Average Error: 14.2 → 8.9
Time: 11.2s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -3.06029486315833535 \cdot 10^{139}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -4.41799484384605494 \cdot 10^{-176}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}}}\\ \end{array}\]
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 -3.06029486315833535 \cdot 10^{139}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}}}\\

\mathbf{elif}\;\frac{h}{\ell} \le -4.41799484384605494 \cdot 10^{-176}:\\
\;\;\;\;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 - \frac{1}{\frac{\ell}{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}}}\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r272084 = w0;
        double r272085 = 1.0;
        double r272086 = M;
        double r272087 = D;
        double r272088 = r272086 * r272087;
        double r272089 = 2.0;
        double r272090 = d;
        double r272091 = r272089 * r272090;
        double r272092 = r272088 / r272091;
        double r272093 = pow(r272092, r272089);
        double r272094 = h;
        double r272095 = l;
        double r272096 = r272094 / r272095;
        double r272097 = r272093 * r272096;
        double r272098 = r272085 - r272097;
        double r272099 = sqrt(r272098);
        double r272100 = r272084 * r272099;
        return r272100;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r272101 = h;
        double r272102 = l;
        double r272103 = r272101 / r272102;
        double r272104 = -3.0602948631583354e+139;
        bool r272105 = r272103 <= r272104;
        double r272106 = w0;
        double r272107 = 1.0;
        double r272108 = 1.0;
        double r272109 = M;
        double r272110 = D;
        double r272111 = r272109 * r272110;
        double r272112 = 2.0;
        double r272113 = d;
        double r272114 = r272112 * r272113;
        double r272115 = r272111 / r272114;
        double r272116 = pow(r272115, r272112);
        double r272117 = r272116 * r272101;
        double r272118 = r272102 / r272117;
        double r272119 = r272108 / r272118;
        double r272120 = r272107 - r272119;
        double r272121 = sqrt(r272120);
        double r272122 = r272106 * r272121;
        double r272123 = -4.417994843846055e-176;
        bool r272124 = r272103 <= r272123;
        double r272125 = 2.0;
        double r272126 = r272112 / r272125;
        double r272127 = pow(r272115, r272126);
        double r272128 = r272127 * r272103;
        double r272129 = r272127 * r272128;
        double r272130 = r272107 - r272129;
        double r272131 = sqrt(r272130);
        double r272132 = r272106 * r272131;
        double r272133 = r272114 / r272110;
        double r272134 = r272109 / r272133;
        double r272135 = pow(r272134, r272126);
        double r272136 = r272135 * r272101;
        double r272137 = r272135 * r272136;
        double r272138 = r272102 / r272137;
        double r272139 = r272108 / r272138;
        double r272140 = r272107 - r272139;
        double r272141 = sqrt(r272140);
        double r272142 = r272106 * r272141;
        double r272143 = r272124 ? r272132 : r272142;
        double r272144 = r272105 ? r272122 : r272143;
        return r272144;
}

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) < -3.0602948631583354e+139

    1. Initial program 34.4

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

    3. Applied associate-*r/19.9

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

    5. Applied clear-num19.9

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

    if -3.0602948631583354e+139 < (/ h l) < -4.417994843846055e-176

    1. Initial program 14.2

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

    3. Applied sqr-pow14.2

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

    4. Applied associate-*l*12.7

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

    if -4.417994843846055e-176 < (/ h l)

    1. Initial program 8.6

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

    3. Applied associate-*r/5.8

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

    5. Applied clear-num5.8

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

    7. Applied associate-/l*5.7

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

    9. Applied sqr-pow5.7

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

    10. Applied associate-*l*3.9

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -3.06029486315833535 \cdot 10^{139}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -4.41799484384605494 \cdot 10^{-176}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{{\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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))))))