Average Error: 14.1 → 8.5
Time: 31.9s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\ell}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\ell}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r6195368 = w0;
        double r6195369 = 1.0;
        double r6195370 = M;
        double r6195371 = D;
        double r6195372 = r6195370 * r6195371;
        double r6195373 = 2.0;
        double r6195374 = d;
        double r6195375 = r6195373 * r6195374;
        double r6195376 = r6195372 / r6195375;
        double r6195377 = pow(r6195376, r6195373);
        double r6195378 = h;
        double r6195379 = l;
        double r6195380 = r6195378 / r6195379;
        double r6195381 = r6195377 * r6195380;
        double r6195382 = r6195369 - r6195381;
        double r6195383 = sqrt(r6195382);
        double r6195384 = r6195368 * r6195383;
        return r6195384;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r6195385 = w0;
        double r6195386 = 1.0;
        double r6195387 = M;
        double r6195388 = D;
        double r6195389 = r6195387 * r6195388;
        double r6195390 = 2.0;
        double r6195391 = d;
        double r6195392 = r6195390 * r6195391;
        double r6195393 = r6195389 / r6195392;
        double r6195394 = 2.0;
        double r6195395 = r6195390 / r6195394;
        double r6195396 = pow(r6195393, r6195395);
        double r6195397 = h;
        double r6195398 = r6195396 * r6195397;
        double r6195399 = l;
        double r6195400 = r6195398 / r6195399;
        double r6195401 = r6195396 * r6195400;
        double r6195402 = r6195386 - r6195401;
        double r6195403 = sqrt(r6195402);
        double r6195404 = r6195385 * r6195403;
        return r6195404;
}

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/10.7

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}}\]
  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*9.2

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

  8. Applied *-un-lft-identity9.2

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

  9. Applied times-frac8.5

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

  10. Simplified8.5

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

  11. Final simplification8.5

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

Reproduce

herbie shell --seed 2019192 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))