Average Error: 14.1 → 8.1
Time: 34.4s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \frac{{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}} \cdot \frac{h \cdot \frac{{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}} \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\sqrt{1 - \frac{{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}} \cdot \frac{h \cdot \frac{{\left(\frac{M \cdot \frac{D}{d}}{2}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}} \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r138479 = w0;
        double r138480 = 1.0;
        double r138481 = M;
        double r138482 = D;
        double r138483 = r138481 * r138482;
        double r138484 = 2.0;
        double r138485 = d;
        double r138486 = r138484 * r138485;
        double r138487 = r138483 / r138486;
        double r138488 = pow(r138487, r138484);
        double r138489 = h;
        double r138490 = l;
        double r138491 = r138489 / r138490;
        double r138492 = r138488 * r138491;
        double r138493 = r138480 - r138492;
        double r138494 = sqrt(r138493);
        double r138495 = r138479 * r138494;
        return r138495;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r138496 = 1.0;
        double r138497 = M;
        double r138498 = D;
        double r138499 = d;
        double r138500 = r138498 / r138499;
        double r138501 = r138497 * r138500;
        double r138502 = 2.0;
        double r138503 = r138501 / r138502;
        double r138504 = 2.0;
        double r138505 = r138502 / r138504;
        double r138506 = pow(r138503, r138505);
        double r138507 = l;
        double r138508 = cbrt(r138507);
        double r138509 = r138506 / r138508;
        double r138510 = h;
        double r138511 = r138510 * r138509;
        double r138512 = r138511 / r138508;
        double r138513 = r138509 * r138512;
        double r138514 = r138496 - r138513;
        double r138515 = sqrt(r138514);
        double r138516 = w0;
        double r138517 = r138515 * r138516;
        return r138517;
}

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. Simplified10.9

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

  4. Applied sqr-pow10.9

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

  5. Applied associate-*r*9.4

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

  6. Simplified10.2

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

  8. Applied add-cube-cbrt10.2

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

  9. Applied times-frac9.4

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

  10. Simplified9.3

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

  11. Simplified8.1

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

  12. Final simplification8.1

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

Reproduce

herbie shell --seed 2019179 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))