Average Error: 14.6 → 8.8
Time: 11.0s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} = -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \le -2.566959543532192688587992548008487318806 \cdot 10^{-296}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]
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 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h\right) \cdot \frac{1}{\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 r254394 = w0;
        double r254395 = 1.0;
        double r254396 = M;
        double r254397 = D;
        double r254398 = r254396 * r254397;
        double r254399 = 2.0;
        double r254400 = d;
        double r254401 = r254399 * r254400;
        double r254402 = r254398 / r254401;
        double r254403 = pow(r254402, r254399);
        double r254404 = h;
        double r254405 = l;
        double r254406 = r254404 / r254405;
        double r254407 = r254403 * r254406;
        double r254408 = r254395 - r254407;
        double r254409 = sqrt(r254408);
        double r254410 = r254394 * r254409;
        return r254410;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r254411 = h;
        double r254412 = l;
        double r254413 = r254411 / r254412;
        double r254414 = -inf.0;
        bool r254415 = r254413 <= r254414;
        double r254416 = w0;
        double r254417 = 1.0;
        double r254418 = M;
        double r254419 = D;
        double r254420 = r254418 * r254419;
        double r254421 = 2.0;
        double r254422 = d;
        double r254423 = r254421 * r254422;
        double r254424 = r254420 / r254423;
        double r254425 = pow(r254424, r254421);
        double r254426 = r254425 * r254411;
        double r254427 = 1.0;
        double r254428 = r254427 / r254412;
        double r254429 = r254426 * r254428;
        double r254430 = r254417 - r254429;
        double r254431 = sqrt(r254430);
        double r254432 = r254416 * r254431;
        double r254433 = -2.5669595435321927e-296;
        bool r254434 = r254413 <= r254433;
        double r254435 = 2.0;
        double r254436 = r254421 / r254435;
        double r254437 = pow(r254424, r254436);
        double r254438 = r254437 * r254413;
        double r254439 = r254437 * r254438;
        double r254440 = r254417 - r254439;
        double r254441 = sqrt(r254440);
        double r254442 = r254416 * r254441;
        double r254443 = sqrt(r254417);
        double r254444 = r254416 * r254443;
        double r254445 = r254434 ? r254442 : r254444;
        double r254446 = r254415 ? r254432 : r254445;
        return r254446;
}

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 div-inv64.0

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

    4. Applied associate-*r*26.9

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

    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

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

Reproduce

herbie shell --seed 2019353 
(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))))))