Average Error: 14.1 → 8.6
Time: 15.9s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{{\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)}}{\frac{\ell}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}}}\]
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{1}{\frac{2 \cdot d}{M \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}}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r388 = w0;
        double r389 = 1.0;
        double r390 = M;
        double r391 = D;
        double r392 = r390 * r391;
        double r393 = 2.0;
        double r394 = d;
        double r395 = r393 * r394;
        double r396 = r392 / r395;
        double r397 = pow(r396, r393);
        double r398 = h;
        double r399 = l;
        double r400 = r398 / r399;
        double r401 = r397 * r400;
        double r402 = r389 - r401;
        double r403 = sqrt(r402);
        double r404 = r388 * r403;
        return r404;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r405 = w0;
        double r406 = 1.0;
        double r407 = 1.0;
        double r408 = 2.0;
        double r409 = d;
        double r410 = r408 * r409;
        double r411 = M;
        double r412 = D;
        double r413 = r411 * r412;
        double r414 = r410 / r413;
        double r415 = r407 / r414;
        double r416 = 2.0;
        double r417 = r408 / r416;
        double r418 = pow(r415, r417);
        double r419 = l;
        double r420 = r413 / r410;
        double r421 = pow(r420, r417);
        double r422 = h;
        double r423 = r421 * r422;
        double r424 = r419 / r423;
        double r425 = r418 / r424;
        double r426 = r406 - r425;
        double r427 = sqrt(r426);
        double r428 = r405 * r427;
        return r428;
}

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

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

  11. Final simplification8.6

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

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(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))))))