Average Error: 14.7 → 8.1
Time: 22.1s
Precision: binary64
\[[M, D]=\mathsf{sort}([M, D])\]
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 4.290998421271188 \cdot 10^{+306}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{\ell}}}\\ \mathbf{elif}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq \infty:\\ \;\;\;\;\frac{w0}{M \cdot \sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25}} \cdot -0.5 - M \cdot \left(w0 \cdot \sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right) \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\ell}}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 4.290998421271188 \cdot 10^{+306}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{\ell}}}\\

\mathbf{elif}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq \infty:\\
\;\;\;\;\frac{w0}{M \cdot \sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25}} \cdot -0.5 - M \cdot \left(w0 \cdot \sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25}\right)\\

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

\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 (<=
      (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
      4.290998421271188e+306)
   (*
    w0
    (sqrt
     (-
      1.0
      (*
       (/ (* (/ (* M D) (* 2.0 d)) h) (* (cbrt l) (cbrt l)))
       (/ (/ (* M D) (* 2.0 d)) (cbrt l))))))
   (if (<=
        (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
        INFINITY)
     (-
      (* (/ w0 (* M (sqrt (* (* (/ h l) (pow (/ D d) 2.0)) -0.25)))) -0.5)
      (* M (* w0 (sqrt (* (* (/ h l) (pow (/ D d) 2.0)) -0.25)))))
     (*
      w0
      (sqrt
       (- 1.0 (* (* (/ (* M D) (* 2.0 d)) h) (/ (/ (* M D) (* 2.0 d)) 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 ((w0 * sqrt(1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l)))) <= 4.290998421271188e+306) {
		tmp = w0 * sqrt(1.0 - (((((M * D) / (2.0 * d)) * h) / (cbrt(l) * cbrt(l))) * (((M * D) / (2.0 * d)) / cbrt(l))));
	} else if ((w0 * sqrt(1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l)))) <= ((double) INFINITY)) {
		tmp = ((w0 / (M * sqrt(((h / l) * pow((D / d), 2.0)) * -0.25))) * -0.5) - (M * (w0 * sqrt(((h / l) * pow((D / d), 2.0)) * -0.25)));
	} else {
		tmp = w0 * sqrt(1.0 - ((((M * D) / (2.0 * d)) * h) * (((M * D) / (2.0 * d)) / 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 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < 4.2909984212711883e306

    1. Initial program 5.4

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

    3. Applied associate-*r/_binary645.5

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

    4. Simplified5.5

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

    6. Applied sqr-pow_binary645.5

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

    7. Applied associate-*r*_binary645.2

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

    8. Simplified5.2

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

    10. Applied add-cube-cbrt_binary645.3

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

    11. Applied times-frac_binary644.7

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

    12. Simplified4.7

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

    if 4.2909984212711883e306 < (*.f64 w0 (sqrt.f64 (-.f64 1 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))))) < +inf.0

    1. Initial program 63.3

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

    2. Taylor expanded around -inf 56.5

      \[\leadsto \color{blue}{-\left(0.5 \cdot \frac{w0}{M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}} + M \cdot \left(\sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}} \cdot w0\right)\right)}\]

    3. Simplified45.0

      \[\leadsto \color{blue}{\frac{w0}{M \cdot \sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25}} \cdot -0.5 - M \cdot \left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot w0\right)}\]

    if +inf.0 < (*.f64 w0 (sqrt.f64 (-.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 associate-*r/_binary6425.6

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

    4. Simplified25.6

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

    6. Applied sqr-pow_binary6425.6

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

    7. Applied associate-*r*_binary6414.7

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

    8. Simplified14.7

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

    10. Applied *-un-lft-identity_binary6414.7

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

    11. Applied times-frac_binary6412.5

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

    12. Simplified12.5

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

    13. Simplified12.5

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

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq 4.290998421271188 \cdot 10^{+306}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{\ell}}}\\ \mathbf{elif}\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \leq \infty:\\ \;\;\;\;\frac{w0}{M \cdot \sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25}} \cdot -0.5 - M \cdot \left(w0 \cdot \sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25}\right)\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right) \cdot \frac{\frac{M \cdot D}{2 \cdot d}}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021144 
(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))))))