Average Error: 14.0 → 8.8
Time: 23.8s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - {\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(h \cdot {\left(\frac{1}{2}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}\right)}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - {\left(\frac{1}{\frac{2 \cdot d}{M \cdot D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(\left(h \cdot {\left(\frac{1}{2}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(\frac{M \cdot D}{d}\right)}^{\left(\frac{2}{2}\right)}}{\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) (1.0 / ((double) (((double) (2.0 * d)) / ((double) (M * D)))))), ((double) (2.0 / 2.0)))) * ((double) (((double) (h * ((double) pow(((double) (1.0 / 2.0)), ((double) (2.0 / 2.0)))))) * ((double) (((double) pow(((double) (((double) (M * D)) / d)), ((double) (2.0 / 2.0)))) / 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 14.0

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

  3. Applied sqr-pow14.0

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

  4. Applied associate-*l*12.6

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

  6. Applied associate-*r/8.8

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

  7. Simplified8.8

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

  9. Applied *-un-lft-identity8.8

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

  10. Applied *-un-lft-identity8.8

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

  11. Applied times-frac8.8

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

  12. Applied unpow-prod-down8.8

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

  13. Applied associate-*r*8.8

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

  14. Applied times-frac8.8

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

  15. Simplified8.8

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

  17. Applied clear-num8.8

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

  18. Final simplification8.8

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

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))