Average Error: 13.9 → 9.1
Time: 9.7s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(h \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \frac{1}{\ell}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(h \cdot {\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \frac{1}{\ell}}
(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
 (*
  w0
  (sqrt
   (-
    1.0
    (*
     (*
      (pow (* (/ D d) (/ M 2.0)) (/ 2.0 2.0))
      (* h (pow (* (/ D d) (/ M 2.0)) (/ 2.0 2.0))))
     (/ 1.0 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) (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) (((double) (((double) pow(((double) ((D / d) * (M / 2.0))), (2.0 / 2.0))) * ((double) (h * ((double) pow(((double) ((D / d) * (M / 2.0))), (2.0 / 2.0))))))) * (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 13.9

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

  3. Applied div-inv_binary6413.9

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

  4. Applied associate-*r*_binary6410.6

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

  5. Simplified10.6

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

  7. Applied times-frac_binary6410.6

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

  9. Applied sqr-pow_binary6410.6

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

  10. Applied associate-*r*_binary649.1

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

  11. Simplified9.1

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

  12. Final simplification9.1

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

Reproduce

herbie shell --seed 2020204 
(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))))))