Average Error: 13.8 → 9.2
Time: 28.0s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -2.30091333523267352 \cdot 10^{119}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{1}{\frac{\ell}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -5.48215042013591528 \cdot 10^{-224}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(M \cdot D\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \end{array}\]
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 -2.30091333523267352 \cdot 10^{119}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{1}{\frac{\ell}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}}}\\

\mathbf{elif}\;\frac{h}{\ell} \le -5.48215042013591528 \cdot 10^{-224}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(M \cdot D\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{1} \cdot w0\\

\end{array}
double f(double w0, double M, double D, double h, double l, double d) {
        double r127330 = w0;
        double r127331 = 1.0;
        double r127332 = M;
        double r127333 = D;
        double r127334 = r127332 * r127333;
        double r127335 = 2.0;
        double r127336 = d;
        double r127337 = r127335 * r127336;
        double r127338 = r127334 / r127337;
        double r127339 = pow(r127338, r127335);
        double r127340 = h;
        double r127341 = l;
        double r127342 = r127340 / r127341;
        double r127343 = r127339 * r127342;
        double r127344 = r127331 - r127343;
        double r127345 = sqrt(r127344);
        double r127346 = r127330 * r127345;
        return r127346;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r127347 = h;
        double r127348 = l;
        double r127349 = r127347 / r127348;
        double r127350 = -2.3009133352326735e+119;
        bool r127351 = r127349 <= r127350;
        double r127352 = w0;
        double r127353 = 1.0;
        double r127354 = M;
        double r127355 = D;
        double r127356 = r127354 * r127355;
        double r127357 = 2.0;
        double r127358 = d;
        double r127359 = r127357 * r127358;
        double r127360 = r127356 / r127359;
        double r127361 = 2.0;
        double r127362 = r127357 / r127361;
        double r127363 = pow(r127360, r127362);
        double r127364 = 1.0;
        double r127365 = r127363 * r127347;
        double r127366 = r127348 / r127365;
        double r127367 = r127364 / r127366;
        double r127368 = r127363 * r127367;
        double r127369 = r127353 - r127368;
        double r127370 = sqrt(r127369);
        double r127371 = r127352 * r127370;
        double r127372 = -5.482150420135915e-224;
        bool r127373 = r127349 <= r127372;
        double r127374 = pow(r127356, r127362);
        double r127375 = r127364 / r127359;
        double r127376 = pow(r127375, r127362);
        double r127377 = r127376 * r127349;
        double r127378 = r127374 * r127377;
        double r127379 = r127363 * r127378;
        double r127380 = r127353 - r127379;
        double r127381 = sqrt(r127380);
        double r127382 = r127352 * r127381;
        double r127383 = sqrt(r127353);
        double r127384 = r127383 * r127352;
        double r127385 = r127373 ? r127382 : r127384;
        double r127386 = r127351 ? r127371 : r127385;
        return r127386;
}

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) < -2.3009133352326735e+119

    1. Initial program 33.0

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

    3. Applied sqr-pow33.0

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

    4. Applied associate-*l*33.0

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

    6. Applied associate-*r/20.7

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

    8. Applied clear-num20.7

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

    if -2.3009133352326735e+119 < (/ h l) < -5.482150420135915e-224

    1. Initial program 13.4

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

    3. Applied sqr-pow13.4

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

    4. Applied associate-*l*12.0

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

    6. Applied div-inv12.0

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

    7. Applied unpow-prod-down12.0

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

    8. Applied associate-*l*12.5

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

    if -5.482150420135915e-224 < (/ h l)

    1. Initial program 7.9

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

    2. Taylor expanded around 0 3.9

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

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \le -2.30091333523267352 \cdot 10^{119}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{1}{\frac{\ell}{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -5.48215042013591528 \cdot 10^{-224}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(M \cdot D\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{1}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1} \cdot w0\\ \end{array}\]

Reproduce

herbie shell --seed 2019198 +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))))))