Average Error: 13.3 → 8.0
Time: 3.5m
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \left(h \cdot \left(\left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right) \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \frac{D \cdot M}{2 \cdot d}\right)\right)\right) \cdot \frac{D \cdot M}{2 \cdot d}} \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\sqrt{1 - \left(h \cdot \left(\left(\sqrt[3]{\frac{1}{\ell}} \cdot \sqrt[3]{\frac{1}{\ell}}\right) \cdot \left(\sqrt[3]{\frac{1}{\ell}} \cdot \frac{D \cdot M}{2 \cdot d}\right)\right)\right) \cdot \frac{D \cdot M}{2 \cdot d}} \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r47462605 = w0;
        double r47462606 = 1.0;
        double r47462607 = M;
        double r47462608 = D;
        double r47462609 = r47462607 * r47462608;
        double r47462610 = 2.0;
        double r47462611 = d;
        double r47462612 = r47462610 * r47462611;
        double r47462613 = r47462609 / r47462612;
        double r47462614 = pow(r47462613, r47462610);
        double r47462615 = h;
        double r47462616 = l;
        double r47462617 = r47462615 / r47462616;
        double r47462618 = r47462614 * r47462617;
        double r47462619 = r47462606 - r47462618;
        double r47462620 = sqrt(r47462619);
        double r47462621 = r47462605 * r47462620;
        return r47462621;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r47462622 = 1.0;
        double r47462623 = h;
        double r47462624 = l;
        double r47462625 = r47462622 / r47462624;
        double r47462626 = cbrt(r47462625);
        double r47462627 = r47462626 * r47462626;
        double r47462628 = D;
        double r47462629 = M;
        double r47462630 = r47462628 * r47462629;
        double r47462631 = 2.0;
        double r47462632 = d;
        double r47462633 = r47462631 * r47462632;
        double r47462634 = r47462630 / r47462633;
        double r47462635 = r47462626 * r47462634;
        double r47462636 = r47462627 * r47462635;
        double r47462637 = r47462623 * r47462636;
        double r47462638 = r47462637 * r47462634;
        double r47462639 = r47462622 - r47462638;
        double r47462640 = sqrt(r47462639);
        double r47462641 = w0;
        double r47462642 = r47462640 * r47462641;
        return r47462642;
}

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 13.3

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

  2. Simplified11.8

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

  4. Applied div-inv11.8

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

  5. Applied associate-*l*8.0

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

  7. Applied add-cube-cbrt8.0

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

  8. Applied associate-*l*8.0

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

  9. Final simplification8.0

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

Reproduce

herbie shell --seed 2019125 +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))))))