Average Error: 14.3 → 9.5
Time: 31.3s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -1.772133656628061046787329400235105824789 \cdot 10^{308}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \mathbf{elif}\;\frac{h}{\ell} \le -2.034203706704132394404794445791386313725 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \end{array}\]
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} \le -1.772133656628061046787329400235105824789 \cdot 10^{308}:\\
\;\;\;\;\sqrt{1} \cdot w0\\

\mathbf{elif}\;\frac{h}{\ell} \le -2.034203706704132394404794445791386313725 \cdot 10^{-311}:\\
\;\;\;\;\sqrt{1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot w0\\

\mathbf{else}:\\
\;\;\;\;\sqrt{1} \cdot w0\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r5145972 = w0;
        double r5145973 = 1.0;
        double r5145974 = M;
        double r5145975 = D;
        double r5145976 = r5145974 * r5145975;
        double r5145977 = 2.0;
        double r5145978 = d;
        double r5145979 = r5145977 * r5145978;
        double r5145980 = r5145976 / r5145979;
        double r5145981 = pow(r5145980, r5145977);
        double r5145982 = h;
        double r5145983 = l;
        double r5145984 = r5145982 / r5145983;
        double r5145985 = r5145981 * r5145984;
        double r5145986 = r5145973 - r5145985;
        double r5145987 = sqrt(r5145986);
        double r5145988 = r5145972 * r5145987;
        return r5145988;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r5145989 = h;
        double r5145990 = l;
        double r5145991 = r5145989 / r5145990;
        double r5145992 = -1.772133656628061e+308;
        bool r5145993 = r5145991 <= r5145992;
        double r5145994 = 1.0;
        double r5145995 = sqrt(r5145994);
        double r5145996 = w0;
        double r5145997 = r5145995 * r5145996;
        double r5145998 = -2.0342037067041e-311;
        bool r5145999 = r5145991 <= r5145998;
        double r5146000 = M;
        double r5146001 = D;
        double r5146002 = r5146000 * r5146001;
        double r5146003 = d;
        double r5146004 = 2.0;
        double r5146005 = r5146003 * r5146004;
        double r5146006 = r5146002 / r5146005;
        double r5146007 = 2.0;
        double r5146008 = r5146004 / r5146007;
        double r5146009 = pow(r5146006, r5146008);
        double r5146010 = r5145991 * r5146009;
        double r5146011 = r5146009 * r5146010;
        double r5146012 = r5145994 - r5146011;
        double r5146013 = sqrt(r5146012);
        double r5146014 = r5146013 * r5145996;
        double r5146015 = r5145999 ? r5146014 : r5145997;
        double r5146016 = r5145993 ? r5145997 : r5146015;
        return r5146016;
}

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) < -1.772133656628061e+308 or -2.0342037067041e-311 < (/ h l)

    1. Initial program 13.1

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

    2. Taylor expanded around 0 5.9

      \[\leadsto w0 \cdot \color{blue}{\sqrt{1}}\]

    if -1.772133656628061e+308 < (/ h l) < -2.0342037067041e-311

    1. Initial program 15.7

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

    3. Applied sqr-pow15.7

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

    4. Applied associate-*l*13.7

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -1.772133656628061046787329400235105824789 \cdot 10^{308}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \mathbf{elif}\;\frac{h}{\ell} \le -2.034203706704132394404794445791386313725 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(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))))))