Average Error: 14.5 → 8.7
Time: 12.4s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
↓
\[\begin{array}{l} t_0 := \frac{D \cdot M}{d \cdot 2}\\ \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{t_0 \cdot \left(h \cdot \left(0.5 \cdot \frac{D \cdot M}{d}\right)\right)}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -6.3510756388060095 \cdot 10^{-292}:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \left(\frac{h}{\ell} \cdot t_0\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
t_0 := \frac{D \cdot M}{d \cdot 2}\\
\mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{t_0 \cdot \left(h \cdot \left(0.5 \cdot \frac{D \cdot M}{d}\right)\right)}{\ell}}\\

\mathbf{elif}\;\frac{h}{\ell} \leq -6.3510756388060095 \cdot 10^{-292}:\\
\;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \left(\frac{h}{\ell} \cdot 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 (/ (* D M) (* d 2.0))))
   (if (<= (/ h l) (- INFINITY))
     (* w0 (sqrt (- 1.0 (/ (* t_0 (* h (* 0.5 (/ (* D M) d)))) l))))
     (if (<= (/ h l) -6.3510756388060095e-292)
       (* w0 (sqrt (- 1.0 (* t_0 (* (/ h l) 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 = (D * M) / (d * 2.0);
	double tmp;
	if ((h / l) <= -((double) INFINITY)) {
		tmp = w0 * sqrt(1.0 - ((t_0 * (h * (0.5 * ((D * M) / d)))) / l));
	} else if ((h / l) <= -6.3510756388060095e-292) {
		tmp = w0 * sqrt(1.0 - (t_0 * ((h / l) * 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. Using strategy rm

    3. Applied div-inv_binary6464.0

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

    4. Applied associate-*r*_binary6425.0

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

    5. Simplified25.0

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

    7. Applied unpow2_binary6425.0

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

    8. Applied associate-*r*_binary6420.5

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

    9. Simplified20.5

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

    10. Taylor expanded in D around 0 20.5

      \[\leadsto w0 \cdot \sqrt{1 - \left(\left(h \cdot \color{blue}{\left(0.5 \cdot \frac{D \cdot M}{d}\right)}\right) \cdot \frac{D \cdot M}{2 \cdot d}\right) \cdot \frac{1}{\ell}} \]

    11. Simplified20.5

      \[\leadsto \color{blue}{w0 \cdot \sqrt{1 - \frac{\frac{D \cdot M}{d \cdot 2} \cdot \left(h \cdot \left(0.5 \cdot \frac{D \cdot M}{d}\right)\right)}{\ell}}} \]

    if -inf.0 < (/.f64 h l) < -6.3510756388060095e-292

    1. Initial program 15.2

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

    3. Applied unpow2_binary6415.2

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

    4. Applied associate-*l*_binary6413.3

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

    5. Simplified13.3

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

    if -6.3510756388060095e-292 < (/.f64 h l)

    1. Initial program 7.9

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

    2. Taylor expanded in M around 0 3.0

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

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\frac{D \cdot M}{d \cdot 2} \cdot \left(h \cdot \left(0.5 \cdot \frac{D \cdot M}{d}\right)\right)}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -6.3510756388060095 \cdot 10^{-292}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{D \cdot M}{d \cdot 2} \cdot \left(\frac{h}{\ell} \cdot \frac{D \cdot M}{d \cdot 2}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Reproduce

herbie shell --seed 2021206 
(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))))))