Average Error: 13.9 → 9.4
Time: 11.9s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \le 1.855277338734028038380413031509408290214 \cdot 10^{-289} \lor \neg \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \le 1.933159505335491079472504373058255984824 \cdot 10^{291}\right):\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\sqrt[3]{\frac{h}{\ell}} \cdot \sqrt[3]{\frac{h}{\ell}}\right)\right) \cdot \sqrt[3]{\frac{h}{\ell}}}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \le 1.855277338734028038380413031509408290214 \cdot 10^{-289} \lor \neg \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \le 1.933159505335491079472504373058255984824 \cdot 10^{291}\right):\\
\;\;\;\;w0 \cdot \sqrt{1}\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\sqrt[3]{\frac{h}{\ell}} \cdot \sqrt[3]{\frac{h}{\ell}}\right)\right) \cdot \sqrt[3]{\frac{h}{\ell}}}\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r285172 = w0;
        double r285173 = 1.0;
        double r285174 = M;
        double r285175 = D;
        double r285176 = r285174 * r285175;
        double r285177 = 2.0;
        double r285178 = d;
        double r285179 = r285177 * r285178;
        double r285180 = r285176 / r285179;
        double r285181 = pow(r285180, r285177);
        double r285182 = h;
        double r285183 = l;
        double r285184 = r285182 / r285183;
        double r285185 = r285181 * r285184;
        double r285186 = r285173 - r285185;
        double r285187 = sqrt(r285186);
        double r285188 = r285172 * r285187;
        return r285188;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r285189 = M;
        double r285190 = D;
        double r285191 = r285189 * r285190;
        double r285192 = 2.0;
        double r285193 = d;
        double r285194 = r285192 * r285193;
        double r285195 = r285191 / r285194;
        double r285196 = pow(r285195, r285192);
        double r285197 = 1.855277338734028e-289;
        bool r285198 = r285196 <= r285197;
        double r285199 = 1.933159505335491e+291;
        bool r285200 = r285196 <= r285199;
        double r285201 = !r285200;
        bool r285202 = r285198 || r285201;
        double r285203 = w0;
        double r285204 = 1.0;
        double r285205 = sqrt(r285204);
        double r285206 = r285203 * r285205;
        double r285207 = h;
        double r285208 = l;
        double r285209 = r285207 / r285208;
        double r285210 = cbrt(r285209);
        double r285211 = r285210 * r285210;
        double r285212 = r285196 * r285211;
        double r285213 = r285212 * r285210;
        double r285214 = r285204 - r285213;
        double r285215 = sqrt(r285214);
        double r285216 = r285203 * r285215;
        double r285217 = r285202 ? r285206 : r285216;
        return r285217;
}

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 (pow (/ (* M D) (* 2.0 d)) 2.0) < 1.855277338734028e-289 or 1.933159505335491e+291 < (pow (/ (* M D) (* 2.0 d)) 2.0)

    1. Initial program 18.2

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

    2. Taylor expanded around 0 11.2

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

    if 1.855277338734028e-289 < (pow (/ (* M D) (* 2.0 d)) 2.0) < 1.933159505335491e+291

    1. Initial program 6.0

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

    3. Applied add-cube-cbrt6.1

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

    4. Applied associate-*r*6.1

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

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \le 1.855277338734028038380413031509408290214 \cdot 10^{-289} \lor \neg \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \le 1.933159505335491079472504373058255984824 \cdot 10^{291}\right):\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\sqrt[3]{\frac{h}{\ell}} \cdot \sqrt[3]{\frac{h}{\ell}}\right)\right) \cdot \sqrt[3]{\frac{h}{\ell}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(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))))))