Average Error: 14.6 → 10.0
Time: 11.3s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -4.0393769549213093 \cdot 10^{257} \lor \neg \left(\frac{h}{\ell} \le 4.30477040146389894 \cdot 10^{222}\right):\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]
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 -4.0393769549213093 \cdot 10^{257} \lor \neg \left(\frac{h}{\ell} \le 4.30477040146389894 \cdot 10^{222}\right):\\
\;\;\;\;w0 \cdot \sqrt{1}\\

\mathbf{else}:\\
\;\;\;\;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)}\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r267448 = w0;
        double r267449 = 1.0;
        double r267450 = M;
        double r267451 = D;
        double r267452 = r267450 * r267451;
        double r267453 = 2.0;
        double r267454 = d;
        double r267455 = r267453 * r267454;
        double r267456 = r267452 / r267455;
        double r267457 = pow(r267456, r267453);
        double r267458 = h;
        double r267459 = l;
        double r267460 = r267458 / r267459;
        double r267461 = r267457 * r267460;
        double r267462 = r267449 - r267461;
        double r267463 = sqrt(r267462);
        double r267464 = r267448 * r267463;
        return r267464;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r267465 = h;
        double r267466 = l;
        double r267467 = r267465 / r267466;
        double r267468 = -4.039376954921309e+257;
        bool r267469 = r267467 <= r267468;
        double r267470 = 4.304770401463899e+222;
        bool r267471 = r267467 <= r267470;
        double r267472 = !r267471;
        bool r267473 = r267469 || r267472;
        double r267474 = w0;
        double r267475 = 1.0;
        double r267476 = sqrt(r267475);
        double r267477 = r267474 * r267476;
        double r267478 = M;
        double r267479 = D;
        double r267480 = r267478 * r267479;
        double r267481 = 2.0;
        double r267482 = d;
        double r267483 = r267481 * r267482;
        double r267484 = r267480 / r267483;
        double r267485 = 2.0;
        double r267486 = r267481 / r267485;
        double r267487 = pow(r267484, r267486);
        double r267488 = r267487 * r267467;
        double r267489 = r267487 * r267488;
        double r267490 = r267475 - r267489;
        double r267491 = sqrt(r267490);
        double r267492 = r267474 * r267491;
        double r267493 = r267473 ? r267477 : r267492;
        return r267493;
}

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) < -4.039376954921309e+257 or 4.304770401463899e+222 < (/ h l)

    1. Initial program 45.3

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

    2. Taylor expanded around 0 21.6

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

    if -4.039376954921309e+257 < (/ h l) < 4.304770401463899e+222

    1. Initial program 10.0

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

    3. Applied sqr-pow10.0

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

    4. Applied associate-*l*8.2

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

  4. Final simplification10.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -4.0393769549213093 \cdot 10^{257} \lor \neg \left(\frac{h}{\ell} \le 4.30477040146389894 \cdot 10^{222}\right):\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 
(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))))))