Average Error: 14.1 → 8.4
Time: 27.7s
Precision: binary64
\[[M, D]=\mathsf{sort}([M, D])\]
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{h \cdot \frac{M}{\frac{2}{\frac{D}{d}}}}{\frac{\ell}{\frac{M}{\frac{2}{\frac{D}{d}}}}}}\]
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 \frac{M}{\frac{2}{\frac{D}{d}}}}{\frac{\ell}{\frac{M}{\frac{2}{\frac{D}{d}}}}}}
(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 (/ (* h (/ M (/ 2.0 (/ D d)))) (/ l (/ M (/ 2.0 (/ D d)))))))))
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 - ((h * (M / (2.0 / (D / d)))) / (l / (M / (2.0 / (D / d))))));
}

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. Using strategy rm

  3. Applied associate-*r/_binary64_104310.6

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

  4. Simplified10.6

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

  6. Applied associate-/l*_binary64_104610.6

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

  7. Simplified10.7

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

  9. Applied sqr-pow_binary64_107310.7

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

  10. Applied associate-*r*_binary64_10419.0

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

  11. Simplified9.0

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

  13. Applied associate-/l*_binary64_10468.4

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

  14. Simplified8.4

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

  15. Final simplification8.4

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

Alternatives

Reproduce

herbie shell --seed 2021104 
(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))))))