Average Error: 14.1 → 8.7
Time: 9.2s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -7.111607855480492 \cdot 10^{+101}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot {\left(\frac{M}{\frac{2}{\frac{D}{d}}}\right)}^{2}\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1.5631206412634186 \cdot 10^{-256}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array}\]
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 -7.111607855480492 \cdot 10^{+101}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot {\left(\frac{M}{\frac{2}{\frac{D}{d}}}\right)}^{2}\right) \cdot \frac{1}{\ell}}\\

\mathbf{elif}\;\frac{h}{\ell} \leq -1.5631206412634186 \cdot 10^{-256}:\\
\;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\\

\mathbf{else}:\\
\;\;\;\;w0\\

\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) -7.111607855480492e+101)
   (* w0 (sqrt (- 1.0 (* (* h (pow (/ M (/ 2.0 (/ D d))) 2.0)) (/ 1.0 l)))))
   (if (<= (/ h l) -1.5631206412634186e-256)
     (*
      w0
      (sqrt
       (- 1.0 (* (/ (* M D) (* 2.0 d)) (* (/ h l) (/ (* M D) (* 2.0 d)))))))
     w0)))
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 ((h / l) <= -7.111607855480492e+101) {
		tmp = w0 * sqrt(1.0 - ((h * pow((M / (2.0 / (D / d))), 2.0)) * (1.0 / l)));
	} else if ((h / l) <= -1.5631206412634186e-256) {
		tmp = w0 * sqrt(1.0 - (((M * D) / (2.0 * d)) * ((h / l) * ((M * D) / (2.0 * d)))));
	} else {
		tmp = w0;
	}
	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 h l) < -7.11160785548049185e101

    1. Initial program 31.1

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

    3. Applied div-inv_binary64_143931.1

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

    4. Applied associate-*r*_binary64_138218.8

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

    5. Simplified18.8

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

    7. Applied associate-/l*_binary64_138718.3

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

    8. Simplified18.3

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

    if -7.11160785548049185e101 < (/.f64 h l) < -1.5631206412634186e-256

    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 unpow2_binary64_150713.7

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

    4. Applied associate-*l*_binary64_138311.6

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

    if -1.5631206412634186e-256 < (/.f64 h l)

    1. Initial program 8.0

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

    2. Taylor expanded around 0 3.2

      \[\leadsto \color{blue}{w0}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification8.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{h}{\ell} \leq -7.111607855480492 \cdot 10^{+101}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \left(h \cdot {\left(\frac{M}{\frac{2}{\frac{D}{d}}}\right)}^{2}\right) \cdot \frac{1}{\ell}}\\ \mathbf{elif}\;\frac{h}{\ell} \leq -1.5631206412634186 \cdot 10^{-256}:\\ \;\;\;\;w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\frac{h}{\ell} \cdot \frac{M \cdot D}{2 \cdot d}\right)}\\ \mathbf{else}:\\ \;\;\;\;w0\\ \end{array}\]

Reproduce

herbie shell --seed 2020342 
(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))))))