Average Error: 14.0 → 8.9
Time: 44.7s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left({\left(\frac{\sqrt[3]{M}}{\sqrt[3]{\frac{2 \cdot d}{D}}}\right)}^{2} \cdot \left({\left(\frac{\sqrt[3]{M} \cdot \sqrt[3]{M}}{\sqrt[3]{\frac{2 \cdot d}{D}} \cdot \sqrt[3]{\frac{2 \cdot d}{D}}}\right)}^{2} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left({\left(\frac{\sqrt[3]{M}}{\sqrt[3]{\frac{2 \cdot d}{D}}}\right)}^{2} \cdot \left({\left(\frac{\sqrt[3]{M} \cdot \sqrt[3]{M}}{\sqrt[3]{\frac{2 \cdot d}{D}} \cdot \sqrt[3]{\frac{2 \cdot d}{D}}}\right)}^{2} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r5178638 = w0;
        double r5178639 = 1.0;
        double r5178640 = M;
        double r5178641 = D;
        double r5178642 = r5178640 * r5178641;
        double r5178643 = 2.0;
        double r5178644 = d;
        double r5178645 = r5178643 * r5178644;
        double r5178646 = r5178642 / r5178645;
        double r5178647 = pow(r5178646, r5178643);
        double r5178648 = h;
        double r5178649 = l;
        double r5178650 = r5178648 / r5178649;
        double r5178651 = r5178647 * r5178650;
        double r5178652 = r5178639 - r5178651;
        double r5178653 = sqrt(r5178652);
        double r5178654 = r5178638 * r5178653;
        return r5178654;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r5178655 = w0;
        double r5178656 = 1.0;
        double r5178657 = h;
        double r5178658 = cbrt(r5178657);
        double r5178659 = l;
        double r5178660 = cbrt(r5178659);
        double r5178661 = r5178658 / r5178660;
        double r5178662 = M;
        double r5178663 = cbrt(r5178662);
        double r5178664 = 2.0;
        double r5178665 = d;
        double r5178666 = r5178664 * r5178665;
        double r5178667 = D;
        double r5178668 = r5178666 / r5178667;
        double r5178669 = cbrt(r5178668);
        double r5178670 = r5178663 / r5178669;
        double r5178671 = pow(r5178670, r5178664);
        double r5178672 = r5178663 * r5178663;
        double r5178673 = r5178669 * r5178669;
        double r5178674 = r5178672 / r5178673;
        double r5178675 = pow(r5178674, r5178664);
        double r5178676 = r5178675 * r5178661;
        double r5178677 = r5178671 * r5178676;
        double r5178678 = r5178661 * r5178677;
        double r5178679 = r5178678 * r5178661;
        double r5178680 = r5178656 - r5178679;
        double r5178681 = sqrt(r5178680);
        double r5178682 = r5178655 * r5178681;
        return r5178682;
}

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 14.0

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

  3. Applied add-cube-cbrt14.1

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

  4. Applied add-cube-cbrt14.1

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

  5. Applied times-frac14.1

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

  6. Applied associate-*r*10.8

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

  7. Simplified9.9

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

  9. Applied add-cube-cbrt10.0

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

  10. Applied add-cube-cbrt10.0

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

  11. Applied times-frac10.0

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

  12. Applied unpow-prod-down10.0

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

  13. Applied associate-*r*8.9

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

  14. Final simplification8.9

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

Reproduce

herbie shell --seed 2019169 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))