Average Error: 13.5 → 8.3
Time: 1.0m
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{\frac{\frac{1}{\ell}}{\frac{\sqrt[3]{d}}{\sqrt[3]{\frac{M \cdot D}{2}}}} \cdot \frac{h}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\frac{M \cdot D}{2}} \cdot \sqrt[3]{\frac{M \cdot D}{2}}}}}{\frac{d}{\frac{M \cdot D}{2}}}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \frac{\frac{\frac{1}{\ell}}{\frac{\sqrt[3]{d}}{\sqrt[3]{\frac{M \cdot D}{2}}}} \cdot \frac{h}{\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\frac{M \cdot D}{2}} \cdot \sqrt[3]{\frac{M \cdot D}{2}}}}}{\frac{d}{\frac{M \cdot D}{2}}}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r5149057 = w0;
        double r5149058 = 1.0;
        double r5149059 = M;
        double r5149060 = D;
        double r5149061 = r5149059 * r5149060;
        double r5149062 = 2.0;
        double r5149063 = d;
        double r5149064 = r5149062 * r5149063;
        double r5149065 = r5149061 / r5149064;
        double r5149066 = pow(r5149065, r5149062);
        double r5149067 = h;
        double r5149068 = l;
        double r5149069 = r5149067 / r5149068;
        double r5149070 = r5149066 * r5149069;
        double r5149071 = r5149058 - r5149070;
        double r5149072 = sqrt(r5149071);
        double r5149073 = r5149057 * r5149072;
        return r5149073;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r5149074 = w0;
        double r5149075 = 1.0;
        double r5149076 = l;
        double r5149077 = r5149075 / r5149076;
        double r5149078 = d;
        double r5149079 = cbrt(r5149078);
        double r5149080 = M;
        double r5149081 = D;
        double r5149082 = r5149080 * r5149081;
        double r5149083 = 2.0;
        double r5149084 = r5149082 / r5149083;
        double r5149085 = cbrt(r5149084);
        double r5149086 = r5149079 / r5149085;
        double r5149087 = r5149077 / r5149086;
        double r5149088 = h;
        double r5149089 = r5149079 * r5149079;
        double r5149090 = r5149085 * r5149085;
        double r5149091 = r5149089 / r5149090;
        double r5149092 = r5149088 / r5149091;
        double r5149093 = r5149087 * r5149092;
        double r5149094 = r5149078 / r5149084;
        double r5149095 = r5149093 / r5149094;
        double r5149096 = r5149075 - r5149095;
        double r5149097 = sqrt(r5149096);
        double r5149098 = r5149074 * r5149097;
        return r5149098;
}

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. Simplified12.0

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

  4. Applied add-cube-cbrt12.0

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

  5. Applied add-cube-cbrt12.0

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

  6. Applied times-frac12.0

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

  7. Applied div-inv12.0

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

  8. Applied times-frac8.3

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

  9. Final simplification8.3

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

Reproduce

herbie shell --seed 2019164 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))