Average Error: 13.8 → 8.3
Time: 31.3s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \left(\left(\left(\left(\sqrt[3]{\frac{D \cdot M}{2 \cdot d}} \cdot \sqrt[3]{\frac{D \cdot M}{2 \cdot d}}\right) \cdot \sqrt[3]{\frac{D \cdot M}{2 \cdot d}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{D \cdot M}{2 \cdot d}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}} \cdot w0\]
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{D \cdot M}{2 \cdot d}} \cdot \sqrt[3]{\frac{D \cdot M}{2 \cdot d}}\right) \cdot \sqrt[3]{\frac{D \cdot M}{2 \cdot d}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{D \cdot M}{2 \cdot d}\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 r4755700 = w0;
        double r4755701 = 1.0;
        double r4755702 = M;
        double r4755703 = D;
        double r4755704 = r4755702 * r4755703;
        double r4755705 = 2.0;
        double r4755706 = d;
        double r4755707 = r4755705 * r4755706;
        double r4755708 = r4755704 / r4755707;
        double r4755709 = pow(r4755708, r4755705);
        double r4755710 = h;
        double r4755711 = l;
        double r4755712 = r4755710 / r4755711;
        double r4755713 = r4755709 * r4755712;
        double r4755714 = r4755701 - r4755713;
        double r4755715 = sqrt(r4755714);
        double r4755716 = r4755700 * r4755715;
        return r4755716;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r4755717 = 1.0;
        double r4755718 = D;
        double r4755719 = M;
        double r4755720 = r4755718 * r4755719;
        double r4755721 = 2.0;
        double r4755722 = d;
        double r4755723 = r4755721 * r4755722;
        double r4755724 = r4755720 / r4755723;
        double r4755725 = cbrt(r4755724);
        double r4755726 = r4755725 * r4755725;
        double r4755727 = r4755726 * r4755725;
        double r4755728 = h;
        double r4755729 = cbrt(r4755728);
        double r4755730 = l;
        double r4755731 = cbrt(r4755730);
        double r4755732 = r4755729 / r4755731;
        double r4755733 = r4755727 * r4755732;
        double r4755734 = r4755732 * r4755724;
        double r4755735 = r4755733 * r4755734;
        double r4755736 = r4755735 * r4755732;
        double r4755737 = r4755717 - r4755736;
        double r4755738 = sqrt(r4755737);
        double r4755739 = w0;
        double r4755740 = r4755738 * r4755739;
        return r4755740;
}

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-cbrt13.8

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

  5. Applied add-cube-cbrt13.8

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

  6. Applied times-frac13.8

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

  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. Simplified8.3

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

  10. Applied add-cube-cbrt8.3

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

  11. Final simplification8.3

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

Reproduce

herbie shell --seed 2019146 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))