Average Error: 13.9 → 8.9
Time: 12.3s
Precision: binary64
\[[M, D] = \mathsf{sort}([M, D]) \\]
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
↓
\[\begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ \mathbf{if}\;{t_0}^{2} \leq 3.96227836372206 \cdot 10^{+300}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right) \cdot \frac{t_0 \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M \cdot w0\right)\\ \end{array} \]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
t_0 := \frac{M \cdot D}{2 \cdot d}\\
\mathbf{if}\;{t_0}^{2} \leq 3.96227836372206 \cdot 10^{+300}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right) \cdot \frac{t_0 \cdot h}{\ell}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M \cdot w0\right)\\


\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 (/ (* M D) (* 2.0 d))))
   (if (<= (pow t_0 2.0) 3.96227836372206e+300)
     (* w0 (sqrt (- 1.0 (* (* (* M D) (/ 1.0 (* 2.0 d))) (/ (* t_0 h) l)))))
     (* (sqrt (* (* (/ h l) (pow (/ D d) 2.0)) -0.25)) (- (* M 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 = (M * D) / (2.0 * d);
	double tmp;
	if (pow(t_0, 2.0) <= 3.96227836372206e+300) {
		tmp = w0 * sqrt(1.0 - (((M * D) * (1.0 / (2.0 * d))) * ((t_0 * h) / l)));
	} else {
		tmp = sqrt(((h / l) * pow((D / d), 2.0)) * -0.25) * -(M * 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 2 regimes

  2. if (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 3.9622783637220599e300

    1. Initial program 6.7

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

    2. Applied div-inv_binary646.7

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

    3. Applied associate-*r*_binary643.4

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

    4. Applied unpow2_binary643.4

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

    5. Applied associate-*l*_binary643.1

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

    6. Applied associate-*l*_binary642.9

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

    7. Simplified2.9

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

    8. Applied div-inv_binary642.9

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

    if 3.9622783637220599e300 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

    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 in M around -inf 57.1

      \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}} \cdot \left(w0 \cdot M\right)\right)} \]

    3. Simplified49.5

      \[\leadsto \color{blue}{\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M \cdot w0\right)} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 3.96227836372206 \cdot 10^{+300}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right) \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M \cdot w0\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022097 
(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))))))