Average Error: 13.8 → 8.1
Time: 32.7s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\left(\sqrt[3]{\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right) \cdot \sqrt[3]{\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}\right) \cdot \left(\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right)}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\left(\sqrt[3]{\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right) \cdot \sqrt[3]{\left(\sqrt[3]{\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot \sqrt[3]{\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right) \cdot \sqrt[3]{\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}\right) \cdot \left(\frac{D \cdot M}{2 \cdot d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right)}
double f(double w0, double M, double D, double h, double l, double d) {
        double r4488097 = w0;
        double r4488098 = 1.0;
        double r4488099 = M;
        double r4488100 = D;
        double r4488101 = r4488099 * r4488100;
        double r4488102 = 2.0;
        double r4488103 = d;
        double r4488104 = r4488102 * r4488103;
        double r4488105 = r4488101 / r4488104;
        double r4488106 = pow(r4488105, r4488102);
        double r4488107 = h;
        double r4488108 = l;
        double r4488109 = r4488107 / r4488108;
        double r4488110 = r4488106 * r4488109;
        double r4488111 = r4488098 - r4488110;
        double r4488112 = sqrt(r4488111);
        double r4488113 = r4488097 * r4488112;
        return r4488113;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r4488114 = w0;
        double r4488115 = 1.0;
        double r4488116 = h;
        double r4488117 = cbrt(r4488116);
        double r4488118 = l;
        double r4488119 = cbrt(r4488118);
        double r4488120 = r4488117 / r4488119;
        double r4488121 = D;
        double r4488122 = M;
        double r4488123 = r4488121 * r4488122;
        double r4488124 = 2.0;
        double r4488125 = d;
        double r4488126 = r4488124 * r4488125;
        double r4488127 = r4488123 / r4488126;
        double r4488128 = r4488127 * r4488120;
        double r4488129 = cbrt(r4488128);
        double r4488130 = r4488129 * r4488129;
        double r4488131 = r4488130 * r4488129;
        double r4488132 = cbrt(r4488131);
        double r4488133 = r4488130 * r4488132;
        double r4488134 = r4488133 * r4488128;
        double r4488135 = r4488120 * r4488134;
        double r4488136 = r4488115 - r4488135;
        double r4488137 = sqrt(r4488136);
        double r4488138 = r4488114 * r4488137;
        return r4488138;
}

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 13.8

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

  2. Simplified13.8

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

  4. Applied add-cube-cbrt13.9

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

  5. Applied add-cube-cbrt13.9

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

  6. Applied times-frac13.9

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

  7. Applied associate-*r*10.6

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

  8. Simplified8.1

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

  10. Applied add-cube-cbrt8.1

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

  12. Applied add-cbrt-cube8.1

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

  13. Final simplification8.1

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

Reproduce

herbie shell --seed 2019143 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))