Average Error: 13.7 → 9.0
Time: 10.1s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right) \cdot \frac{1}{\ell}}\]
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{M \cdot D}{2 \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}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r215347 = w0;
        double r215348 = 1.0;
        double r215349 = M;
        double r215350 = D;
        double r215351 = r215349 * r215350;
        double r215352 = 2.0;
        double r215353 = d;
        double r215354 = r215352 * r215353;
        double r215355 = r215351 / r215354;
        double r215356 = pow(r215355, r215352);
        double r215357 = h;
        double r215358 = l;
        double r215359 = r215357 / r215358;
        double r215360 = r215356 * r215359;
        double r215361 = r215348 - r215360;
        double r215362 = sqrt(r215361);
        double r215363 = r215347 * r215362;
        return r215363;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r215364 = w0;
        double r215365 = 1.0;
        double r215366 = M;
        double r215367 = D;
        double r215368 = r215366 * r215367;
        double r215369 = 2.0;
        double r215370 = d;
        double r215371 = r215369 * r215370;
        double r215372 = r215368 / r215371;
        double r215373 = 2.0;
        double r215374 = r215369 / r215373;
        double r215375 = pow(r215372, r215374);
        double r215376 = 1.0;
        double r215377 = r215371 / r215368;
        double r215378 = r215376 / r215377;
        double r215379 = pow(r215378, r215374);
        double r215380 = h;
        double r215381 = r215379 * r215380;
        double r215382 = r215375 * r215381;
        double r215383 = l;
        double r215384 = r215376 / r215383;
        double r215385 = r215382 * r215384;
        double r215386 = r215365 - r215385;
        double r215387 = sqrt(r215386);
        double r215388 = r215364 * r215387;
        return r215388;
}

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 13.7

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

  3. Applied div-inv13.7

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

  4. Applied associate-*r*10.5

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

  6. Applied sqr-pow10.5

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

  7. Applied associate-*l*9.0

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

  9. Applied clear-num9.0

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

  10. Final simplification9.0

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

Reproduce

herbie shell --seed 2020039 
(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))))))