Average Error: 14.2 → 8.4
Time: 9.1s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\frac{\ell}{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \frac{h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\frac{\ell}{{\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\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) (M * D)) / ((double) (2.0 * d))), 2.0)) * (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) (h * ((double) pow(((double) (M * (D / ((double) (2.0 * d))))), (2.0 / 2.0))))) / (l / ((double) pow(((double) (M * (D / ((double) (2.0 * d))))), (2.0 / 2.0)))))))))));
}

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 Error: 14.2 bits

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

  2. SimplifiedError: 14.3 bits

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

  4. Applied associate-*r/Error: 10.9 bits

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

  5. SimplifiedError: 10.9 bits

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

  7. Applied sqr-powError: 10.9 bits

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

  8. Applied associate-*r*Error: 9.2 bits

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

  10. Applied associate-/l*Error: 8.4 bits

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

  11. Final simplificationError: 8.4 bits

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

Reproduce

herbie shell --seed 2020200 
(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))))))