Average Error: 14.0 → 9.1
Time: 17.4s
Precision: binary64
\[[M, D]=\mathsf{sort}([M, D])\]
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;M \leq -7.585162659511325 \cdot 10^{+212} \lor \neg \left(M \leq -8.913855756833514 \cdot 10^{+196}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)\right) \cdot \frac{0.5 \cdot \frac{M \cdot D}{d}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{w0}{M \cdot \sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25}} \cdot -0.5 - M \cdot \left(w0 \cdot \sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25}\right)\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;M \leq -7.585162659511325 \cdot 10^{+212} \lor \neg \left(M \leq -8.913855756833514 \cdot 10^{+196}\right):\\
\;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)\right) \cdot \frac{0.5 \cdot \frac{M \cdot D}{d}}{\ell}}\\

\mathbf{else}:\\
\;\;\;\;\frac{w0}{M \cdot \sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25}} \cdot -0.5 - M \cdot \left(w0 \cdot \sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25}\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 (<= M -7.585162659511325e+212) (not (<= M -8.913855756833514e+196)))
   (*
    w0
    (sqrt (- 1.0 (* (* h (* (* M D) (/ 0.5 d))) (/ (* 0.5 (/ (* M D) d)) l)))))
   (-
    (* (/ w0 (* M (sqrt (* (* (/ h l) (pow (/ D d) 2.0)) -0.25)))) -0.5)
    (* M (* w0 (sqrt (* (* (/ h l) (pow (/ D d) 2.0)) -0.25)))))))
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 <= -7.585162659511325e+212) || !(M <= -8.913855756833514e+196)) {
		tmp = w0 * sqrt(1.0 - ((h * ((M * D) * (0.5 / d))) * ((0.5 * ((M * D) / d)) / l)));
	} else {
		tmp = ((w0 / (M * sqrt(((h / l) * pow((D / d), 2.0)) * -0.25))) * -0.5) - (M * (w0 * sqrt(((h / l) * pow((D / d), 2.0)) * -0.25)));
	}
	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 M < -7.58516265951132523e212 or -8.91385575683351356e196 < M

    1. Initial program 13.7

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

    3. Applied div-inv_binary6413.7

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

    4. Applied associate-*r*_binary6410.7

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

    5. Simplified10.7

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

    7. Applied unpow2_binary6410.7

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

    8. Applied associate-*r*_binary649.1

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

    9. Simplified9.1

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

    11. Applied div-inv_binary649.1

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

    12. Simplified9.1

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

    14. Applied associate-*l*_binary648.4

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

    15. Simplified8.4

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

    if -7.58516265951132523e212 < M < -8.91385575683351356e196

    1. Initial program 28.9

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

    2. Taylor expanded around -inf 54.3

      \[\leadsto \color{blue}{-\left(0.5 \cdot \frac{w0}{M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}} + M \cdot \left(\sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}} \cdot w0\right)\right)}\]

    3. Simplified49.3

      \[\leadsto \color{blue}{\frac{w0}{M \cdot \sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25}} \cdot -0.5 - M \cdot \left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot w0\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;M \leq -7.585162659511325 \cdot 10^{+212} \lor \neg \left(M \leq -8.913855756833514 \cdot 10^{+196}\right):\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \left(\left(M \cdot D\right) \cdot \frac{0.5}{d}\right)\right) \cdot \frac{0.5 \cdot \frac{M \cdot D}{d}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{w0}{M \cdot \sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25}} \cdot -0.5 - M \cdot \left(w0 \cdot \sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021174 
(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))))))