Average Error: 14.2 → 8.4
Time: 9.2s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -8.232614490771766 \cdot 10^{-146}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(h \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)}{\ell}}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}}\\

\mathbf{elif}\;\frac{h}{\ell} \leq -8.232614490771766 \cdot 10^{-146}:\\
\;\;\;\;w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)}\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(h \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)}{\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 (<= (/ h l) (- INFINITY))
   (*
    w0
    (sqrt
     (-
      1.0
      (/
       (*
        (* h (pow (* M (/ D (* 2.0 d))) (/ 2.0 2.0)))
        (pow (/ (* M D) (* 2.0 d)) (/ 2.0 2.0)))
       l))))
   (if (<= (/ h l) -8.232614490771766e-146)
     (*
      w0
      (sqrt
       (-
        1.0
        (*
         (pow (* M (/ D (* 2.0 d))) (/ 2.0 2.0))
         (* (/ h l) (pow (* M (/ D (* 2.0 d))) (/ 2.0 2.0)))))))
     (*
      w0
      (sqrt
       (-
        1.0
        (/
         (*
          (pow (/ (* M D) (* 2.0 d)) (/ 2.0 2.0))
          (* h (pow (/ (* M D) (* 2.0 d)) (/ 2.0 2.0))))
         l)))))))
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 tmp;
	if (((h / l) <= ((double) -(((double) INFINITY))))) {
		tmp = ((double) (w0 * ((double) sqrt(((double) (1.0 - (((double) (((double) (h * ((double) pow(((double) (M * (D / ((double) (2.0 * d))))), (2.0 / 2.0))))) * ((double) pow((((double) (M * D)) / ((double) (2.0 * d))), (2.0 / 2.0))))) / l)))))));
	} else {
		double tmp_1;
		if (((h / l) <= -8.232614490771766e-146)) {
			tmp_1 = ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) pow(((double) (M * (D / ((double) (2.0 * d))))), (2.0 / 2.0))) * ((double) ((h / l) * ((double) pow(((double) (M * (D / ((double) (2.0 * d))))), (2.0 / 2.0)))))))))))));
		} else {
			tmp_1 = ((double) (w0 * ((double) sqrt(((double) (1.0 - (((double) (((double) pow((((double) (M * D)) / ((double) (2.0 * d))), (2.0 / 2.0))) * ((double) (h * ((double) pow((((double) (M * D)) / ((double) (2.0 * d))), (2.0 / 2.0))))))) / l)))))));
		}
		tmp = tmp_1;
	}
	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 (/ h l) < -inf.0

    1. Initial program Error: 64.0 bits

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

    2. SimplifiedError: 64.0 bits

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

    4. Applied associate-*r/Error: 26.8 bits

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

    5. SimplifiedError: 26.8 bits

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

    7. Applied sqr-powError: 26.8 bits

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

    8. Applied associate-*r*Error: 23.5 bits

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

    10. Applied associate-*r/Error: 25.4 bits

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

    if -inf.0 < (/ h l) < -8.2326144907717664e-146

    1. Initial program Error: 13.7 bits

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

    2. SimplifiedError: 13.3 bits

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

    4. Applied sqr-powError: 13.3 bits

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

    5. Applied associate-*l*Error: 12.7 bits

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

    6. SimplifiedError: 12.7 bits

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

    if -8.2326144907717664e-146 < (/ h l)

    1. Initial program Error: 9.3 bits

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

    2. SimplifiedError: 9.5 bits

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

    4. Applied associate-*r/Error: 6.4 bits

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

    5. SimplifiedError: 6.4 bits

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

    7. Applied sqr-powError: 6.4 bits

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

    8. Applied associate-*r*Error: 4.2 bits

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

    10. Applied associate-*r/Error: 5.0 bits

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

    12. Applied associate-*r/Error: 4.0 bits

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

  4. Final simplificationError: 8.4 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -\infty:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{\left(h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -8.232614490771766 \cdot 10^{-146}:\\ \;\;\;\;w0 \cdot \sqrt{1 - {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\frac{h}{\ell} \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(h \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)}{\ell}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020203 
(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))))))