Average Error: 13.6 → 8.8
Time: 22.0s
Precision: binary64
\[[M, D]=\mathsf{sort}([M, D])\]
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2.4125158115012737 \cdot 10^{+176}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M\right)\right)\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2.4125158115012737 \cdot 10^{+176}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)}}}\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M\right)\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 (<= (/ (* M D) (* 2.0 d)) 2.4125158115012737e+176)
   (*
    w0
    (sqrt
     (-
      1.0
      (/ 1.0 (/ l (* (/ (* M D) (* 2.0 d)) (* (/ (* M D) (* 2.0 d)) h)))))))
   (* w0 (* (sqrt (* (* (/ h l) (pow (/ D d) 2.0)) -0.25)) (- M)))))
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)) <= 2.4125158115012737e+176) {
		tmp = w0 * sqrt(1.0 - (1.0 / (l / (((M * D) / (2.0 * d)) * (((M * D) / (2.0 * d)) * h)))));
	} else {
		tmp = w0 * (sqrt(((h / l) * pow((D / d), 2.0)) * -0.25) * -M);
	}
	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)) < 2.41251581150127374e176

    1. Initial program 10.7

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

    3. Applied associate-*r/_binary64_10437.7

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

    4. Simplified7.7

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

    6. Applied unpow2_binary64_11667.7

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

    7. Applied associate-*r*_binary64_10416.5

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

    9. Applied clear-num_binary64_11006.5

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

    if 2.41251581150127374e176 < (/.f64 (*.f64 M D) (*.f64 2 d))

    1. Initial program 64.0

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

    2. Taylor expanded around -inf 58.7

      \[\leadsto w0 \cdot \color{blue}{\left(-1 \cdot \left(M \cdot \sqrt{-0.25 \cdot \frac{{D}^{2} \cdot h}{{d}^{2} \cdot \ell}}\right)\right)}\]

    3. Simplified50.3

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq 2.4125158115012737 \cdot 10^{+176}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{1}{\frac{\ell}{\frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot h\right)}}}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \left(\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(-M\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021050 
(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))))))