Average Error: 13.7 → 9.2
Time: 35.6s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \left(\sqrt[3]{h} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\sqrt[3]{h}}{\ell} \cdot \left(\sqrt[3]{h} \cdot \frac{M \cdot D}{d \cdot 2}\right)\right)} \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\sqrt{1 - \left(\sqrt[3]{h} \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \left(\frac{\sqrt[3]{h}}{\ell} \cdot \left(\sqrt[3]{h} \cdot \frac{M \cdot D}{d \cdot 2}\right)\right)} \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r4510817 = w0;
        double r4510818 = 1.0;
        double r4510819 = M;
        double r4510820 = D;
        double r4510821 = r4510819 * r4510820;
        double r4510822 = 2.0;
        double r4510823 = d;
        double r4510824 = r4510822 * r4510823;
        double r4510825 = r4510821 / r4510824;
        double r4510826 = pow(r4510825, r4510822);
        double r4510827 = h;
        double r4510828 = l;
        double r4510829 = r4510827 / r4510828;
        double r4510830 = r4510826 * r4510829;
        double r4510831 = r4510818 - r4510830;
        double r4510832 = sqrt(r4510831);
        double r4510833 = r4510817 * r4510832;
        return r4510833;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r4510834 = 1.0;
        double r4510835 = h;
        double r4510836 = cbrt(r4510835);
        double r4510837 = M;
        double r4510838 = D;
        double r4510839 = r4510837 * r4510838;
        double r4510840 = d;
        double r4510841 = 2.0;
        double r4510842 = r4510840 * r4510841;
        double r4510843 = r4510839 / r4510842;
        double r4510844 = r4510836 * r4510843;
        double r4510845 = l;
        double r4510846 = r4510836 / r4510845;
        double r4510847 = r4510846 * r4510844;
        double r4510848 = r4510844 * r4510847;
        double r4510849 = r4510834 - r4510848;
        double r4510850 = sqrt(r4510849);
        double r4510851 = w0;
        double r4510852 = r4510850 * r4510851;
        return r4510852;
}

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 13.7

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

  2. Simplified13.7

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

  4. Applied *-un-lft-identity13.7

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

  5. Applied add-cube-cbrt13.7

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

  6. Applied times-frac13.7

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

  7. Applied associate-*r*11.4

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

  8. Simplified10.3

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

  10. Applied associate-*l*9.2

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

  11. Final simplification9.2

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

Reproduce

herbie shell --seed 2019129 +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))))))