Average Error: 14.8 → 9.0
Time: 15.1s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left(h \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \frac{\frac{M \cdot D}{d \cdot 2}}{\ell}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \left(h \cdot \frac{M \cdot D}{d \cdot 2}\right) \cdot \frac{\frac{M \cdot D}{d \cdot 2}}{\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 (* (* h (/ (* M D) (* d 2.0))) (/ (/ (* M D) (* d 2.0)) 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 - ((h * ((M * D) / (d * 2.0))) * (((M * D) / (d * 2.0)) / 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 14.8

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

  3. Applied associate-*r/_binary64_104311.3

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

  4. Simplified11.3

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

  6. Applied unpow2_binary64_116611.3

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

  7. Applied associate-*r*_binary64_10419.6

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

  9. Applied *-un-lft-identity_binary64_11019.6

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

  10. Applied times-frac_binary64_11079.0

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

  11. Simplified9.0

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

  12. Simplified9.0

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

  13. Final simplification9.0

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

Reproduce

herbie shell --seed 2021015 
(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))))))