Average Error: 13.5 → 9.0
Time: 28.8s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left(\left(\frac{M}{d} \cdot \frac{D}{2}\right) \cdot \left(\sqrt[3]{h \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)} \cdot \left(\sqrt[3]{h \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)} \cdot \sqrt[3]{h \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)}\right)\right)\right) \cdot \frac{1}{\ell}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \left(\left(\frac{M}{d} \cdot \frac{D}{2}\right) \cdot \left(\sqrt[3]{h \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)} \cdot \left(\sqrt[3]{h \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)} \cdot \sqrt[3]{h \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)}\right)\right)\right) \cdot \frac{1}{\ell}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r6274657 = w0;
        double r6274658 = 1.0;
        double r6274659 = M;
        double r6274660 = D;
        double r6274661 = r6274659 * r6274660;
        double r6274662 = 2.0;
        double r6274663 = d;
        double r6274664 = r6274662 * r6274663;
        double r6274665 = r6274661 / r6274664;
        double r6274666 = pow(r6274665, r6274662);
        double r6274667 = h;
        double r6274668 = l;
        double r6274669 = r6274667 / r6274668;
        double r6274670 = r6274666 * r6274669;
        double r6274671 = r6274658 - r6274670;
        double r6274672 = sqrt(r6274671);
        double r6274673 = r6274657 * r6274672;
        return r6274673;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r6274674 = w0;
        double r6274675 = 1.0;
        double r6274676 = M;
        double r6274677 = d;
        double r6274678 = r6274676 / r6274677;
        double r6274679 = D;
        double r6274680 = 2.0;
        double r6274681 = r6274679 / r6274680;
        double r6274682 = r6274678 * r6274681;
        double r6274683 = h;
        double r6274684 = r6274683 * r6274682;
        double r6274685 = cbrt(r6274684);
        double r6274686 = r6274685 * r6274685;
        double r6274687 = r6274685 * r6274686;
        double r6274688 = r6274682 * r6274687;
        double r6274689 = l;
        double r6274690 = r6274675 / r6274689;
        double r6274691 = r6274688 * r6274690;
        double r6274692 = r6274675 - r6274691;
        double r6274693 = sqrt(r6274692);
        double r6274694 = r6274674 * r6274693;
        return r6274694;
}

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 13.5

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

  3. Applied div-inv13.5

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

  4. Applied associate-*r*10.4

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

  5. Simplified8.9

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

  7. Applied add-cube-cbrt9.0

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

  8. Final simplification9.0

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

Reproduce

herbie shell --seed 2019158 +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))))))