Average Error: 14.1 → 9.3
Time: 13.2s
Precision: binary64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{d \cdot 2} \cdot \sqrt[3]{h}\right) \cdot \left(\left(\frac{M \cdot D}{d \cdot 2} \cdot \sqrt[3]{h}\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{d \cdot 2} \cdot \sqrt[3]{h}\right) \cdot \left(\left(\frac{M \cdot D}{d \cdot 2} \cdot \sqrt[3]{h}\right) \cdot \frac{\sqrt[3]{h}}{\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
    (*
     (* (/ (* M D) (* d 2.0)) (cbrt h))
     (* (* (/ (* M D) (* d 2.0)) (cbrt h)) (/ (cbrt h) 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 - ((((M * D) / (d * 2.0)) * cbrt(h)) * ((((M * D) / (d * 2.0)) * cbrt(h)) * (cbrt(h) / 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 14.1

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

  3. Applied *-un-lft-identity_binary6414.1

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

  4. Applied add-cube-cbrt_binary6414.2

    \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}{1 \cdot \ell}}\]

  5. Applied times-frac_binary6414.1

    \[\leadsto w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{\left(\frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{1} \cdot \frac{\sqrt[3]{h}}{\ell}\right)}}\]

  6. Applied associate-*r*_binary6411.5

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{1}\right) \cdot \frac{\sqrt[3]{h}}{\ell}}}\]

  7. Simplified11.5

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)\right)} \cdot \frac{\sqrt[3]{h}}{\ell}}\]
  8. Using strategy rm

  9. Applied pow2_binary6411.5

    \[\leadsto w0 \cdot \sqrt{1 - \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \color{blue}{{\left(\sqrt[3]{h}\right)}^{2}}\right) \cdot \frac{\sqrt[3]{h}}{\ell}}\]

  10. Applied pow-prod-down_binary6410.4

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{{\left(\frac{M \cdot D}{2 \cdot d} \cdot \sqrt[3]{h}\right)}^{2}} \cdot \frac{\sqrt[3]{h}}{\ell}}\]

  11. Simplified10.4

    \[\leadsto w0 \cdot \sqrt{1 - {\color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \sqrt[3]{h}\right)}}^{2} \cdot \frac{\sqrt[3]{h}}{\ell}}\]
  12. Using strategy rm

  13. Applied unpow2_binary6410.4

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\left(\frac{M \cdot D}{d \cdot 2} \cdot \sqrt[3]{h}\right) \cdot \left(\frac{M \cdot D}{d \cdot 2} \cdot \sqrt[3]{h}\right)\right)} \cdot \frac{\sqrt[3]{h}}{\ell}}\]

  14. Applied associate-*l*_binary649.3

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\left(\frac{M \cdot D}{d \cdot 2} \cdot \sqrt[3]{h}\right) \cdot \left(\left(\frac{M \cdot D}{d \cdot 2} \cdot \sqrt[3]{h}\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)}}\]

  15. Final simplification9.3

    \[\leadsto w0 \cdot \sqrt{1 - \left(\frac{M \cdot D}{d \cdot 2} \cdot \sqrt[3]{h}\right) \cdot \left(\left(\frac{M \cdot D}{d \cdot 2} \cdot \sqrt[3]{h}\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)}\]

Reproduce

herbie shell --seed 2021149 
(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))))))