Average Error: 14.1 → 8.5
Time: 12.1s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}} \]
↓
\[w0 \cdot \sqrt{1 - \frac{\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}}{\frac{\ell}{\frac{M \cdot D}{2 \cdot d} \cdot h}}} \]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \frac{\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}}{\frac{\ell}{\frac{M \cdot D}{2 \cdot d} \cdot h}}}

(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
    (/ (* (* M D) (/ 1.0 (* 2.0 d))) (/ l (* (/ (* M D) (* 2.0 d)) h)))))))
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 - (((M * D) * (1.0 / (2.0 * d))) / (l / (((M * D) / (2.0 * d)) * h))));
}

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 14.1

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

  2. Applied associate-*r/_binary6410.7

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

  3. Applied unpow2_binary6410.7

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

  4. Applied associate-*l*_binary649.2

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

  5. Applied associate-/l*_binary648.5

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

  6. Applied div-inv_binary648.5

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

  7. Final simplification8.5

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

Reproduce

herbie shell --seed 2022121 
(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))))))