Average Error: 13.6 → 8.6
Time: 12.4s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right) \cdot \frac{1}{\ell}\right)}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right) \cdot \frac{1}{\ell}\right)}
double code(double w0, double M, double D, double h, double l, double d) {
	return ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) pow(((double) (((double) (M * D)) / ((double) (2.0 * d)))), 2.0)) * ((double) (h / l))))))))));
}
↓
double code(double w0, double M, double D, double h, double l, double d) {
	return ((double) (w0 * ((double) sqrt(((double) (1.0 - ((double) (((double) pow(((double) (((double) (M * D)) / ((double) (2.0 * d)))), ((double) (2.0 / 2.0)))) * ((double) (((double) (((double) pow(((double) (((double) (M * D)) / ((double) (2.0 * d)))), ((double) (2.0 / 2.0)))) * h)) * ((double) (1.0 / 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 13.6

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

  3. Applied div-inv13.6

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

  4. Applied associate-*r*10.6

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

  6. Applied sqr-pow10.6

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

  7. Applied associate-*l*9.2

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

  9. Applied associate-*l*8.6

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

  10. Final simplification8.6

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

Reproduce

herbie shell --seed 2020171 
(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))))))