Average Error: 14.1 → 8.6
Time: 28.6s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty \lor \neg \left(\frac{h}{\ell} \le -1.137897252452376107719566610395472751847 \cdot 10^{-223}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} = -\infty \lor \neg \left(\frac{h}{\ell} \le -1.137897252452376107719566610395472751847 \cdot 10^{-223}\right):\\
\;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right) \cdot \frac{1}{\ell}}\\

\mathbf{else}:\\
\;\;\;\;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}{\ell}\right)}\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r153704 = w0;
        double r153705 = 1.0;
        double r153706 = M;
        double r153707 = D;
        double r153708 = r153706 * r153707;
        double r153709 = 2.0;
        double r153710 = d;
        double r153711 = r153709 * r153710;
        double r153712 = r153708 / r153711;
        double r153713 = pow(r153712, r153709);
        double r153714 = h;
        double r153715 = l;
        double r153716 = r153714 / r153715;
        double r153717 = r153713 * r153716;
        double r153718 = r153705 - r153717;
        double r153719 = sqrt(r153718);
        double r153720 = r153704 * r153719;
        return r153720;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r153721 = h;
        double r153722 = l;
        double r153723 = r153721 / r153722;
        double r153724 = -inf.0;
        bool r153725 = r153723 <= r153724;
        double r153726 = -1.1378972524523761e-223;
        bool r153727 = r153723 <= r153726;
        double r153728 = !r153727;
        bool r153729 = r153725 || r153728;
        double r153730 = w0;
        double r153731 = 1.0;
        double r153732 = M;
        double r153733 = D;
        double r153734 = r153732 * r153733;
        double r153735 = 2.0;
        double r153736 = d;
        double r153737 = r153735 * r153736;
        double r153738 = r153734 / r153737;
        double r153739 = 2.0;
        double r153740 = r153735 / r153739;
        double r153741 = pow(r153738, r153740);
        double r153742 = 1.0;
        double r153743 = r153742 / r153737;
        double r153744 = r153734 * r153743;
        double r153745 = pow(r153744, r153740);
        double r153746 = r153745 * r153721;
        double r153747 = r153741 * r153746;
        double r153748 = r153742 / r153722;
        double r153749 = r153747 * r153748;
        double r153750 = r153731 - r153749;
        double r153751 = sqrt(r153750);
        double r153752 = r153730 * r153751;
        double r153753 = r153741 * r153723;
        double r153754 = r153741 * r153753;
        double r153755 = r153731 - r153754;
        double r153756 = sqrt(r153755);
        double r153757 = r153730 * r153756;
        double r153758 = r153729 ? r153752 : r153757;
        return r153758;
}

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 2 regimes

  2. if (/ h l) < -inf.0 or -1.1378972524523761e-223 < (/ h l)

    1. Initial program 13.9

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

    3. Applied div-inv13.9

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

    4. Applied associate-*r*7.5

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

    6. Applied sqr-pow7.5

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

    7. Applied associate-*l*5.4

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

    9. Applied div-inv5.4

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

    if -inf.0 < (/ h l) < -1.1378972524523761e-223

    1. Initial program 14.4

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

    3. Applied sqr-pow14.4

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

    4. Applied associate-*l*13.0

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty \lor \neg \left(\frac{h}{\ell} \le -1.137897252452376107719566610395472751847 \cdot 10^{-223}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +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))))))