Average Error: 14.5 → 10.0
Time: 28.6s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -1.510341556579411067921188404923562874146 \cdot 10^{201}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M}{2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -8.676239891926242557896224383038014757216 \cdot 10^{-146}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]
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} \le -1.510341556579411067921188404923562874146 \cdot 10^{201}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M}{2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right)\right) \cdot \frac{1}{\ell}}\\

\mathbf{elif}\;\frac{h}{\ell} \le -8.676239891926242557896224383038014757216 \cdot 10^{-146}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{h}{\ell}}\\

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

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r128183 = w0;
        double r128184 = 1.0;
        double r128185 = M;
        double r128186 = D;
        double r128187 = r128185 * r128186;
        double r128188 = 2.0;
        double r128189 = d;
        double r128190 = r128188 * r128189;
        double r128191 = r128187 / r128190;
        double r128192 = pow(r128191, r128188);
        double r128193 = h;
        double r128194 = l;
        double r128195 = r128193 / r128194;
        double r128196 = r128192 * r128195;
        double r128197 = r128184 - r128196;
        double r128198 = sqrt(r128197);
        double r128199 = r128183 * r128198;
        return r128199;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r128200 = h;
        double r128201 = l;
        double r128202 = r128200 / r128201;
        double r128203 = -1.510341556579411e+201;
        bool r128204 = r128202 <= r128203;
        double r128205 = w0;
        double r128206 = 1.0;
        double r128207 = M;
        double r128208 = D;
        double r128209 = r128207 * r128208;
        double r128210 = 2.0;
        double r128211 = d;
        double r128212 = r128210 * r128211;
        double r128213 = r128209 / r128212;
        double r128214 = 2.0;
        double r128215 = r128210 / r128214;
        double r128216 = pow(r128213, r128215);
        double r128217 = r128207 / r128210;
        double r128218 = pow(r128217, r128215);
        double r128219 = r128208 / r128211;
        double r128220 = pow(r128219, r128215);
        double r128221 = r128220 * r128200;
        double r128222 = r128218 * r128221;
        double r128223 = r128216 * r128222;
        double r128224 = 1.0;
        double r128225 = r128224 / r128201;
        double r128226 = r128223 * r128225;
        double r128227 = r128206 - r128226;
        double r128228 = sqrt(r128227);
        double r128229 = r128205 * r128228;
        double r128230 = -8.676239891926243e-146;
        bool r128231 = r128202 <= r128230;
        double r128232 = r128212 / r128208;
        double r128233 = r128207 / r128232;
        double r128234 = pow(r128233, r128210);
        double r128235 = r128234 * r128202;
        double r128236 = r128206 - r128235;
        double r128237 = sqrt(r128236);
        double r128238 = r128205 * r128237;
        double r128239 = sqrt(r128206);
        double r128240 = r128205 * r128239;
        double r128241 = r128231 ? r128238 : r128240;
        double r128242 = r128204 ? r128229 : r128241;
        return r128242;
}

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) < -1.510341556579411e+201

    1. Initial program 40.9

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

    3. Applied div-inv40.9

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

    4. Applied associate-*r*21.6

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

    6. Applied sqr-pow21.6

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

    7. Applied associate-*l*20.3

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

    9. Applied times-frac21.2

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

    10. Applied unpow-prod-down21.2

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

    11. Applied associate-*l*23.7

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

    if -1.510341556579411e+201 < (/ h l) < -8.676239891926243e-146

    1. Initial program 13.2

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

    3. Applied associate-/l*12.9

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

    if -8.676239891926243e-146 < (/ h 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 div-inv9.9

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

    4. Applied associate-*r*7.0

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}}\]
    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*5.0

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

    9. Applied add-cube-cbrt5.0

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

    10. Applied associate-*r*5.0

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

    11. Taylor expanded around 0 5.9

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -1.510341556579411067921188404923562874146 \cdot 10^{201}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M}{2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{D}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -8.676239891926242557896224383038014757216 \cdot 10^{-146}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(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))))))