Average Error: 14.8 → 9.7
Time: 9.5s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 8.919970551884597 \cdot 10^{+113}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\ell}}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 9.961765663830628 \cdot 10^{+260}:\\ \;\;\;\;\sqrt{\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \cdot \left(w0 \cdot \sqrt{\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 8.919970551884597 \cdot 10^{+113}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\ell}}\\

\mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 9.961765663830628 \cdot 10^{+260}:\\
\;\;\;\;\sqrt{\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \cdot \left(w0 \cdot \sqrt{\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}}\right)\\

\mathbf{else}:\\
\;\;\;\;w0\\

\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 (<= (pow (/ (* M D) (* 2.0 d)) 2.0) 8.919970551884597e+113)
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* D (/ M (* 2.0 d))) 2.0)) l))))
   (if (<= (pow (/ (* M D) (* 2.0 d)) 2.0) 9.961765663830628e+260)
     (*
      (sqrt (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))
      (* w0 (sqrt (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l)))))))
     w0)))
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 (pow(((M * D) / (2.0 * d)), 2.0) <= 8.919970551884597e+113) {
		tmp = w0 * sqrt(1.0 - ((h * pow((D * (M / (2.0 * d))), 2.0)) / l));
	} else if (pow(((M * D) / (2.0 * d)), 2.0) <= 9.961765663830628e+260) {
		tmp = sqrt(sqrt(1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l)))) * (w0 * sqrt(sqrt(1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l)))));
	} else {
		tmp = w0;
	}
	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 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 8.9199705518845974e113

    1. Initial program 6.9

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

    3. Applied associate-*r/_binary641.7

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

    4. Simplified1.7

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

    6. Applied associate-/l*_binary641.7

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

    7. Simplified1.8

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

    9. Applied div-inv_binary641.8

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

    10. Applied *-un-lft-identity_binary641.8

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

    11. Applied times-frac_binary641.7

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

    12. Applied *-un-lft-identity_binary641.7

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

    13. Applied times-frac_binary641.7

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

    14. Simplified1.7

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

    15. Simplified1.7

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

    if 8.9199705518845974e113 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 9.96176566383062843e260

    1. Initial program 12.7

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

    3. Applied add-sqr-sqrt_binary6412.8

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

    4. Applied associate-*r*_binary6412.8

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

    5. Simplified12.8

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

    if 9.96176566383062843e260 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

    1. Initial program 59.6

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

    2. Taylor expanded around 0 52.4

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 8.919970551884597 \cdot 10^{+113}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(D \cdot \frac{M}{2 \cdot d}\right)}^{2}}{\ell}}\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 9.961765663830628 \cdot 10^{+260}:\\ \;\;\;\;\sqrt{\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}} \cdot \left(w0 \cdot \sqrt{\sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}}\right)\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array}\]

Reproduce

herbie shell --seed 2020268 
(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))))))