Average Error: 13.6 → 8.5
Time: 10.4s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -2.7954846401797472 \cdot 10^{302}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -9.77445148014032578 \cdot 10^{-249}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]
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} \le -2.7954846401797472 \cdot 10^{302}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\\

\mathbf{elif}\;\frac{h}{\ell} \le -9.77445148014032578 \cdot 10^{-249}:\\
\;\;\;\;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 r212075 = w0;
        double r212076 = 1.0;
        double r212077 = M;
        double r212078 = D;
        double r212079 = r212077 * r212078;
        double r212080 = 2.0;
        double r212081 = d;
        double r212082 = r212080 * r212081;
        double r212083 = r212079 / r212082;
        double r212084 = pow(r212083, r212080);
        double r212085 = h;
        double r212086 = l;
        double r212087 = r212085 / r212086;
        double r212088 = r212084 * r212087;
        double r212089 = r212076 - r212088;
        double r212090 = sqrt(r212089);
        double r212091 = r212075 * r212090;
        return r212091;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r212092 = h;
        double r212093 = l;
        double r212094 = r212092 / r212093;
        double r212095 = -2.7954846401797472e+302;
        bool r212096 = r212094 <= r212095;
        double r212097 = w0;
        double r212098 = 1.0;
        double r212099 = M;
        double r212100 = 2.0;
        double r212101 = d;
        double r212102 = r212100 * r212101;
        double r212103 = D;
        double r212104 = r212102 / r212103;
        double r212105 = r212099 / r212104;
        double r212106 = pow(r212105, r212100);
        double r212107 = r212106 * r212092;
        double r212108 = 1.0;
        double r212109 = r212108 / r212093;
        double r212110 = r212107 * r212109;
        double r212111 = r212098 - r212110;
        double r212112 = sqrt(r212111);
        double r212113 = r212097 * r212112;
        double r212114 = -9.774451480140326e-249;
        bool r212115 = r212094 <= r212114;
        double r212116 = r212099 * r212103;
        double r212117 = r212116 / r212102;
        double r212118 = 2.0;
        double r212119 = r212100 / r212118;
        double r212120 = pow(r212117, r212119);
        double r212121 = r212120 * r212094;
        double r212122 = r212120 * r212121;
        double r212123 = r212098 - r212122;
        double r212124 = sqrt(r212123);
        double r212125 = r212097 * r212124;
        double r212126 = sqrt(r212098);
        double r212127 = r212097 * r212126;
        double r212128 = r212115 ? r212125 : r212127;
        double r212129 = r212096 ? r212113 : r212128;
        return r212129;
}

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) < -2.7954846401797472e+302

    1. Initial program 61.4

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

    3. Applied div-inv61.4

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

    4. Applied associate-*r*24.7

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

    6. Applied associate-/l*26.3

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

    if -2.7954846401797472e+302 < (/ h l) < -9.774451480140326e-249

    1. Initial program 13.4

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

    3. Applied sqr-pow13.4

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

    4. Applied associate-*l*11.9

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

    if -9.774451480140326e-249 < (/ 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. Using strategy rm

    3. Applied div-inv8.0

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

    4. Applied associate-*r*5.0

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

    5. Taylor expanded around 0 3.6

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

  4. Final simplification8.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -2.7954846401797472 \cdot 10^{302}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -9.77445148014032578 \cdot 10^{-249}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(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))))))