Average Error: 14.4 → 8.6
Time: 26.1s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -4.677179237977214472415692974466936692272 \cdot 10^{305}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -1.507520128176834291251451178459305145259 \cdot 10^{-281}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \end{array}\]
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 -4.677179237977214472415692974466936692272 \cdot 10^{305}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right) \cdot \frac{1}{\ell}}\\

\mathbf{elif}\;\frac{h}{\ell} \le -1.507520128176834291251451178459305145259 \cdot 10^{-281}:\\
\;\;\;\;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}:\\
\;\;\;\;\sqrt{1} \cdot w0\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r176149 = w0;
        double r176150 = 1.0;
        double r176151 = M;
        double r176152 = D;
        double r176153 = r176151 * r176152;
        double r176154 = 2.0;
        double r176155 = d;
        double r176156 = r176154 * r176155;
        double r176157 = r176153 / r176156;
        double r176158 = pow(r176157, r176154);
        double r176159 = h;
        double r176160 = l;
        double r176161 = r176159 / r176160;
        double r176162 = r176158 * r176161;
        double r176163 = r176150 - r176162;
        double r176164 = sqrt(r176163);
        double r176165 = r176149 * r176164;
        return r176165;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r176166 = h;
        double r176167 = l;
        double r176168 = r176166 / r176167;
        double r176169 = -4.6771792379772145e+305;
        bool r176170 = r176168 <= r176169;
        double r176171 = w0;
        double r176172 = 1.0;
        double r176173 = M;
        double r176174 = D;
        double r176175 = r176173 * r176174;
        double r176176 = 2.0;
        double r176177 = d;
        double r176178 = r176176 * r176177;
        double r176179 = r176175 / r176178;
        double r176180 = 2.0;
        double r176181 = r176176 / r176180;
        double r176182 = pow(r176179, r176181);
        double r176183 = 1.0;
        double r176184 = r176178 / r176175;
        double r176185 = r176183 / r176184;
        double r176186 = pow(r176185, r176181);
        double r176187 = r176186 * r176166;
        double r176188 = r176182 * r176187;
        double r176189 = r176183 / r176167;
        double r176190 = r176188 * r176189;
        double r176191 = r176172 - r176190;
        double r176192 = sqrt(r176191);
        double r176193 = r176171 * r176192;
        double r176194 = -1.5075201281768343e-281;
        bool r176195 = r176168 <= r176194;
        double r176196 = r176182 * r176168;
        double r176197 = r176182 * r176196;
        double r176198 = r176172 - r176197;
        double r176199 = sqrt(r176198);
        double r176200 = r176171 * r176199;
        double r176201 = sqrt(r176172);
        double r176202 = r176201 * r176171;
        double r176203 = r176195 ? r176200 : r176202;
        double r176204 = r176170 ? r176193 : r176203;
        return r176204;
}

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) < -4.6771792379772145e+305

    1. Initial program 62.9

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

    3. Applied div-inv62.9

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

    4. Applied associate-*r*27.5

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

    6. Applied sqr-pow27.5

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

    7. Applied associate-*l*24.0

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

    9. Applied clear-num24.0

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

    if -4.6771792379772145e+305 < (/ h l) < -1.5075201281768343e-281

    1. Initial program 15.0

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

    3. Applied sqr-pow15.0

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

    4. Applied associate-*l*13.0

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

    if -1.5075201281768343e-281 < (/ h l)

    1. Initial program 8.0

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

    2. Taylor expanded around 0 2.9

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -4.677179237977214472415692974466936692272 \cdot 10^{305}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -1.507520128176834291251451178459305145259 \cdot 10^{-281}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(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))))))