Average Error: 14.6 → 9.7
Time: 14.0s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -4.434769257142831 \cdot 10^{294} \lor \neg \left(\frac{h}{\ell} \le -3.0353158251 \cdot 10^{-314}\right):\\ \;\;\;\;\sqrt{1} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{d \cdot \frac{2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]
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 -4.434769257142831 \cdot 10^{294} \lor \neg \left(\frac{h}{\ell} \le -3.0353158251 \cdot 10^{-314}\right):\\
\;\;\;\;\sqrt{1} \cdot w0\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{d \cdot \frac{2}{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)}\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r183125 = w0;
        double r183126 = 1.0;
        double r183127 = M;
        double r183128 = D;
        double r183129 = r183127 * r183128;
        double r183130 = 2.0;
        double r183131 = d;
        double r183132 = r183130 * r183131;
        double r183133 = r183129 / r183132;
        double r183134 = pow(r183133, r183130);
        double r183135 = h;
        double r183136 = l;
        double r183137 = r183135 / r183136;
        double r183138 = r183134 * r183137;
        double r183139 = r183126 - r183138;
        double r183140 = sqrt(r183139);
        double r183141 = r183125 * r183140;
        return r183141;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r183142 = h;
        double r183143 = l;
        double r183144 = r183142 / r183143;
        double r183145 = -4.434769257142831e+294;
        bool r183146 = r183144 <= r183145;
        double r183147 = -3.0353158251021e-314;
        bool r183148 = r183144 <= r183147;
        double r183149 = !r183148;
        bool r183150 = r183146 || r183149;
        double r183151 = 1.0;
        double r183152 = sqrt(r183151);
        double r183153 = w0;
        double r183154 = r183152 * r183153;
        double r183155 = M;
        double r183156 = d;
        double r183157 = 2.0;
        double r183158 = D;
        double r183159 = r183157 / r183158;
        double r183160 = r183156 * r183159;
        double r183161 = r183155 / r183160;
        double r183162 = 2.0;
        double r183163 = r183157 / r183162;
        double r183164 = pow(r183161, r183163);
        double r183165 = r183155 * r183158;
        double r183166 = r183157 * r183156;
        double r183167 = r183165 / r183166;
        double r183168 = pow(r183167, r183163);
        double r183169 = r183168 * r183144;
        double r183170 = r183164 * r183169;
        double r183171 = r183151 - r183170;
        double r183172 = sqrt(r183171);
        double r183173 = r183153 * r183172;
        double r183174 = r183150 ? r183154 : r183173;
        return r183174;
}

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 (/ h l) < -4.434769257142831e+294 or -3.0353158251021e-314 < (/ h l)

    1. Initial program 14.5

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

    2. Taylor expanded around 0 6.5

      \[\leadsto \color{blue}{\sqrt{1} \cdot w0}\]

    if -4.434769257142831e+294 < (/ h l) < -3.0353158251021e-314

    1. Initial program 14.6

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

    3. Applied sqr-pow14.6

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

    4. Applied associate-*l*12.6

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

    6. Applied associate-/l*13.5

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

    7. Simplified13.5

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -4.434769257142831 \cdot 10^{294} \lor \neg \left(\frac{h}{\ell} \le -3.0353158251 \cdot 10^{-314}\right):\\ \;\;\;\;\sqrt{1} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M}{d \cdot \frac{2}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))))))