Average Error: 14.2 → 8.9
Time: 12.3s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
↓
\[w0 \cdot \sqrt{1 - \frac{1}{\frac{2 \cdot d}{M \cdot D}} \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{M \cdot D}{d}\right)\right) \cdot \frac{1}{\ell}\right)} \]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \frac{1}{\frac{2 \cdot d}{M \cdot D}} \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{M \cdot D}{d}\right)\right) \cdot \frac{1}{\ell}\right)}

(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
    (*
     (/ 1.0 (/ (* 2.0 d) (* M D)))
     (* (* h (* 0.5 (/ (* M D) d))) (/ 1.0 l)))))))
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt(1.0 - (pow(((M * D) / (2.0 * d)), 2.0) * (h / l)));
}
↓
double code(double w0, double M, double D, double h, double l, double d) {
	return w0 * sqrt(1.0 - ((1.0 / ((2.0 * d) / (M * D))) * ((h * (0.5 * ((M * D) / d))) * (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 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 unpow2_binary6414.2

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

  4. Applied associate-*l*_binary6412.8

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

  5. Simplified12.8

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

  7. Applied div-inv_binary6412.8

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

  8. Applied associate-*r*_binary648.9

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

  9. Simplified8.9

    \[\leadsto w0 \cdot \sqrt{1 - \frac{M \cdot D}{2 \cdot d} \cdot \left(\color{blue}{\left(h \cdot \left(0.5 \cdot \frac{D \cdot M}{d}\right)\right)} \cdot \frac{1}{\ell}\right)} \]
  10. Using strategy rm

  11. Applied clear-num_binary648.9

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{1}{\frac{2 \cdot d}{M \cdot D}}} \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{D \cdot M}{d}\right)\right) \cdot \frac{1}{\ell}\right)} \]

  12. Final simplification8.9

    \[\leadsto w0 \cdot \sqrt{1 - \frac{1}{\frac{2 \cdot d}{M \cdot D}} \cdot \left(\left(h \cdot \left(0.5 \cdot \frac{M \cdot D}{d}\right)\right) \cdot \frac{1}{\ell}\right)} \]

Reproduce

herbie shell --seed 2021202 
(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))))))