52913107 = r52913091 * r52913106;
        return r52913107;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r52913108 = 1.0;
        double r52913109 = h;
        double r52913110 = cbrt(r52913109);
        double r52913111 = l;
        double r52913112 = cbrt(r52913111);
        double r52913113 = r52913110 / r52913112;
        double r52913114 = M;
        double r52913115 = 2.0;
        double r52913116 = d;
        double r52913117 = r52913115 * r52913116;
        double r52913118 = r52913114 / r52913117;
        double r52913119 = D;
        double r52913120 = r52913118 * r52913119;
        double r52913121 = r52913113 * r52913120;
        double r52913122 = r52913110 * r52913120;
        double r52913123 = r52913122 / r52913112;
        double r52913124 = r52913121 * r52913123;
        double r52913125 = r52913113 * r52913124;
        double r52913126 = r52913108 - r52913125;
        double r52913127 = sqrt(r52913126);
        double r52913128 = w0;
        double r52913129 = r52913127 * r52913128;
        return r52913129;
}

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 13.3

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

  2. Simplified13.3

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

  4. Applied add-cube-cbrt13.3

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

  5. Applied add-cube-cbrt13.3

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

  6. Applied times-frac13.3

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

  7. Applied associate-*r*10.2

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

  8. Simplified7.9

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

  10. Applied associate-*r/7.9

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

  11. Final simplification7.9

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

Reproduce

herbie shell --seed 2019125 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))