Average Error: 13.1 → 7.6
Time: 3.7m
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\left|\sqrt{1 - \left(\frac{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}{\frac{\frac{d \cdot 2}{M}}{D}} \cdot \frac{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}{\frac{\frac{d \cdot 2}{M}}{D}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right| \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\left|\sqrt{1 - \left(\frac{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}{\frac{\frac{d \cdot 2}{M}}{D}} \cdot \frac{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}{\frac{\frac{d \cdot 2}{M}}{D}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\right| \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r41875422 = w0;
        double r41875423 = 1.0;
        double r41875424 = M;
        double r41875425 = D;
        double r41875426 = r41875424 * r41875425;
        double r41875427 = 2.0;
        double r41875428 = d;
        double r41875429 = r41875427 * r41875428;
        double r41875430 = r41875426 / r41875429;
        double r41875431 = pow(r41875430, r41875427);
        double r41875432 = h;
        double r41875433 = l;
        double r41875434 = r41875432 / r41875433;
        double r41875435 = r41875431 * r41875434;
        double r41875436 = r41875423 - r41875435;
        double r41875437 = sqrt(r41875436);
        double r41875438 = r41875422 * r41875437;
        return r41875438;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r41875439 = 1.0;
        double r41875440 = h;
        double r41875441 = cbrt(r41875440);
        double r41875442 = l;
        double r41875443 = cbrt(r41875442);
        double r41875444 = r41875441 / r41875443;
        double r41875445 = d;
        double r41875446 = 2.0;
        double r41875447 = r41875445 * r41875446;
        double r41875448 = M;
        double r41875449 = r41875447 / r41875448;
        double r41875450 = D;
        double r41875451 = r41875449 / r41875450;
        double r41875452 = r41875444 / r41875451;
        double r41875453 = r41875452 * r41875452;
        double r41875454 = r41875453 * r41875444;
        double r41875455 = r41875439 - r41875454;
        double r41875456 = sqrt(r41875455);
        double r41875457 = fabs(r41875456);
        double r41875458 = w0;
        double r41875459 = r41875457 * r41875458;
        return r41875459;
}

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 13.1

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

  2. Simplified11.8

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

  4. Applied add-cube-cbrt11.8

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

  5. Applied add-cube-cbrt11.8

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

  6. Applied times-frac11.8

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

  7. Applied associate-*l*8.4

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

  8. Simplified8.4

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

  10. Applied add-sqr-sqrt8.4

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

  11. Applied rem-sqrt-square8.4

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

  12. Simplified7.6

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

  13. Final simplification7.6

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

Reproduce

herbie shell --seed 2019121 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))