Average Error: 13.9 → 8.7
Time: 28.8s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\ell \le -6500259796587717 \lor \neg \left(\ell \le 6.37480860988696922644694869262999114928 \cdot 10^{140}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\sqrt[3]{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h} \cdot \sqrt[3]{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}\right) \cdot \sqrt[3]{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}\right)\right) \cdot \frac{1}{\ell}}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\ell \le -6500259796587717 \lor \neg \left(\ell \le 6.37480860988696922644694869262999114928 \cdot 10^{140}\right):\\
\;\;\;\;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 - \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\sqrt[3]{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h} \cdot \sqrt[3]{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}\right) \cdot \sqrt[3]{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}\right)\right) \cdot \frac{1}{\ell}}\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r151179 = w0;
        double r151180 = 1.0;
        double r151181 = M;
        double r151182 = D;
        double r151183 = r151181 * r151182;
        double r151184 = 2.0;
        double r151185 = d;
        double r151186 = r151184 * r151185;
        double r151187 = r151183 / r151186;
        double r151188 = pow(r151187, r151184);
        double r151189 = h;
        double r151190 = l;
        double r151191 = r151189 / r151190;
        double r151192 = r151188 * r151191;
        double r151193 = r151180 - r151192;
        double r151194 = sqrt(r151193);
        double r151195 = r151179 * r151194;
        return r151195;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r151196 = l;
        double r151197 = -6500259796587717.0;
        bool r151198 = r151196 <= r151197;
        double r151199 = 6.374808609886969e+140;
        bool r151200 = r151196 <= r151199;
        double r151201 = !r151200;
        bool r151202 = r151198 || r151201;
        double r151203 = w0;
        double r151204 = 1.0;
        double r151205 = M;
        double r151206 = D;
        double r151207 = r151205 * r151206;
        double r151208 = 2.0;
        double r151209 = d;
        double r151210 = r151208 * r151209;
        double r151211 = r151207 / r151210;
        double r151212 = 2.0;
        double r151213 = r151208 / r151212;
        double r151214 = pow(r151211, r151213);
        double r151215 = h;
        double r151216 = r151215 / r151196;
        double r151217 = r151214 * r151216;
        double r151218 = r151214 * r151217;
        double r151219 = r151204 - r151218;
        double r151220 = sqrt(r151219);
        double r151221 = r151203 * r151220;
        double r151222 = r151205 / r151208;
        double r151223 = r151206 / r151209;
        double r151224 = r151222 * r151223;
        double r151225 = pow(r151224, r151213);
        double r151226 = r151225 * r151215;
        double r151227 = cbrt(r151226);
        double r151228 = r151227 * r151227;
        double r151229 = r151228 * r151227;
        double r151230 = r151225 * r151229;
        double r151231 = 1.0;
        double r151232 = r151231 / r151196;
        double r151233 = r151230 * r151232;
        double r151234 = r151204 - r151233;
        double r151235 = sqrt(r151234);
        double r151236 = r151203 * r151235;
        double r151237 = r151202 ? r151221 : r151236;
        return r151237;
}

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 l < -6500259796587717.0 or 6.374808609886969e+140 < l

    1. Initial program 9.9

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

    3. Applied sqr-pow9.9

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

    4. Applied associate-*l*7.3

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

    if -6500259796587717.0 < l < 6.374808609886969e+140

    1. Initial program 16.9

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

    3. Applied div-inv16.9

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

    4. Applied associate-*r*10.6

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

    6. Applied times-frac10.7

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

    8. Applied sqr-pow10.7

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

    9. Applied associate-*l*9.7

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

    11. Applied add-cube-cbrt9.7

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \le -6500259796587717 \lor \neg \left(\ell \le 6.37480860988696922644694869262999114928 \cdot 10^{140}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(\sqrt[3]{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h} \cdot \sqrt[3]{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}\right) \cdot \sqrt[3]{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}\right)\right) \cdot \frac{1}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 
(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))))))