Average Error: 13.4 → 8.0
Time: 33.2s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\left(\sqrt[3]{\frac{\frac{M}{d} \cdot D}{2}} \cdot \sqrt[3]{\frac{\frac{M}{d} \cdot D}{2}}\right) \cdot \left(\sqrt[3]{\frac{\frac{M}{d} \cdot D}{2}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{h} \cdot \frac{\frac{M}{d} \cdot D}{2}}{\sqrt[3]{\ell}}\right)}\]
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{\frac{M}{d} \cdot D}{2}} \cdot \sqrt[3]{\frac{\frac{M}{d} \cdot D}{2}}\right) \cdot \left(\sqrt[3]{\frac{\frac{M}{d} \cdot D}{2}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{h} \cdot \frac{\frac{M}{d} \cdot D}{2}}{\sqrt[3]{\ell}}\right)}
double f(double w0, double M, double D, double h, double l, double d) {
        double r4525717 = w0;
        double r4525718 = 1.0;
        double r4525719 = M;
        double r4525720 = D;
        double r4525721 = r4525719 * r4525720;
        double r4525722 = 2.0;
        double r4525723 = d;
        double r4525724 = r4525722 * r4525723;
        double r4525725 = r4525721 / r4525724;
        double r4525726 = pow(r4525725, r4525722);
        double r4525727 = h;
        double r4525728 = l;
        double r4525729 = r4525727 / r4525728;
        double r4525730 = r4525726 * r4525729;
        double r4525731 = r4525718 - r4525730;
        double r4525732 = sqrt(r4525731);
        double r4525733 = r4525717 * r4525732;
        return r4525733;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r4525734 = w0;
        double r4525735 = 1.0;
        double r4525736 = h;
        double r4525737 = cbrt(r4525736);
        double r4525738 = l;
        double r4525739 = cbrt(r4525738);
        double r4525740 = r4525737 / r4525739;
        double r4525741 = M;
        double r4525742 = d;
        double r4525743 = r4525741 / r4525742;
        double r4525744 = D;
        double r4525745 = r4525743 * r4525744;
        double r4525746 = 2.0;
        double r4525747 = r4525745 / r4525746;
        double r4525748 = cbrt(r4525747);
        double r4525749 = r4525748 * r4525748;
        double r4525750 = r4525748 * r4525740;
        double r4525751 = r4525749 * r4525750;
        double r4525752 = r4525737 * r4525747;
        double r4525753 = r4525752 / r4525739;
        double r4525754 = r4525751 * r4525753;
        double r4525755 = r4525740 * r4525754;
        double r4525756 = r4525735 - r4525755;
        double r4525757 = sqrt(r4525756);
        double r4525758 = r4525734 * r4525757;
        return r4525758;
}

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-cbrt13.4

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

  5. Applied add-cube-cbrt13.4

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

  6. Applied times-frac13.4

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

  7. Applied associate-*r*10.4

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

  8. Simplified8.0

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

  10. Applied associate-*r/8.0

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

  12. Applied add-cube-cbrt8.0

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

  13. Applied associate-*l*8.0

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

  14. Final simplification8.0

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

Reproduce

herbie shell --seed 2019133 +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))))))