Average Error: 13.7 → 7.8
Time: 10.5s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{\left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt[3]{h}}{\ell}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{\left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \sqrt[3]{h}}{\ell}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r250398 = w0;
        double r250399 = 1.0;
        double r250400 = M;
        double r250401 = D;
        double r250402 = r250400 * r250401;
        double r250403 = 2.0;
        double r250404 = d;
        double r250405 = r250403 * r250404;
        double r250406 = r250402 / r250405;
        double r250407 = pow(r250406, r250403);
        double r250408 = h;
        double r250409 = l;
        double r250410 = r250408 / r250409;
        double r250411 = r250407 * r250410;
        double r250412 = r250399 - r250411;
        double r250413 = sqrt(r250412);
        double r250414 = r250398 * r250413;
        return r250414;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r250415 = w0;
        double r250416 = 1.0;
        double r250417 = h;
        double r250418 = cbrt(r250417);
        double r250419 = M;
        double r250420 = D;
        double r250421 = r250419 * r250420;
        double r250422 = 2.0;
        double r250423 = d;
        double r250424 = r250422 * r250423;
        double r250425 = r250421 / r250424;
        double r250426 = 2.0;
        double r250427 = r250422 / r250426;
        double r250428 = pow(r250425, r250427);
        double r250429 = r250418 * r250428;
        double r250430 = r250429 * r250418;
        double r250431 = l;
        double r250432 = r250430 / r250431;
        double r250433 = r250429 * r250432;
        double r250434 = r250416 - r250433;
        double r250435 = sqrt(r250434);
        double r250436 = r250415 * r250435;
        return r250436;
}

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 13.7

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

  3. Applied *-un-lft-identity13.7

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

  4. Applied add-cube-cbrt13.8

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

  5. Applied times-frac13.8

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

  6. Applied associate-*r*11.4

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

  7. Simplified11.4

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

  9. Applied sqr-pow11.4

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

  10. Applied unswap-sqr10.3

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

  12. Applied associate-*l*9.1

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

  14. Applied associate-*r/7.8

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

  15. Final simplification7.8

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

Reproduce

herbie shell --seed 2020057 +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))))))