Average Error: 14.1 → 8.4
Time: 27.0s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\sqrt[3]{\ell}}}\]
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)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\sqrt[3]{\ell}}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r175471 = w0;
        double r175472 = 1.0;
        double r175473 = M;
        double r175474 = D;
        double r175475 = r175473 * r175474;
        double r175476 = 2.0;
        double r175477 = d;
        double r175478 = r175476 * r175477;
        double r175479 = r175475 / r175478;
        double r175480 = pow(r175479, r175476);
        double r175481 = h;
        double r175482 = l;
        double r175483 = r175481 / r175482;
        double r175484 = r175480 * r175483;
        double r175485 = r175472 - r175484;
        double r175486 = sqrt(r175485);
        double r175487 = r175471 * r175486;
        return r175487;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r175488 = w0;
        double r175489 = 1.0;
        double r175490 = M;
        double r175491 = D;
        double r175492 = r175490 * r175491;
        double r175493 = 2.0;
        double r175494 = d;
        double r175495 = r175493 * r175494;
        double r175496 = r175492 / r175495;
        double r175497 = 2.0;
        double r175498 = r175493 / r175497;
        double r175499 = pow(r175496, r175498);
        double r175500 = l;
        double r175501 = cbrt(r175500);
        double r175502 = r175501 * r175501;
        double r175503 = r175499 / r175502;
        double r175504 = h;
        double r175505 = r175499 * r175504;
        double r175506 = r175505 / r175501;
        double r175507 = r175503 * r175506;
        double r175508 = r175489 - r175507;
        double r175509 = sqrt(r175508);
        double r175510 = r175488 * r175509;
        return r175510;
}

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/10.9

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

  5. Applied sqr-pow10.9

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

  6. Applied associate-*l*9.3

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

  8. Applied add-cube-cbrt9.4

    \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}}\]

  9. Applied times-frac8.4

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\sqrt[3]{\ell}}}}\]

  10. Final simplification8.4

    \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\sqrt[3]{\ell}}}\]

Reproduce

herbie shell --seed 2019303 +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))))))