Average Error: 13.9 → 8.8
Time: 8.8s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq -1.9500665433669326 \cdot 10^{+41} \lor \neg \left(\frac{M \cdot D}{2 \cdot d} \leq 1.062766902089649 \cdot 10^{-68}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq -1.9500665433669326 \cdot 10^{+41} \lor \neg \left(\frac{M \cdot D}{2 \cdot d} \leq 1.062766902089649 \cdot 10^{-68}\right):\\
\;\;\;\;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 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}}\\

\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 (or (<= (/ (* M D) (* 2.0 d)) -1.9500665433669326e+41)
         (not (<= (/ (* M D) (* 2.0 d)) 1.062766902089649e-68)))
   (*
    w0
    (sqrt (- 1.0 (* (/ (* M D) (* 2.0 d)) (* (/ (* M D) (* 2.0 d)) (/ h l))))))
   (* w0 (sqrt (- 1.0 (/ (* h (pow (* (* M D) (/ 0.5 d)) 2.0)) l))))))
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 ((((M * D) / (2.0 * d)) <= -1.9500665433669326e+41) || !(((M * D) / (2.0 * d)) <= 1.062766902089649e-68)) {
		tmp = w0 * sqrt(1.0 - (((M * D) / (2.0 * d)) * (((M * D) / (2.0 * d)) * (h / l))));
	} else {
		tmp = w0 * sqrt(1.0 - ((h * pow(((M * D) * (0.5 / d)), 2.0)) / l));
	}
	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 (/.f64 (*.f64 M D) (*.f64 2 d)) < -1.9500665433669326e41 or 1.06276690208964906e-68 < (/.f64 (*.f64 M D) (*.f64 2 d))

    1. Initial program 32.0

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

    3. Applied unpow2_binary64_151632.0

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

    4. Applied associate-*l*_binary64_163726.8

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

    if -1.9500665433669326e41 < (/.f64 (*.f64 M D) (*.f64 2 d)) < 1.06276690208964906e-68

    1. Initial program 6.2

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

    3. Applied associate-*r/_binary64_16381.0

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

    4. Simplified1.0

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

    6. Applied div-inv_binary64_15781.0

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

    7. Simplified1.0

      \[\leadsto w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(M \cdot D\right) \cdot \color{blue}{\frac{0.5}{d}}\right)}^{2}}{\ell}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq -1.9500665433669326 \cdot 10^{+41} \lor \neg \left(\frac{M \cdot D}{2 \cdot d} \leq 1.062766902089649 \cdot 10^{-68}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{h \cdot {\left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)}^{2}}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020262 
(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))))))