Average Error: 14.2 → 9.0
Time: 9.4s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left(h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(2 \cdot \frac{1}{2}\right)}\right) \cdot \frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(2 \cdot \frac{1}{2}\right)}}{\ell}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \left(h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(2 \cdot \frac{1}{2}\right)}\right) \cdot \frac{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(2 \cdot \frac{1}{2}\right)}}{\ell}}
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) (h * ((double) pow(((double) (M * ((double) (D / ((double) (2.0 * d)))))), ((double) (2.0 * 0.5)))))) * ((double) (((double) pow(((double) (M * ((double) (D / ((double) (2.0 * d)))))), ((double) (2.0 * 0.5)))) / 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. Using strategy rm

  3. Applied associate-*r/11.0

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

  4. Simplified11.1

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

  6. Applied sqr-pow11.1

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

  7. Applied associate-*r*9.6

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

  8. Simplified9.6

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

  10. Applied *-un-lft-identity9.6

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

  11. Applied times-frac9.0

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

  12. Simplified9.0

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

  13. Simplified9.0

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

  14. Final simplification9.0

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

Reproduce

herbie shell --seed 2020184 
(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))))))