Average Error: 14.1 → 8.6
Time: 15.8s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\frac{\ell}{{\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h}}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\frac{\ell}{{\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h}}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r413 = w0;
        double r414 = 1.0;
        double r415 = M;
        double r416 = D;
        double r417 = r415 * r416;
        double r418 = 2.0;
        double r419 = d;
        double r420 = r418 * r419;
        double r421 = r417 / r420;
        double r422 = pow(r421, r418);
        double r423 = h;
        double r424 = l;
        double r425 = r423 / r424;
        double r426 = r422 * r425;
        double r427 = r414 - r426;
        double r428 = sqrt(r427);
        double r429 = r413 * r428;
        return r429;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r430 = w0;
        double r431 = 1.0;
        double r432 = M;
        double r433 = D;
        double r434 = r432 * r433;
        double r435 = 2.0;
        double r436 = d;
        double r437 = r435 * r436;
        double r438 = r434 / r437;
        double r439 = 2.0;
        double r440 = r435 / r439;
        double r441 = pow(r438, r440);
        double r442 = l;
        double r443 = 1.0;
        double r444 = r437 / r434;
        double r445 = r443 / r444;
        double r446 = pow(r445, r440);
        double r447 = h;
        double r448 = r446 * r447;
        double r449 = r442 / r448;
        double r450 = r441 / r449;
        double r451 = r431 - r450;
        double r452 = sqrt(r451);
        double r453 = r430 * r452;
        return r453;
}

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 14.1

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

  3. Applied associate-*r/10.8

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

  5. Applied sqr-pow10.8

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

  6. Applied associate-*l*9.4

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

  8. Applied associate-/l*8.6

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

  10. Applied clear-num8.6

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

  11. Final simplification8.6

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

Reproduce

herbie shell --seed 2020025 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))