Average Error: 14.6 → 8.8
Time: 10.6s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -2.566959543532192688587992548008487318806 \cdot 10^{-296}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]
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:\\
\;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\\

\mathbf{elif}\;\frac{h}{\ell} \le -2.566959543532192688587992548008487318806 \cdot 10^{-296}:\\
\;\;\;\;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)}\\

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

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r255280 = w0;
        double r255281 = 1.0;
        double r255282 = M;
        double r255283 = D;
        double r255284 = r255282 * r255283;
        double r255285 = 2.0;
        double r255286 = d;
        double r255287 = r255285 * r255286;
        double r255288 = r255284 / r255287;
        double r255289 = pow(r255288, r255285);
        double r255290 = h;
        double r255291 = l;
        double r255292 = r255290 / r255291;
        double r255293 = r255289 * r255292;
        double r255294 = r255281 - r255293;
        double r255295 = sqrt(r255294);
        double r255296 = r255280 * r255295;
        return r255296;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r255297 = h;
        double r255298 = l;
        double r255299 = r255297 / r255298;
        double r255300 = -inf.0;
        bool r255301 = r255299 <= r255300;
        double r255302 = w0;
        double r255303 = 1.0;
        double r255304 = M;
        double r255305 = D;
        double r255306 = r255304 * r255305;
        double r255307 = 2.0;
        double r255308 = d;
        double r255309 = r255307 * r255308;
        double r255310 = r255306 / r255309;
        double r255311 = pow(r255310, r255307);
        double r255312 = r255311 * r255297;
        double r255313 = 1.0;
        double r255314 = r255313 / r255298;
        double r255315 = r255312 * r255314;
        double r255316 = r255303 - r255315;
        double r255317 = sqrt(r255316);
        double r255318 = r255302 * r255317;
        double r255319 = -2.5669595435321927e-296;
        bool r255320 = r255299 <= r255319;
        double r255321 = 2.0;
        double r255322 = r255307 / r255321;
        double r255323 = pow(r255310, r255322);
        double r255324 = r255323 * r255299;
        double r255325 = r255323 * r255324;
        double r255326 = r255303 - r255325;
        double r255327 = sqrt(r255326);
        double r255328 = r255302 * r255327;
        double r255329 = sqrt(r255303);
        double r255330 = r255302 * r255329;
        double r255331 = r255320 ? r255328 : r255330;
        double r255332 = r255301 ? r255318 : r255331;
        return r255332;
}

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 (/ h l) < -inf.0

    1. Initial program 64.0

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

    3. Applied div-inv64.0

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

    4. Applied associate-*r*26.9

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

    if -inf.0 < (/ h l) < -2.5669595435321927e-296

    1. Initial program 14.5

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

    3. Applied sqr-pow14.5

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

    4. Applied associate-*l*12.5

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

    if -2.5669595435321927e-296 < (/ h l)

    1. Initial program 8.3

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

    2. Taylor expanded around 0 3.0

      \[\leadsto w0 \cdot \color{blue}{\sqrt{1}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -2.566959543532192688587992548008487318806 \cdot 10^{-296}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(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))))))