Average Error: 13.9 → 9.1
Time: 10.7s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 + \left({\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot \frac{-1}{\ell}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 + \left({\left(M \cdot \frac{D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(h \cdot {\left(M \cdot \frac{D}{2 \cdot d}\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 (* M (/ D (* 2.0 d))) (/ 2.0 2.0))
      (* h (pow (* M (/ D (* 2.0 d))) (/ 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) (M * (D / ((double) (2.0 * d))))), (2.0 / 2.0))) * ((double) (h * ((double) pow(((double) (M * (D / ((double) (2.0 * d))))), (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 Error: 13.9 bits

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

  2. SimplifiedError: 14.0 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: 10.6 bits

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

  5. SimplifiedError: 10.6 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: 10.6 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: 9.1 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 div-invError: 9.1 bits

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

  11. Final simplificationError: 9.1 bits

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

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))))))