Average Error: 14.6 → 8.8
Time: 10.8s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -2.566959543532192688587992548008487318806 \cdot 10^{-296}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]
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} = -\infty:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}\\

\mathbf{elif}\;\frac{h}{\ell} \le -2.566959543532192688587992548008487318806 \cdot 10^{-296}:\\
\;\;\;\;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}\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r249198 = w0;
        double r249199 = 1.0;
        double r249200 = M;
        double r249201 = D;
        double r249202 = r249200 * r249201;
        double r249203 = 2.0;
        double r249204 = d;
        double r249205 = r249203 * r249204;
        double r249206 = r249202 / r249205;
        double r249207 = pow(r249206, r249203);
        double r249208 = h;
        double r249209 = l;
        double r249210 = r249208 / r249209;
        double r249211 = r249207 * r249210;
        double r249212 = r249199 - r249211;
        double r249213 = sqrt(r249212);
        double r249214 = r249198 * r249213;
        return r249214;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r249215 = h;
        double r249216 = l;
        double r249217 = r249215 / r249216;
        double r249218 = -inf.0;
        bool r249219 = r249217 <= r249218;
        double r249220 = w0;
        double r249221 = 1.0;
        double r249222 = M;
        double r249223 = D;
        double r249224 = r249222 * r249223;
        double r249225 = 2.0;
        double r249226 = d;
        double r249227 = r249225 * r249226;
        double r249228 = r249224 / r249227;
        double r249229 = pow(r249228, r249225);
        double r249230 = r249229 * r249215;
        double r249231 = r249230 / r249216;
        double r249232 = r249221 - r249231;
        double r249233 = sqrt(r249232);
        double r249234 = r249220 * r249233;
        double r249235 = -2.5669595435321927e-296;
        bool r249236 = r249217 <= r249235;
        double r249237 = 2.0;
        double r249238 = r249225 / r249237;
        double r249239 = pow(r249228, r249238);
        double r249240 = r249239 * r249217;
        double r249241 = r249239 * r249240;
        double r249242 = r249221 - r249241;
        double r249243 = sqrt(r249242);
        double r249244 = r249220 * r249243;
        double r249245 = sqrt(r249221);
        double r249246 = r249220 * r249245;
        double r249247 = r249236 ? r249244 : r249246;
        double r249248 = r249219 ? r249234 : r249247;
        return r249248;
}

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) < -inf.0

    1. Initial program 64.0

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

    3. Applied associate-*r/26.9

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

    if -inf.0 < (/ h l) < -2.5669595435321927e-296

    1. Initial program 14.5

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

    3. Applied sqr-pow14.5

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

    4. Applied associate-*l*12.5

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

    if -2.5669595435321927e-296 < (/ h l)

    1. Initial program 8.3

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

    2. Taylor expanded around 0 3.0

      \[\leadsto w0 \cdot \color{blue}{\sqrt{1}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -2.566959543532192688587992548008487318806 \cdot 10^{-296}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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))))))