Average Error: 14.0 → 9.1
Time: 26.7s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} = -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le -2.459544082082623877126440069328057953432 \cdot 10^{-105} \lor \neg \left(\frac{M \cdot D}{2 \cdot d} \le 2.475593378518339803291364317838566404249 \cdot 10^{-226}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{M \cdot D}{2 \cdot d} = -\infty:\\
\;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\\

\mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le -2.459544082082623877126440069328057953432 \cdot 10^{-105} \lor \neg \left(\frac{M \cdot D}{2 \cdot d} \le 2.475593378518339803291364317838566404249 \cdot 10^{-226}\right):\\
\;\;\;\;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 r183377 = w0;
        double r183378 = 1.0;
        double r183379 = M;
        double r183380 = D;
        double r183381 = r183379 * r183380;
        double r183382 = 2.0;
        double r183383 = d;
        double r183384 = r183382 * r183383;
        double r183385 = r183381 / r183384;
        double r183386 = pow(r183385, r183382);
        double r183387 = h;
        double r183388 = l;
        double r183389 = r183387 / r183388;
        double r183390 = r183386 * r183389;
        double r183391 = r183378 - r183390;
        double r183392 = sqrt(r183391);
        double r183393 = r183377 * r183392;
        return r183393;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r183394 = M;
        double r183395 = D;
        double r183396 = r183394 * r183395;
        double r183397 = 2.0;
        double r183398 = d;
        double r183399 = r183397 * r183398;
        double r183400 = r183396 / r183399;
        double r183401 = -inf.0;
        bool r183402 = r183400 <= r183401;
        double r183403 = w0;
        double r183404 = 1.0;
        double r183405 = r183399 / r183395;
        double r183406 = r183394 / r183405;
        double r183407 = pow(r183406, r183397);
        double r183408 = h;
        double r183409 = cbrt(r183408);
        double r183410 = r183409 * r183409;
        double r183411 = l;
        double r183412 = cbrt(r183411);
        double r183413 = r183412 * r183412;
        double r183414 = r183410 / r183413;
        double r183415 = r183407 * r183414;
        double r183416 = r183409 / r183412;
        double r183417 = r183415 * r183416;
        double r183418 = r183404 - r183417;
        double r183419 = sqrt(r183418);
        double r183420 = r183403 * r183419;
        double r183421 = -2.459544082082624e-105;
        bool r183422 = r183400 <= r183421;
        double r183423 = 2.4755933785183398e-226;
        bool r183424 = r183400 <= r183423;
        double r183425 = !r183424;
        bool r183426 = r183422 || r183425;
        double r183427 = 2.0;
        double r183428 = r183397 / r183427;
        double r183429 = pow(r183400, r183428);
        double r183430 = r183408 / r183411;
        double r183431 = r183429 * r183430;
        double r183432 = r183429 * r183431;
        double r183433 = r183404 - r183432;
        double r183434 = sqrt(r183433);
        double r183435 = r183403 * r183434;
        double r183436 = sqrt(r183404);
        double r183437 = r183403 * r183436;
        double r183438 = r183426 ? r183435 : r183437;
        double r183439 = r183402 ? r183420 : r183438;
        return r183439;
}

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 (/ (* M D) (* 2.0 d)) < -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 add-cube-cbrt64.0

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

    4. Applied add-cube-cbrt64.0

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

    5. Applied times-frac64.0

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

    6. Applied associate-*r*64.0

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}\]
    7. Using strategy rm

    8. Applied associate-/l*56.3

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

    if -inf.0 < (/ (* M D) (* 2.0 d)) < -2.459544082082624e-105 or 2.4755933785183398e-226 < (/ (* M D) (* 2.0 d))

    1. Initial program 17.3

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

    3. Applied sqr-pow17.3

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

    4. Applied associate-*l*14.3

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

    if -2.459544082082624e-105 < (/ (* M D) (* 2.0 d)) < 2.4755933785183398e-226

    1. Initial program 7.5

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

    3. Applied add-cube-cbrt7.5

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

    4. Applied add-cube-cbrt7.5

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

    5. Applied times-frac7.5

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

    6. Applied associate-*r*2.2

      \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}}\]
    7. Using strategy rm

    8. Applied sqr-pow2.2

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

    9. Applied associate-*l*2.2

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

    10. Taylor expanded around 0 1.0

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} = -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{\frac{2 \cdot d}{D}}\right)}^{2} \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}}\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \le -2.459544082082623877126440069328057953432 \cdot 10^{-105} \lor \neg \left(\frac{M \cdot D}{2 \cdot d} \le 2.475593378518339803291364317838566404249 \cdot 10^{-226}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))))))