Average Error: 14.4 → 9.5
Time: 11.4s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left({\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{\left(2 \cdot d\right) \cdot \frac{1}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right) \cdot \frac{1}{\ell}}\]
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{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{\left(2 \cdot d\right) \cdot \frac{1}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right) \cdot \frac{1}{\ell}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r266409 = w0;
        double r266410 = 1.0;
        double r266411 = M;
        double r266412 = D;
        double r266413 = r266411 * r266412;
        double r266414 = 2.0;
        double r266415 = d;
        double r266416 = r266414 * r266415;
        double r266417 = r266413 / r266416;
        double r266418 = pow(r266417, r266414);
        double r266419 = h;
        double r266420 = l;
        double r266421 = r266419 / r266420;
        double r266422 = r266418 * r266421;
        double r266423 = r266410 - r266422;
        double r266424 = sqrt(r266423);
        double r266425 = r266409 * r266424;
        return r266425;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r266426 = w0;
        double r266427 = 1.0;
        double r266428 = 1.0;
        double r266429 = 2.0;
        double r266430 = d;
        double r266431 = r266429 * r266430;
        double r266432 = M;
        double r266433 = D;
        double r266434 = r266432 * r266433;
        double r266435 = r266431 / r266434;
        double r266436 = r266428 / r266435;
        double r266437 = 2.0;
        double r266438 = r266429 / r266437;
        double r266439 = pow(r266436, r266438);
        double r266440 = r266428 / r266434;
        double r266441 = r266431 * r266440;
        double r266442 = r266428 / r266441;
        double r266443 = pow(r266442, r266438);
        double r266444 = h;
        double r266445 = r266443 * r266444;
        double r266446 = r266439 * r266445;
        double r266447 = l;
        double r266448 = r266428 / r266447;
        double r266449 = r266446 * r266448;
        double r266450 = r266427 - r266449;
        double r266451 = sqrt(r266450);
        double r266452 = r266426 * r266451;
        return r266452;
}

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*11.0

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

  6. Applied clear-num11.0

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

  8. Applied sqr-pow11.0

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

  9. Applied associate-*l*9.5

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

  11. Applied div-inv9.5

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

  12. Final simplification9.5

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

Reproduce

herbie shell --seed 2019356 
(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))))))