Average Error: 14.1 → 8.6
Time: 1.2m
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - {\left(\frac{\frac{M}{\frac{2}{D}}}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\frac{\ell}{{\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{\left(\frac{2}{2}\right)}}}} \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\sqrt{1 - {\left(\frac{\frac{M}{\frac{2}{D}}}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\frac{\ell}{{\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{\left(\frac{2}{2}\right)}}}} \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r215519 = w0;
        double r215520 = 1.0;
        double r215521 = M;
        double r215522 = D;
        double r215523 = r215521 * r215522;
        double r215524 = 2.0;
        double r215525 = d;
        double r215526 = r215524 * r215525;
        double r215527 = r215523 / r215526;
        double r215528 = pow(r215527, r215524);
        double r215529 = h;
        double r215530 = l;
        double r215531 = r215529 / r215530;
        double r215532 = r215528 * r215531;
        double r215533 = r215520 - r215532;
        double r215534 = sqrt(r215533);
        double r215535 = r215519 * r215534;
        return r215535;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r215536 = 1.0;
        double r215537 = M;
        double r215538 = 2.0;
        double r215539 = D;
        double r215540 = r215538 / r215539;
        double r215541 = r215537 / r215540;
        double r215542 = d;
        double r215543 = r215541 / r215542;
        double r215544 = 2.0;
        double r215545 = r215538 / r215544;
        double r215546 = pow(r215543, r215545);
        double r215547 = h;
        double r215548 = l;
        double r215549 = 0.5;
        double r215550 = r215539 * r215537;
        double r215551 = r215550 / r215542;
        double r215552 = r215549 * r215551;
        double r215553 = pow(r215552, r215545);
        double r215554 = r215548 / r215553;
        double r215555 = r215547 / r215554;
        double r215556 = r215546 * r215555;
        double r215557 = r215536 - r215556;
        double r215558 = sqrt(r215557);
        double r215559 = w0;
        double r215560 = r215558 * r215559;
        return r215560;
}

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/10.9

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

  4. Simplified10.6

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

  6. Applied sqr-pow10.6

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

  7. Applied associate-*l*9.1

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

  8. Simplified9.0

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

  10. Applied *-un-lft-identity9.0

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

  11. Applied times-frac8.3

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

  12. Simplified9.4

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

  13. Simplified8.6

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

  14. Taylor expanded around 0 8.6

    \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{\frac{M}{\frac{2}{D}}}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\frac{\ell}{{\color{blue}{\left(0.5 \cdot \frac{M \cdot D}{d}\right)}}^{\left(\frac{2}{2}\right)}}}}\]

  15. Final simplification8.6

    \[\leadsto \sqrt{1 - {\left(\frac{\frac{M}{\frac{2}{D}}}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\frac{\ell}{{\left(0.5 \cdot \frac{D \cdot M}{d}\right)}^{\left(\frac{2}{2}\right)}}}} \cdot w0\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(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))))))