Average Error: 13.9 → 9.3
Time: 17.5s
Precision: binary64
\[[M, D] = \mathsf{sort}([M, D]) \\]
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
↓
\[\begin{array}{l} t_0 := \frac{2 \cdot d}{D}\\ \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{2 \cdot d}\right)}^{2} \cdot \left(h \cdot {D}^{2}\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -3.034501557897143 \cdot 10^{-241}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{t_0}{M}} \cdot \left(\frac{h}{\ell} \cdot \frac{M}{t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
t_0 := \frac{2 \cdot d}{D}\\
\mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\
\;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{2 \cdot d}\right)}^{2} \cdot \left(h \cdot {D}^{2}\right)\right) \cdot \frac{1}{\ell}}\\

\mathbf{elif}\;\frac{h}{\ell} \leq -3.034501557897143 \cdot 10^{-241}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{t_0}{M}} \cdot \left(\frac{h}{\ell} \cdot \frac{M}{t_0}\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
 (let* ((t_0 (/ (* 2.0 d) D)))
   (if (<= (/ h l) (- INFINITY))
     (*
      w0
      (sqrt
       (- 1.0 (* (* (pow (/ M (* 2.0 d)) 2.0) (* h (pow D 2.0))) (/ 1.0 l)))))
     (if (<= (/ h l) -3.034501557897143e-241)
       (* w0 (sqrt (- 1.0 (* (/ 1.0 (/ t_0 M)) (* (/ h l) (/ M t_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 t_0 = (2.0 * d) / D;
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * sqrt((1.0 - ((pow((M / (2.0 * d)), 2.0) * (h * pow(D, 2.0))) * (1.0 / l))));
	} else if ((h / l) <= -3.034501557897143e-241) {
		tmp = w0 * sqrt((1.0 - ((1.0 / (t_0 / M)) * ((h / l) * (M / t_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. Applied associate-/l*_binary6464.0

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

    3. Applied div-inv_binary6464.0

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

    4. Applied associate-*r*_binary6425.6

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

    5. Applied associate-/r/_binary6425.2

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

    6. Applied unpow-prod-down_binary6431.5

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

    7. Applied associate-*l*_binary6433.8

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

    if -inf.0 < (/.f64 h l) < -3.034501557897143e-241

    1. Initial program 13.7

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

    2. Applied associate-/l*_binary6413.6

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

    3. Applied unpow2_binary6413.6

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

    4. Applied associate-*l*_binary6412.2

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

    5. Applied clear-num_binary6412.2

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

    if -3.034501557897143e-241 < (/.f64 h l)

    1. Initial program 8.2

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

    2. Taylor expanded in M around 0 3.9

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left({\left(\frac{M}{2 \cdot d}\right)}^{2} \cdot \left(h \cdot {D}^{2}\right)\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -3.034501557897143 \cdot 10^{-241}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{\frac{2 \cdot d}{D}}{M}} \cdot \left(\frac{h}{\ell} \cdot \frac{M}{\frac{2 \cdot d}{D}}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Reproduce

herbie shell --seed 2022125 
(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))))))