Average Error: 14.1 → 9.0
Time: 36.3s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -1.772435279212265745841411522771178725987 \cdot 10^{308}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \mathbf{elif}\;\frac{h}{\ell} \le -9.283098232840349567842386362959585009642 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \end{array}\]
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 -1.772435279212265745841411522771178725987 \cdot 10^{308}:\\
\;\;\;\;\sqrt{1} \cdot w0\\

\mathbf{elif}\;\frac{h}{\ell} \le -9.283098232840349567842386362959585009642 \cdot 10^{-319}:\\
\;\;\;\;\sqrt{1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot w0\\

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

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r7663082 = w0;
        double r7663083 = 1.0;
        double r7663084 = M;
        double r7663085 = D;
        double r7663086 = r7663084 * r7663085;
        double r7663087 = 2.0;
        double r7663088 = d;
        double r7663089 = r7663087 * r7663088;
        double r7663090 = r7663086 / r7663089;
        double r7663091 = pow(r7663090, r7663087);
        double r7663092 = h;
        double r7663093 = l;
        double r7663094 = r7663092 / r7663093;
        double r7663095 = r7663091 * r7663094;
        double r7663096 = r7663083 - r7663095;
        double r7663097 = sqrt(r7663096);
        double r7663098 = r7663082 * r7663097;
        return r7663098;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r7663099 = h;
        double r7663100 = l;
        double r7663101 = r7663099 / r7663100;
        double r7663102 = -1.7724352792122657e+308;
        bool r7663103 = r7663101 <= r7663102;
        double r7663104 = 1.0;
        double r7663105 = sqrt(r7663104);
        double r7663106 = w0;
        double r7663107 = r7663105 * r7663106;
        double r7663108 = -9.2830982328403e-319;
        bool r7663109 = r7663101 <= r7663108;
        double r7663110 = M;
        double r7663111 = D;
        double r7663112 = r7663110 * r7663111;
        double r7663113 = d;
        double r7663114 = 2.0;
        double r7663115 = r7663113 * r7663114;
        double r7663116 = r7663112 / r7663115;
        double r7663117 = 2.0;
        double r7663118 = r7663114 / r7663117;
        double r7663119 = pow(r7663116, r7663118);
        double r7663120 = r7663101 * r7663119;
        double r7663121 = r7663119 * r7663120;
        double r7663122 = r7663104 - r7663121;
        double r7663123 = sqrt(r7663122);
        double r7663124 = r7663123 * r7663106;
        double r7663125 = r7663109 ? r7663124 : r7663107;
        double r7663126 = r7663103 ? r7663107 : r7663125;
        return r7663126;
}

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) < -1.7724352792122657e+308 or -9.2830982328403e-319 < (/ h l)

    1. Initial program 13.7

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

    2. Taylor expanded around 0 6.1

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

    if -1.7724352792122657e+308 < (/ h l) < -9.2830982328403e-319

    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*12.4

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

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -1.772435279212265745841411522771178725987 \cdot 10^{308}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \mathbf{elif}\;\frac{h}{\ell} \le -9.283098232840349567842386362959585009642 \cdot 10^{-319}:\\ \;\;\;\;\sqrt{1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))