Average Error: 14.1 → 10.3
Time: 9.4s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right)}^{2} \cdot \frac{\sqrt[3]{h}}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -4.787574581286617 \cdot 10^{-223}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right)}^{2} \cdot \frac{\sqrt[3]{h}}{\ell}}\\

\mathbf{elif}\;\frac{h}{\ell} \leq -4.787574581286617 \cdot 10^{-223}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}\\

\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 (<= (/ h l) (- INFINITY))
   (*
    w0
    (sqrt
     (- 1.0 (* (pow (* (/ (* M D) (* 2.0 d)) (cbrt h)) 2.0) (/ (cbrt h) l)))))
   (if (<= (/ h l) -4.787574581286617e-223)
     (* w0 (sqrt (- 1.0 (* (/ h l) (pow (* (/ M 2.0) (/ D d)) 2.0)))))
     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 ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * sqrt(1.0 - (pow((((M * D) / (2.0 * d)) * cbrt(h)), 2.0) * (cbrt(h) / l)));
	} else if ((h / l) <= -4.787574581286617e-223) {
		tmp = w0 * sqrt(1.0 - ((h / l) * pow(((M / 2.0) * (D / d)), 2.0)));
	} 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 (/.f64 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 *-un-lft-identity_binary6464.0

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

    4. Applied add-cube-cbrt_binary6464.0

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

    5. Applied times-frac_binary6464.0

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

    6. Applied associate-*r*_binary6436.7

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

    7. Simplified36.7

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

    9. Applied pow2_binary6436.7

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

    10. Applied pow-prod-down_binary6434.6

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

    if -inf.0 < (/.f64 h l) < -4.7875745812866172e-223

    1. Initial program 14.0

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

    3. Applied times-frac_binary6414.1

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

    if -4.7875745812866172e-223 < (/.f64 h l)

    1. Initial program 8.4

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

    2. Taylor expanded around 0 4.6

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

  4. Final simplification10.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right)}^{2} \cdot \frac{\sqrt[3]{h}}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -4.787574581286617 \cdot 10^{-223}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h}{\ell} \cdot {\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array}\]

Reproduce

herbie shell --seed 2020270 
(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))))))