Average Error: 14.8 → 9.0
Time: 1.3m
Precision: 64
\[w0 \cdot \sqrt{1.0 - {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1.0 - \frac{{\left(\frac{D}{d} \cdot \frac{M}{2.0}\right)}^{\left(\frac{2.0}{2}\right)} \cdot h}{\sqrt[3]{\ell}} \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{2.0}\right)}^{\left(\frac{2.0}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot w0\]
w0 \cdot \sqrt{1.0 - {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0} \cdot \frac{h}{\ell}}
↓
\sqrt{1.0 - \frac{{\left(\frac{D}{d} \cdot \frac{M}{2.0}\right)}^{\left(\frac{2.0}{2}\right)} \cdot h}{\sqrt[3]{\ell}} \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{2.0}\right)}^{\left(\frac{2.0}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r5859814 = w0;
        double r5859815 = 1.0;
        double r5859816 = M;
        double r5859817 = D;
        double r5859818 = r5859816 * r5859817;
        double r5859819 = 2.0;
        double r5859820 = d;
        double r5859821 = r5859819 * r5859820;
        double r5859822 = r5859818 / r5859821;
        double r5859823 = pow(r5859822, r5859819);
        double r5859824 = h;
        double r5859825 = l;
        double r5859826 = r5859824 / r5859825;
        double r5859827 = r5859823 * r5859826;
        double r5859828 = r5859815 - r5859827;
        double r5859829 = sqrt(r5859828);
        double r5859830 = r5859814 * r5859829;
        return r5859830;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r5859831 = 1.0;
        double r5859832 = D;
        double r5859833 = d;
        double r5859834 = r5859832 / r5859833;
        double r5859835 = M;
        double r5859836 = 2.0;
        double r5859837 = r5859835 / r5859836;
        double r5859838 = r5859834 * r5859837;
        double r5859839 = 2.0;
        double r5859840 = r5859836 / r5859839;
        double r5859841 = pow(r5859838, r5859840);
        double r5859842 = h;
        double r5859843 = r5859841 * r5859842;
        double r5859844 = l;
        double r5859845 = cbrt(r5859844);
        double r5859846 = r5859843 / r5859845;
        double r5859847 = r5859845 * r5859845;
        double r5859848 = r5859841 / r5859847;
        double r5859849 = r5859846 * r5859848;
        double r5859850 = r5859831 - r5859849;
        double r5859851 = sqrt(r5859850);
        double r5859852 = w0;
        double r5859853 = r5859851 * r5859852;
        return r5859853;
}

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 14.8

    \[w0 \cdot \sqrt{1.0 - {\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0} \cdot \frac{h}{\ell}}\]
  2. Using strategy rm

  3. Applied associate-*r/11.7

    \[\leadsto w0 \cdot \sqrt{1.0 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2.0 \cdot d}\right)}^{2.0} \cdot h}{\ell}}}\]
  4. Using strategy rm

  5. Applied times-frac11.6

    \[\leadsto w0 \cdot \sqrt{1.0 - \frac{{\color{blue}{\left(\frac{M}{2.0} \cdot \frac{D}{d}\right)}}^{2.0} \cdot h}{\ell}}\]
  6. Using strategy rm

  7. Applied sqr-pow11.6

    \[\leadsto w0 \cdot \sqrt{1.0 - \frac{\color{blue}{\left({\left(\frac{M}{2.0} \cdot \frac{D}{d}\right)}^{\left(\frac{2.0}{2}\right)} \cdot {\left(\frac{M}{2.0} \cdot \frac{D}{d}\right)}^{\left(\frac{2.0}{2}\right)}\right)} \cdot h}{\ell}}\]

  8. Applied associate-*l*9.9

    \[\leadsto w0 \cdot \sqrt{1.0 - \frac{\color{blue}{{\left(\frac{M}{2.0} \cdot \frac{D}{d}\right)}^{\left(\frac{2.0}{2}\right)} \cdot \left({\left(\frac{M}{2.0} \cdot \frac{D}{d}\right)}^{\left(\frac{2.0}{2}\right)} \cdot h\right)}}{\ell}}\]
  9. Using strategy rm

  10. Applied add-cube-cbrt10.0

    \[\leadsto w0 \cdot \sqrt{1.0 - \frac{{\left(\frac{M}{2.0} \cdot \frac{D}{d}\right)}^{\left(\frac{2.0}{2}\right)} \cdot \left({\left(\frac{M}{2.0} \cdot \frac{D}{d}\right)}^{\left(\frac{2.0}{2}\right)} \cdot h\right)}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}\]

  11. Applied times-frac9.0

    \[\leadsto w0 \cdot \sqrt{1.0 - \color{blue}{\frac{{\left(\frac{M}{2.0} \cdot \frac{D}{d}\right)}^{\left(\frac{2.0}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\frac{M}{2.0} \cdot \frac{D}{d}\right)}^{\left(\frac{2.0}{2}\right)} \cdot h}{\sqrt[3]{\ell}}}}\]

  12. Final simplification9.0

    \[\leadsto \sqrt{1.0 - \frac{{\left(\frac{D}{d} \cdot \frac{M}{2.0}\right)}^{\left(\frac{2.0}{2}\right)} \cdot h}{\sqrt[3]{\ell}} \cdot \frac{{\left(\frac{D}{d} \cdot \frac{M}{2.0}\right)}^{\left(\frac{2.0}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot w0\]

Reproduce

herbie shell --seed 2019165 +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))))))