Average Error: 14.2 → 8.8
Time: 16.0s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
↓
\[\begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ w0 \cdot \sqrt{1 - t_0 \cdot \frac{t_0 \cdot h}{\ell}} \end{array} \]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
t_0 := \frac{M \cdot D}{2 \cdot d}\\
w0 \cdot \sqrt{1 - t_0 \cdot \frac{t_0 \cdot h}{\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
 (let* ((t_0 (/ (* M D) (* 2.0 d))))
   (* w0 (sqrt (- 1.0 (* t_0 (/ (* t_0 h) 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 t_0 = (M * D) / (2.0 * d);
	return w0 * sqrt(1.0 - (t_0 * ((t_0 * h) / l)));
}

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. Initial program 14.2

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

  2. Applied div-inv_binary6414.2

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

  3. Applied associate-*r*_binary6410.9

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

  4. Applied unpow2_binary6410.9

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

  5. Applied associate-*l*_binary649.4

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

  6. Applied associate-*l*_binary648.8

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

  7. Simplified8.8

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

  8. Final simplification8.8

    \[\leadsto w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \frac{\frac{M \cdot D}{2 \cdot d} \cdot h}{\ell}} \]

Reproduce

herbie shell --seed 2022094 
(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))))))