Average Error: 14.0 → 8.4
Time: 8.0s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 2.032656749707616 \cdot 10^{-191} \lor \neg \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 3.1589775460686644 \cdot 10^{+261}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot h\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 2.032656749707616 \cdot 10^{-191} \lor \neg \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 3.1589775460686644 \cdot 10^{+261}\right):\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot h\right)}{\ell}}\\

\mathbf{else}:\\
\;\;\;\;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)}\\

\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 (<= (pow (/ (* M D) (* 2.0 d)) 2.0) 2.032656749707616e-191)
         (not (<= (pow (/ (* M D) (* 2.0 d)) 2.0) 3.1589775460686644e+261)))
   (*
    w0
    (sqrt (- 1.0 (/ (* (* (/ M 2.0) (/ D d)) (* (* (/ M 2.0) (/ D d)) h)) l))))
   (*
    w0
    (sqrt
     (- 1.0 (* (/ (* M D) (* 2.0 d)) (* (/ (* M D) (* 2.0 d)) (/ h 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 ((pow(((M * D) / (2.0 * d)), 2.0) <= 2.032656749707616e-191) || !(pow(((M * D) / (2.0 * d)), 2.0) <= 3.1589775460686644e+261)) {
		tmp = w0 * sqrt(1.0 - ((((M / 2.0) * (D / d)) * (((M / 2.0) * (D / d)) * h)) / l));
	} else {
		tmp = w0 * sqrt(1.0 - (((M * D) / (2.0 * d)) * (((M * D) / (2.0 * d)) * (h / 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 (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 2.03265674970761598e-191 or 3.15897754606866437e261 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2)

    1. Initial program 17.0

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

    3. Applied associate-*r/_binary6411.8

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

    4. Simplified11.8

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

    6. Applied times-frac_binary6411.3

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

    8. Applied unpow2_binary6411.3

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

    9. Applied associate-*r*_binary649.3

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

    10. Simplified9.3

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

    if 2.03265674970761598e-191 < (pow.f64 (/.f64 (*.f64 M D) (*.f64 2 d)) 2) < 3.15897754606866437e261

    1. Initial program 5.9

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

    3. Applied unpow2_binary645.9

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

    4. Applied associate-*l*_binary645.9

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

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 2.032656749707616 \cdot 10^{-191} \lor \neg \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \leq 3.1589775460686644 \cdot 10^{+261}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot h\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020233 
(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))))))