Average Error: 13.7 → 8.0
Time: 24.8s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -1.383958076153428267352270978205041932919 \cdot 10^{195}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -3.118272253432409186567015566500838854162 \cdot 10^{-70}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right) \cdot \frac{1}{\ell}\right)}\\ \end{array}\]
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} \le -1.383958076153428267352270978205041932919 \cdot 10^{195}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \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}}\\

\mathbf{elif}\;\frac{h}{\ell} \le -3.118272253432409186567015566500838854162 \cdot 10^{-70}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\\

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

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r145226 = w0;
        double r145227 = 1.0;
        double r145228 = M;
        double r145229 = D;
        double r145230 = r145228 * r145229;
        double r145231 = 2.0;
        double r145232 = d;
        double r145233 = r145231 * r145232;
        double r145234 = r145230 / r145233;
        double r145235 = pow(r145234, r145231);
        double r145236 = h;
        double r145237 = l;
        double r145238 = r145236 / r145237;
        double r145239 = r145235 * r145238;
        double r145240 = r145227 - r145239;
        double r145241 = sqrt(r145240);
        double r145242 = r145226 * r145241;
        return r145242;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r145243 = h;
        double r145244 = l;
        double r145245 = r145243 / r145244;
        double r145246 = -1.3839580761534283e+195;
        bool r145247 = r145245 <= r145246;
        double r145248 = w0;
        double r145249 = 1.0;
        double r145250 = M;
        double r145251 = D;
        double r145252 = r145250 * r145251;
        double r145253 = 2.0;
        double r145254 = d;
        double r145255 = r145253 * r145254;
        double r145256 = r145252 / r145255;
        double r145257 = 2.0;
        double r145258 = r145253 / r145257;
        double r145259 = pow(r145256, r145258);
        double r145260 = r145259 * r145243;
        double r145261 = r145259 * r145260;
        double r145262 = 1.0;
        double r145263 = r145262 / r145244;
        double r145264 = r145261 * r145263;
        double r145265 = r145249 - r145264;
        double r145266 = sqrt(r145265);
        double r145267 = r145248 * r145266;
        double r145268 = -3.118272253432409e-70;
        bool r145269 = r145245 <= r145268;
        double r145270 = r145250 / r145253;
        double r145271 = r145251 / r145254;
        double r145272 = r145270 * r145271;
        double r145273 = pow(r145272, r145253);
        double r145274 = r145273 * r145245;
        double r145275 = r145249 - r145274;
        double r145276 = sqrt(r145275);
        double r145277 = r145248 * r145276;
        double r145278 = r145260 * r145263;
        double r145279 = r145259 * r145278;
        double r145280 = r145249 - r145279;
        double r145281 = sqrt(r145280);
        double r145282 = r145248 * r145281;
        double r145283 = r145269 ? r145277 : r145282;
        double r145284 = r145247 ? r145267 : r145283;
        return r145284;
}

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) < -1.3839580761534283e+195

    1. Initial program 39.0

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

    3. Applied div-inv39.0

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

    4. Applied associate-*r*21.1

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

    6. Applied sqr-pow21.1

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

    7. Applied associate-*l*18.9

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

    if -1.3839580761534283e+195 < (/ h l) < -3.118272253432409e-70

    1. Initial program 13.0

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

    3. Applied times-frac13.0

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

    if -3.118272253432409e-70 < (/ h l)

    1. Initial program 9.4

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

    3. Applied div-inv9.4

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

    4. Applied associate-*r*7.0

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

    6. Applied sqr-pow7.0

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

    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 associate-*l*4.4

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

    10. Simplified4.4

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

    12. Applied div-inv4.4

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

  4. Final simplification8.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -1.383958076153428267352270978205041932919 \cdot 10^{195}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -3.118272253432409186567015566500838854162 \cdot 10^{-70}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right) \cdot \frac{1}{\ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(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))))))