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 r249431 = w0;
        double r249432 = 1.0;
        double r249433 = M;
        double r249434 = D;
        double r249435 = r249433 * r249434;
        double r249436 = 2.0;
        double r249437 = d;
        double r249438 = r249436 * r249437;
        double r249439 = r249435 / r249438;
        double r249440 = pow(r249439, r249436);
        double r249441 = h;
        double r249442 = l;
        double r249443 = r249441 / r249442;
        double r249444 = r249440 * r249443;
        double r249445 = r249432 - r249444;
        double r249446 = sqrt(r249445);
        double r249447 = r249431 * r249446;
        return r249447;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r249448 = h;
        double r249449 = l;
        double r249450 = r249448 / r249449;
        double r249451 = -inf.0;
        bool r249452 = r249450 <= r249451;
        double r249453 = w0;
        double r249454 = 1.0;
        double r249455 = M;
        double r249456 = D;
        double r249457 = r249455 * r249456;
        double r249458 = 2.0;
        double r249459 = d;
        double r249460 = r249458 * r249459;
        double r249461 = r249457 / r249460;
        double r249462 = pow(r249461, r249458);
        double r249463 = r249462 * r249448;
        double r249464 = r249463 / r249449;
        double r249465 = r249454 - r249464;
        double r249466 = sqrt(r249465);
        double r249467 = r249453 * r249466;
        double r249468 = -2.5669595435321927e-296;
        bool r249469 = r249450 <= r249468;
        double r249470 = 2.0;
        double r249471 = r249458 / r249470;
        double r249472 = pow(r249461, r249471);
        double r249473 = r249472 * r249450;
        double r249474 = r249472 * r249473;
        double r249475 = r249454 - r249474;
        double r249476 = sqrt(r249475);
        double r249477 = r249453 * r249476;
        double r249478 = sqrt(r249454);
        double r249479 = r249453 * r249478;
        double r249480 = r249469 ? r249477 : r249479;
        double r249481 = r249452 ? r249467 : r249480;
        return r249481;
}

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))))))