Average Error: 14.5 → 9.0
Time: 16.3s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq -2.7167738327420827 \cdot 10^{+267}:\\ \;\;\;\;\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(\left(-M\right) \cdot w0\right)\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \leq -4.5146335076926557 \cdot 10^{+117}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \frac{1}{\ell}\right)}\\ \end{array}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
\mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq -2.7167738327420827 \cdot 10^{+267}:\\
\;\;\;\;\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(\left(-M\right) \cdot w0\right)\\

\mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \leq -4.5146335076926557 \cdot 10^{+117}:\\
\;\;\;\;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)}\\

\mathbf{else}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \frac{1}{\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 (<= (/ (* M D) (* 2.0 d)) -2.7167738327420827e+267)
   (* (sqrt (* (* (/ h l) (pow (/ D d) 2.0)) -0.25)) (* (- M) w0))
   (if (<= (/ (* M D) (* 2.0 d)) -4.5146335076926557e+117)
     (*
      w0
      (sqrt
       (- 1.0 (* (/ (* M D) (* 2.0 d)) (* (/ (* M D) (* 2.0 d)) (/ h l))))))
     (*
      w0
      (sqrt
       (-
        1.0
        (*
         (* h (* (/ D d) (/ M 2.0)))
         (* (* (/ D d) (/ M 2.0)) (/ 1.0 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 (((M * D) / (2.0 * d)) <= -2.7167738327420827e+267) {
		tmp = sqrt(((h / l) * pow((D / d), 2.0)) * -0.25) * (-M * w0);
	} else if (((M * D) / (2.0 * d)) <= -4.5146335076926557e+117) {
		tmp = w0 * sqrt(1.0 - (((M * D) / (2.0 * d)) * (((M * D) / (2.0 * d)) * (h / l))));
	} else {
		tmp = w0 * sqrt(1.0 - ((h * ((D / d) * (M / 2.0))) * (((D / d) * (M / 2.0)) * (1.0 / 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 3 regimes

  2. if (/.f64 (*.f64 M D) (*.f64 2 d)) < -2.7167738327420827e267

    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 61.9

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

    3. Simplified57.6

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

    if -2.7167738327420827e267 < (/.f64 (*.f64 M D) (*.f64 2 d)) < -4.5146335076926557e117

    1. Initial program 47.7

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

    3. Applied unpow2_binary64_116647.7

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

    4. Applied associate-*l*_binary64_104229.8

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

    if -4.5146335076926557e117 < (/.f64 (*.f64 M D) (*.f64 2 d))

    1. Initial program 10.9

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

    3. Applied div-inv_binary64_109810.9

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

    4. Applied associate-*r*_binary64_10417.2

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

    5. Simplified7.2

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

    7. Applied times-frac_binary64_11077.4

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

    9. Applied unpow2_binary64_11667.4

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

    10. Applied associate-*r*_binary64_10416.6

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

    12. Applied associate-*l*_binary64_10426.0

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

  4. Final simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{M \cdot D}{2 \cdot d} \leq -2.7167738327420827 \cdot 10^{+267}:\\ \;\;\;\;\sqrt{\left(\frac{h}{\ell} \cdot {\left(\frac{D}{d}\right)}^{2}\right) \cdot -0.25} \cdot \left(\left(-M\right) \cdot w0\right)\\ \mathbf{elif}\;\frac{M \cdot D}{2 \cdot d} \leq -4.5146335076926557 \cdot 10^{+117}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{h}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)\right) \cdot \left(\left(\frac{D}{d} \cdot \frac{M}{2}\right) \cdot \frac{1}{\ell}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021024 
(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))))))