Average Error: 14.3 → 9.9
Time: 16.6s
Precision: binary64
[M D]: =\mathsf{sort}([M D])
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 1.2899148596621315 \cdot 10^{+297}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\\ \mathbf{elif}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq \infty:\\ \;\;\;\;\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(w0 \cdot \left(-M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 1.2899148596621315 \cdot 10^{+297}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\\

\mathbf{elif}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq \infty:\\
\;\;\;\;\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(w0 \cdot \left(-M\right)\right)\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)}\\

\end{array}
(FPCore (w0 M D h l d)
 :precision binary64
 (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))
↓
(FPCore (w0 M D h l d)
 :precision binary64
 (if (<=
      (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))
      1.2899148596621315e+297)
   (*
    w0
    (sqrt (- 1.0 (* (/ (* M D) (* 2.0 d)) (* (/ (* M D) (* 2.0 d)) (/ h l))))))
   (if (<= (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))) INFINITY)
     (* (sqrt (* (* (/ h l) (pow (/ D d) 2.0)) -0.25)) (* w0 (- M)))
     (*
      w0
      (sqrt
       (-
        1.0
        (*
         (* (/ (* M D) (* 2.0 d)) (cbrt h))
         (* (* (/ (* M D) (* 2.0 d)) (cbrt h)) (/ (cbrt h) l)))))))))
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt(1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l)));
}
↓
double code(double w0, double M, double D, double h, double l, double d) {
	double tmp;
	if ((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= 1.2899148596621315e+297) {
		tmp = w0 * sqrt(1.0 - (((M * D) / (2.0 * d)) * (((M * D) / (2.0 * d)) * (h / l))));
	} else if ((1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l))) <= ((double) INFINITY)) {
		tmp = sqrt(((h / l) * pow((D / d), 2.0)) * -0.25) * (w0 * -M);
	} else {
		tmp = w0 * sqrt(1.0 - ((((M * D) / (2.0 * d)) * cbrt(h)) * ((((M * D) / (2.0 * d)) * cbrt(h)) * (cbrt(h) / l))));
	}
	return tmp;
}

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 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < 1.28991485966213148e297

    1. Initial program 0.2

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

    3. Applied unpow2_binary64_8250.2

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

    4. Applied associate-*l*_binary64_7010.2

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

    if 1.28991485966213148e297 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))) < +inf.0

    1. Initial program 63.2

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

    2. Taylor expanded around -inf 60.3

      \[\leadsto \color{blue}{-1 \cdot \left(M \cdot \left(\sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}} \cdot w0\right)\right)}\]

    3. Simplified55.2

      \[\leadsto \color{blue}{\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(\left(-M\right) \cdot w0\right)}\]

    if +inf.0 < (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)))

    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 *-un-lft-identity_binary64_76064.0

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

    4. Applied add-cube-cbrt_binary64_79564.0

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

    5. Applied times-frac_binary64_76664.0

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

    6. Applied associate-*r*_binary64_70038.1

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

    7. Simplified38.1

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

    9. Applied pow2_binary64_84138.1

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

    10. Applied pow-prod-down_binary64_83129.6

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

    12. Applied unpow2_binary64_82529.6

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

    13. Applied associate-*l*_binary64_70126.5

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

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq 1.2899148596621315 \cdot 10^{+297}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\\ \mathbf{elif}\;1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq \infty:\\ \;\;\;\;\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(w0 \cdot \left(-M\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right) \cdot \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021019 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))