Average Error: 14.2 → 8.1
Time: 14.8s
Precision: binary64
\[[M, D]=\mathsf{sort}([M, D])\]
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
↓
\[\begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ t_1 := {t_0}^{2} \cdot \frac{h}{\ell}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M \cdot w0\right)\\ \mathbf{elif}\;t_1 \leq -1690.3678749542453:\\ \;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \left(t_0 \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
t_0 := \frac{M \cdot D}{2 \cdot d}\\
t_1 := {t_0}^{2} \cdot \frac{h}{\ell}\\
\mathbf{if}\;t_1 \leq -\infty:\\
\;\;\;\;\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M \cdot w0\right)\\

\mathbf{elif}\;t_1 \leq -1690.3678749542453:\\
\;\;\;\;w0 \cdot \sqrt{1 - t_0 \cdot \left(t_0 \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
 (let* ((t_0 (/ (* M D) (* 2.0 d))) (t_1 (* (pow t_0 2.0) (/ h l))))
   (if (<= t_1 (- INFINITY))
     (* (sqrt (* (* (/ h l) (pow (/ D d) 2.0)) -0.25)) (- (* M w0)))
     (if (<= t_1 -1690.3678749542453)
       (* w0 (sqrt (- 1.0 (* t_0 (* t_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 t_0 = (M * D) / (2.0 * d);
	double t_1 = pow(t_0, 2.0) * (h / l);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = sqrt(((h / l) * pow((D / d), 2.0)) * -0.25) * -(M * w0);
	} else if (t_1 <= -1690.3678749542453) {
		tmp = w0 * sqrt(1.0 - (t_0 * (t_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 (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.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. Taylor expanded in M around -inf 56.3

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

    3. Simplified48.6

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

    if -inf.0 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l)) < -1690.36787495424528

    1. Initial program 0.5

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

    2. Applied unpow2_binary640.5

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

    3. Applied associate-*l*_binary640.5

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

    if -1690.36787495424528 < (*.f64 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) (/.f64 h l))

    1. Initial program 7.0

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

    2. Taylor expanded in M around 0 1.9

      \[\leadsto w0 \cdot \color{blue}{1} \]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -\infty:\\ \;\;\;\;\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M \cdot w0\right)\\ \mathbf{elif}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell} \leq -1690.3678749542453:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array} \]

Reproduce

herbie shell --seed 2021280 
(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))))))