Average Error: 14.4 → 8.7
Time: 32.3s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right) \cdot \frac{1}{\ell}\right)}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right) \cdot \frac{1}{\ell}\right)}
double f(double w0, double M, double D, double h, double l, double d) {
        double r6244959 = w0;
        double r6244960 = 1.0;
        double r6244961 = M;
        double r6244962 = D;
        double r6244963 = r6244961 * r6244962;
        double r6244964 = 2.0;
        double r6244965 = d;
        double r6244966 = r6244964 * r6244965;
        double r6244967 = r6244963 / r6244966;
        double r6244968 = pow(r6244967, r6244964);
        double r6244969 = h;
        double r6244970 = l;
        double r6244971 = r6244969 / r6244970;
        double r6244972 = r6244968 * r6244971;
        double r6244973 = r6244960 - r6244972;
        double r6244974 = sqrt(r6244973);
        double r6244975 = r6244959 * r6244974;
        return r6244975;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r6244976 = w0;
        double r6244977 = 1.0;
        double r6244978 = M;
        double r6244979 = D;
        double r6244980 = r6244978 * r6244979;
        double r6244981 = 2.0;
        double r6244982 = d;
        double r6244983 = r6244981 * r6244982;
        double r6244984 = r6244980 / r6244983;
        double r6244985 = 2.0;
        double r6244986 = r6244981 / r6244985;
        double r6244987 = pow(r6244984, r6244986);
        double r6244988 = h;
        double r6244989 = r6244987 * r6244988;
        double r6244990 = 1.0;
        double r6244991 = l;
        double r6244992 = r6244990 / r6244991;
        double r6244993 = r6244989 * r6244992;
        double r6244994 = r6244987 * r6244993;
        double r6244995 = r6244977 - r6244994;
        double r6244996 = sqrt(r6244995);
        double r6244997 = r6244976 * r6244996;
        return r6244997;
}

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 14.4

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

  3. Applied div-inv14.4

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

  4. Applied associate-*r*10.9

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

  6. Applied sqr-pow10.9

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

  7. Applied associate-*l*9.2

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

  9. Applied associate-*l*8.7

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

  10. Final simplification8.7

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

Reproduce

herbie shell --seed 2019172 
(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))))))