Average Error: 14.3 → 8.7
Time: 10.8s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\frac{\frac{\ell}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}}{h}}}\]
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{\frac{\ell}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}}{h}}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r193373 = w0;
        double r193374 = 1.0;
        double r193375 = M;
        double r193376 = D;
        double r193377 = r193375 * r193376;
        double r193378 = 2.0;
        double r193379 = d;
        double r193380 = r193378 * r193379;
        double r193381 = r193377 / r193380;
        double r193382 = pow(r193381, r193378);
        double r193383 = h;
        double r193384 = l;
        double r193385 = r193383 / r193384;
        double r193386 = r193382 * r193385;
        double r193387 = r193374 - r193386;
        double r193388 = sqrt(r193387);
        double r193389 = r193373 * r193388;
        return r193389;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r193390 = w0;
        double r193391 = 1.0;
        double r193392 = M;
        double r193393 = D;
        double r193394 = r193392 * r193393;
        double r193395 = 2.0;
        double r193396 = d;
        double r193397 = r193395 * r193396;
        double r193398 = r193394 / r193397;
        double r193399 = 2.0;
        double r193400 = r193395 / r193399;
        double r193401 = pow(r193398, r193400);
        double r193402 = l;
        double r193403 = r193402 / r193401;
        double r193404 = h;
        double r193405 = r193403 / r193404;
        double r193406 = r193401 / r193405;
        double r193407 = r193391 - r193406;
        double r193408 = sqrt(r193407);
        double r193409 = r193390 * r193408;
        return r193409;
}

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 14.3

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

  3. Applied associate-*r/10.7

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

  5. Applied sqr-pow10.7

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

  6. Applied associate-*l*9.1

    \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}}{\ell}}\]
  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 associate-/r*8.7

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

  11. Final simplification8.7

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

Reproduce

herbie shell --seed 2020064 +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))))))