Average Error: 14.4 → 8.8
Time: 12.5s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
↓
\[\begin{array}{l} t_0 := \frac{M \cdot D}{2 \cdot d}\\ w0 \cdot \sqrt{1 - t_0 \cdot \left(h \cdot \frac{t_0}{\ell}\right)} \end{array} \]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\begin{array}{l}
t_0 := \frac{M \cdot D}{2 \cdot d}\\
w0 \cdot \sqrt{1 - t_0 \cdot \left(h \cdot \frac{t_0}{\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
 (let* ((t_0 (/ (* M D) (* 2.0 d))))
   (* w0 (sqrt (- 1.0 (* t_0 (* h (/ t_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 t_0 = (M * D) / (2.0 * d);
	return w0 * sqrt((1.0 - (t_0 * (h * (t_0 / 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 14.4

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

  2. Applied unpow2_binary6414.4

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

  3. Applied associate-*l*_binary6413.0

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

  4. Applied associate-*r/_binary648.8

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

  5. Applied add-cbrt-cube_binary6413.1

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

  6. Simplified13.1

    \[\leadsto w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{\color{blue}{{\left(\frac{h \cdot \frac{D \cdot M}{d \cdot 2}}{\ell}\right)}^{3}}}} \]

  7. Applied *-un-lft-identity_binary6413.1

    \[\leadsto w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{{\left(\frac{h \cdot \frac{D \cdot M}{d \cdot 2}}{\color{blue}{1 \cdot \ell}}\right)}^{3}}} \]

  8. Applied times-frac_binary6413.1

    \[\leadsto w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{{\color{blue}{\left(\frac{h}{1} \cdot \frac{\frac{D \cdot M}{d \cdot 2}}{\ell}\right)}}^{3}}} \]

  9. Applied unpow-prod-down_binary6428.4

    \[\leadsto w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{\color{blue}{{\left(\frac{h}{1}\right)}^{3} \cdot {\left(\frac{\frac{D \cdot M}{d \cdot 2}}{\ell}\right)}^{3}}}} \]

  10. Applied cbrt-prod_binary6428.0

    \[\leadsto w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \color{blue}{\left(\sqrt[3]{{\left(\frac{h}{1}\right)}^{3}} \cdot \sqrt[3]{{\left(\frac{\frac{D \cdot M}{d \cdot 2}}{\ell}\right)}^{3}}\right)}} \]

  11. Simplified13.8

    \[\leadsto w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\color{blue}{h} \cdot \sqrt[3]{{\left(\frac{\frac{D \cdot M}{d \cdot 2}}{\ell}\right)}^{3}}\right)} \]

  12. Simplified8.8

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

  13. Final simplification8.8

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

Reproduce

herbie shell --seed 2022160 
(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))))))