Average Error: 14.2 → 9.2
Time: 11.7s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right) \cdot \frac{1}{\ell}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right) \cdot \frac{1}{\ell}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r223689 = w0;
        double r223690 = 1.0;
        double r223691 = M;
        double r223692 = D;
        double r223693 = r223691 * r223692;
        double r223694 = 2.0;
        double r223695 = d;
        double r223696 = r223694 * r223695;
        double r223697 = r223693 / r223696;
        double r223698 = pow(r223697, r223694);
        double r223699 = h;
        double r223700 = l;
        double r223701 = r223699 / r223700;
        double r223702 = r223698 * r223701;
        double r223703 = r223690 - r223702;
        double r223704 = sqrt(r223703);
        double r223705 = r223689 * r223704;
        return r223705;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r223706 = w0;
        double r223707 = 1.0;
        double r223708 = M;
        double r223709 = D;
        double r223710 = r223708 * r223709;
        double r223711 = 2.0;
        double r223712 = d;
        double r223713 = r223711 * r223712;
        double r223714 = r223710 / r223713;
        double r223715 = 2.0;
        double r223716 = r223711 / r223715;
        double r223717 = pow(r223714, r223716);
        double r223718 = 1.0;
        double r223719 = r223713 / r223710;
        double r223720 = r223718 / r223719;
        double r223721 = pow(r223720, r223716);
        double r223722 = h;
        double r223723 = r223721 * r223722;
        double r223724 = r223717 * r223723;
        double r223725 = l;
        double r223726 = r223718 / r223725;
        double r223727 = r223724 * r223726;
        double r223728 = r223707 - r223727;
        double r223729 = sqrt(r223728);
        double r223730 = r223706 * r223729;
        return r223730;
}

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 14.2

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

  3. Applied div-inv14.2

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

  4. Applied associate-*r*10.7

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

  6. Applied sqr-pow10.7

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

  7. Applied associate-*l*9.2

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

  9. Applied clear-num9.2

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

  10. Final simplification9.2

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

Reproduce

herbie shell --seed 2020021 +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))))))