Average Error: 14.3 → 9.5
Time: 41.7s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -1.772133656628061046787329400235105824789 \cdot 10^{308}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \mathbf{elif}\;\frac{h}{\ell} \le -2.034203706704132394404794445791386313725 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \end{array}\]
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 -1.772133656628061046787329400235105824789 \cdot 10^{308}:\\
\;\;\;\;\sqrt{1} \cdot w0\\

\mathbf{elif}\;\frac{h}{\ell} \le -2.034203706704132394404794445791386313725 \cdot 10^{-311}:\\
\;\;\;\;\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 r5955782 = w0;
        double r5955783 = 1.0;
        double r5955784 = M;
        double r5955785 = D;
        double r5955786 = r5955784 * r5955785;
        double r5955787 = 2.0;
        double r5955788 = d;
        double r5955789 = r5955787 * r5955788;
        double r5955790 = r5955786 / r5955789;
        double r5955791 = pow(r5955790, r5955787);
        double r5955792 = h;
        double r5955793 = l;
        double r5955794 = r5955792 / r5955793;
        double r5955795 = r5955791 * r5955794;
        double r5955796 = r5955783 - r5955795;
        double r5955797 = sqrt(r5955796);
        double r5955798 = r5955782 * r5955797;
        return r5955798;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r5955799 = h;
        double r5955800 = l;
        double r5955801 = r5955799 / r5955800;
        double r5955802 = -1.772133656628061e+308;
        bool r5955803 = r5955801 <= r5955802;
        double r5955804 = 1.0;
        double r5955805 = sqrt(r5955804);
        double r5955806 = w0;
        double r5955807 = r5955805 * r5955806;
        double r5955808 = -2.0342037067041e-311;
        bool r5955809 = r5955801 <= r5955808;
        double r5955810 = M;
        double r5955811 = D;
        double r5955812 = r5955810 * r5955811;
        double r5955813 = d;
        double r5955814 = 2.0;
        double r5955815 = r5955813 * r5955814;
        double r5955816 = r5955812 / r5955815;
        double r5955817 = 2.0;
        double r5955818 = r5955814 / r5955817;
        double r5955819 = pow(r5955816, r5955818);
        double r5955820 = r5955801 * r5955819;
        double r5955821 = r5955819 * r5955820;
        double r5955822 = r5955804 - r5955821;
        double r5955823 = sqrt(r5955822);
        double r5955824 = r5955823 * r5955806;
        double r5955825 = r5955809 ? r5955824 : r5955807;
        double r5955826 = r5955803 ? r5955807 : r5955825;
        return r5955826;
}

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) < -1.772133656628061e+308 or -2.0342037067041e-311 < (/ h l)

    1. Initial program 13.1

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

    2. Taylor expanded around 0 5.9

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

    if -1.772133656628061e+308 < (/ h l) < -2.0342037067041e-311

    1. Initial program 15.7

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

    3. Applied sqr-pow15.7

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

    4. Applied associate-*l*13.7

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

  4. Final simplification9.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -1.772133656628061046787329400235105824789 \cdot 10^{308}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \mathbf{elif}\;\frac{h}{\ell} \le -2.034203706704132394404794445791386313725 \cdot 10^{-311}:\\ \;\;\;\;\sqrt{1 - {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(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))))))