Average Error: 14.6 → 10.7
Time: 10.4s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;2 \cdot d \le -4.9978883920778511 \cdot 10^{159}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{elif}\;2 \cdot d \le -8.4818635748686101 \cdot 10^{-74}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(0.5 \cdot \left(\frac{h \cdot \left(M \cdot D\right)}{d \cdot \ell} \cdot {\left(\frac{1}{{-1}^{2}}\right)}^{1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\sqrt[3]{h} \cdot \left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;2 \cdot d \le -4.9978883920778511 \cdot 10^{159}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\\

\mathbf{elif}\;2 \cdot d \le -8.4818635748686101 \cdot 10^{-74}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(0.5 \cdot \left(\frac{h \cdot \left(M \cdot D\right)}{d \cdot \ell} \cdot {\left(\frac{1}{{-1}^{2}}\right)}^{1}\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\sqrt[3]{h} \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}\right)}\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r176689 = w0;
        double r176690 = 1.0;
        double r176691 = M;
        double r176692 = D;
        double r176693 = r176691 * r176692;
        double r176694 = 2.0;
        double r176695 = d;
        double r176696 = r176694 * r176695;
        double r176697 = r176693 / r176696;
        double r176698 = pow(r176697, r176694);
        double r176699 = h;
        double r176700 = l;
        double r176701 = r176699 / r176700;
        double r176702 = r176698 * r176701;
        double r176703 = r176690 - r176702;
        double r176704 = sqrt(r176703);
        double r176705 = r176689 * r176704;
        return r176705;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r176706 = 2.0;
        double r176707 = d;
        double r176708 = r176706 * r176707;
        double r176709 = -4.997888392077851e+159;
        bool r176710 = r176708 <= r176709;
        double r176711 = w0;
        double r176712 = 1.0;
        double r176713 = M;
        double r176714 = r176713 / r176706;
        double r176715 = D;
        double r176716 = r176715 / r176707;
        double r176717 = r176714 * r176716;
        double r176718 = pow(r176717, r176706);
        double r176719 = h;
        double r176720 = l;
        double r176721 = r176719 / r176720;
        double r176722 = r176718 * r176721;
        double r176723 = r176712 - r176722;
        double r176724 = sqrt(r176723);
        double r176725 = r176711 * r176724;
        double r176726 = -8.48186357486861e-74;
        bool r176727 = r176708 <= r176726;
        double r176728 = r176713 * r176715;
        double r176729 = r176728 / r176708;
        double r176730 = 2.0;
        double r176731 = r176706 / r176730;
        double r176732 = pow(r176729, r176731);
        double r176733 = 0.5;
        double r176734 = r176719 * r176728;
        double r176735 = r176707 * r176720;
        double r176736 = r176734 / r176735;
        double r176737 = 1.0;
        double r176738 = -1.0;
        double r176739 = pow(r176738, r176706);
        double r176740 = r176737 / r176739;
        double r176741 = pow(r176740, r176712);
        double r176742 = r176736 * r176741;
        double r176743 = r176733 * r176742;
        double r176744 = r176732 * r176743;
        double r176745 = r176712 - r176744;
        double r176746 = sqrt(r176745);
        double r176747 = r176711 * r176746;
        double r176748 = cbrt(r176719);
        double r176749 = r176748 * r176732;
        double r176750 = r176748 * r176749;
        double r176751 = r176748 / r176720;
        double r176752 = r176750 * r176751;
        double r176753 = r176732 * r176752;
        double r176754 = r176712 - r176753;
        double r176755 = sqrt(r176754);
        double r176756 = r176711 * r176755;
        double r176757 = r176727 ? r176747 : r176756;
        double r176758 = r176710 ? r176725 : r176757;
        return r176758;
}

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. Split input into 3 regimes

  2. if (* 2.0 d) < -4.997888392077851e+159

    1. Initial program 10.5

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

    3. Applied times-frac8.9

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

    if -4.997888392077851e+159 < (* 2.0 d) < -8.48186357486861e-74

    1. Initial program 12.3

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

    3. Applied sqr-pow12.3

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

    4. Applied associate-*l*10.9

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

    5. Taylor expanded around -inf 8.6

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

    if -8.48186357486861e-74 < (* 2.0 d)

    1. Initial program 16.5

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

    3. Applied sqr-pow16.5

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

    4. Applied associate-*l*14.6

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

    6. Applied *-un-lft-identity14.6

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

    7. Applied add-cube-cbrt14.6

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

    8. Applied times-frac14.6

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

    9. Applied associate-*r*11.8

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

    10. Simplified11.8

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

    12. Applied associate-*l*11.8

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

  4. Final simplification10.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot d \le -4.9978883920778511 \cdot 10^{159}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{elif}\;2 \cdot d \le -8.4818635748686101 \cdot 10^{-74}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(0.5 \cdot \left(\frac{h \cdot \left(M \cdot D\right)}{d \cdot \ell} \cdot {\left(\frac{1}{{-1}^{2}}\right)}^{1}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\sqrt[3]{h} \cdot \left(\sqrt[3]{h} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 +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))))))