Average Error: 13.4 → 8.1
Time: 2.1m
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \left(\left(\left(\left(\sqrt[3]{\frac{M}{2 \cdot d} \cdot D} \cdot \sqrt[3]{\frac{M}{2 \cdot d} \cdot D}\right) \cdot \sqrt[3]{\frac{M}{2 \cdot d} \cdot D}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\sqrt[3]{D} \cdot \left(\frac{M}{2 \cdot d} \cdot \left(\sqrt[3]{D} \cdot \sqrt[3]{D}\right)\right)\right)\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\sqrt{1 - \left(\left(\left(\left(\sqrt[3]{\frac{M}{2 \cdot d} \cdot D} \cdot \sqrt[3]{\frac{M}{2 \cdot d} \cdot D}\right) \cdot \sqrt[3]{\frac{M}{2 \cdot d} \cdot D}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\sqrt[3]{D} \cdot \left(\frac{M}{2 \cdot d} \cdot \left(\sqrt[3]{D} \cdot \sqrt[3]{D}\right)\right)\right)\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r45215085 = w0;
        double r45215086 = 1.0;
        double r45215087 = M;
        double r45215088 = D;
        double r45215089 = r45215087 * r45215088;
        double r45215090 = 2.0;
        double r45215091 = d;
        double r45215092 = r45215090 * r45215091;
        double r45215093 = r45215089 / r45215092;
        double r45215094 = pow(r45215093, r45215090);
        double r45215095 = h;
        double r45215096 = l;
        double r45215097 = r45215095 / r45215096;
        double r45215098 = r45215094 * r45215097;
        double r45215099 = r45215086 - r45215098;
        double r45215100 = sqrt(r45215099);
        double r45215101 = r45215085 * r45215100;
        return r45215101;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r45215102 = 1.0;
        double r45215103 = M;
        double r45215104 = 2.0;
        double r45215105 = d;
        double r45215106 = r45215104 * r45215105;
        double r45215107 = r45215103 / r45215106;
        double r45215108 = D;
        double r45215109 = r45215107 * r45215108;
        double r45215110 = cbrt(r45215109);
        double r45215111 = r45215110 * r45215110;
        double r45215112 = r45215111 * r45215110;
        double r45215113 = h;
        double r45215114 = cbrt(r45215113);
        double r45215115 = l;
        double r45215116 = cbrt(r45215115);
        double r45215117 = r45215114 / r45215116;
        double r45215118 = r45215112 * r45215117;
        double r45215119 = cbrt(r45215108);
        double r45215120 = r45215119 * r45215119;
        double r45215121 = r45215107 * r45215120;
        double r45215122 = r45215119 * r45215121;
        double r45215123 = r45215117 * r45215122;
        double r45215124 = r45215118 * r45215123;
        double r45215125 = r45215124 * r45215117;
        double r45215126 = r45215102 - r45215125;
        double r45215127 = sqrt(r45215126);
        double r45215128 = w0;
        double r45215129 = r45215127 * r45215128;
        return r45215129;
}

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 13.4

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

  2. Simplified13.4

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

  4. Applied add-cube-cbrt13.5

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

  5. Applied add-cube-cbrt13.5

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

  6. Applied times-frac13.5

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

  7. Applied associate-*r*10.5

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

  8. Simplified8.1

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

  10. Applied add-cube-cbrt8.1

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

  12. Applied add-cube-cbrt8.1

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

  13. Applied associate-*r*8.1

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

  14. Final simplification8.1

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

Reproduce

herbie shell --seed 2019128 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))