Average Error: 14.7 → 9.4
Time: 7.9s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;M \cdot D \le 4.4576118192580824 \cdot 10^{-92}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{\left(\frac{2}{2}\right)}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;M \cdot D \le 4.4576118192580824 \cdot 10^{-92}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{\left(\frac{2}{2}\right)}}{\ell}}\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\

\end{array}
double code(double w0, double M, double D, double h, double l, double d) {
	return ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) pow((((double) (M * D)) / ((double) (2.0 * d))), 2.0)) * (h / l)))))))));
}
↓
double code(double w0, double M, double D, double h, double l, double d) {
	double VAR;
	if ((((double) (M * D)) <= 4.4576118192580824e-92)) {
		VAR = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) (h * ((double) pow(((double) (0.5 * ((double) (D * (M / d))))), (2.0 / 2.0))))) * (((double) pow(((double) (0.5 * ((double) (D * (M / d))))), (2.0 / 2.0))) / l)))))))));
	} else {
		VAR = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) pow(((double) (M * (D / ((double) (d * 2.0))))), (2.0 / 2.0))) * ((double) (((double) pow(((double) (M * (D / ((double) (d * 2.0))))), (2.0 / 2.0))) * (h / l)))))))))));
	}
	return VAR;
}

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 (* M D) < 4.4576118192580824e-92

    1. Initial program 12.4

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

    2. Simplified12.4

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

    4. Applied div-inv12.4

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

    5. Applied associate-*r*8.0

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

    6. Simplified8.0

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

    7. Taylor expanded around 0 8.0

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

    8. Simplified8.1

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

    10. Applied sqr-pow8.1

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

    11. Applied associate-*r*7.0

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

    13. Applied associate-*l*6.4

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

    14. Simplified6.4

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

    if 4.4576118192580824e-92 < (* M D)

    1. Initial program 21.5

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

    2. Simplified21.3

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

    4. Applied sqr-pow21.3

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

    5. Applied associate-*l*18.2

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

    6. Simplified18.2

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

  4. Final simplification9.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \le 4.4576118192580824 \cdot 10^{-92}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot {\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(0.5 \cdot \left(D \cdot \frac{M}{d}\right)\right)}^{\left(\frac{2}{2}\right)}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{h}{\ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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))))))