Average Error: 13.9 → 8.7
Time: 11.8s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right) \cdot \frac{1}{\ell}\right)}\]
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 \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right) \cdot \frac{1}{\ell}\right)}
double f(double w0, double M, double D, double h, double l, double d) {
        double r212484 = w0;
        double r212485 = 1.0;
        double r212486 = M;
        double r212487 = D;
        double r212488 = r212486 * r212487;
        double r212489 = 2.0;
        double r212490 = d;
        double r212491 = r212489 * r212490;
        double r212492 = r212488 / r212491;
        double r212493 = pow(r212492, r212489);
        double r212494 = h;
        double r212495 = l;
        double r212496 = r212494 / r212495;
        double r212497 = r212493 * r212496;
        double r212498 = r212485 - r212497;
        double r212499 = sqrt(r212498);
        double r212500 = r212484 * r212499;
        return r212500;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r212501 = w0;
        double r212502 = 1.0;
        double r212503 = M;
        double r212504 = D;
        double r212505 = r212503 * r212504;
        double r212506 = 2.0;
        double r212507 = d;
        double r212508 = r212506 * r212507;
        double r212509 = r212505 / r212508;
        double r212510 = 2.0;
        double r212511 = r212506 / r212510;
        double r212512 = pow(r212509, r212511);
        double r212513 = h;
        double r212514 = r212512 * r212513;
        double r212515 = 1.0;
        double r212516 = l;
        double r212517 = r212515 / r212516;
        double r212518 = r212514 * r212517;
        double r212519 = r212512 * r212518;
        double r212520 = r212502 - r212519;
        double r212521 = sqrt(r212520);
        double r212522 = r212501 * r212521;
        return r212522;
}

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 13.9

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

  3. Applied sqr-pow13.9

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

  4. Applied associate-*l*12.7

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

  6. Applied div-inv12.7

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

  7. Applied associate-*r*8.7

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

  8. Final simplification8.7

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

Reproduce

herbie shell --seed 2019354 +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))))))