Average Error: 14.1 → 8.4
Time: 18.3s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{{\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\ell}}\]
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{1}{\frac{2 \cdot d}{M \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 r256524 = w0;
        double r256525 = 1.0;
        double r256526 = M;
        double r256527 = D;
        double r256528 = r256526 * r256527;
        double r256529 = 2.0;
        double r256530 = d;
        double r256531 = r256529 * r256530;
        double r256532 = r256528 / r256531;
        double r256533 = pow(r256532, r256529);
        double r256534 = h;
        double r256535 = l;
        double r256536 = r256534 / r256535;
        double r256537 = r256533 * r256536;
        double r256538 = r256525 - r256537;
        double r256539 = sqrt(r256538);
        double r256540 = r256524 * r256539;
        return r256540;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r256541 = w0;
        double r256542 = 1.0;
        double r256543 = M;
        double r256544 = D;
        double r256545 = r256543 * r256544;
        double r256546 = 2.0;
        double r256547 = d;
        double r256548 = r256546 * r256547;
        double r256549 = r256545 / r256548;
        double r256550 = 2.0;
        double r256551 = r256546 / r256550;
        double r256552 = pow(r256549, r256551);
        double r256553 = 1.0;
        double r256554 = r256548 / r256545;
        double r256555 = r256553 / r256554;
        double r256556 = pow(r256555, r256551);
        double r256557 = h;
        double r256558 = r256556 * r256557;
        double r256559 = l;
        double r256560 = r256558 / r256559;
        double r256561 = r256552 * r256560;
        double r256562 = r256542 - r256561;
        double r256563 = sqrt(r256562);
        double r256564 = r256541 * r256563;
        return r256564;
}

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/10.5

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

  5. Applied sqr-pow10.5

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

  6. Applied associate-*l*8.9

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

  8. Applied *-un-lft-identity8.9

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

  9. Applied times-frac8.4

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

  10. Simplified8.4

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

  12. Applied clear-num8.4

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

  13. Final simplification8.4

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

Reproduce

herbie shell --seed 2020046 +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))))))