Average Error: 26.2 → 13.1
Time: 18.4s
Precision: binary64
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + h \cdot \left(\frac{1}{2} \cdot \left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{-1}{\ell}\right)\right)\right)\right)\right)\]
\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)
↓
\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}} \cdot \frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left({\left(\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 + h \cdot \left(\frac{1}{2} \cdot \left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(M \cdot \frac{D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{-1}{\ell}\right)\right)\right)\right)\right)
double code(double d, double h, double l, double M, double D) {
	return ((double) (((double) (((double) pow(((double) (d / h)), ((double) (1.0 / 2.0)))) * ((double) pow(((double) (d / l)), ((double) (1.0 / 2.0)))))) * ((double) (1.0 - ((double) (((double) (((double) (1.0 / 2.0)) * ((double) pow(((double) (((double) (M * D)) / ((double) (2.0 * d)))), 2.0)))) * ((double) (h / l))))))));
}
↓
double code(double d, double h, double l, double M, double D) {
	return ((double) (((double) (((double) pow(((double) (((double) (((double) cbrt(d)) / ((double) cbrt(h)))) * ((double) (((double) cbrt(d)) / ((double) cbrt(h)))))), ((double) (1.0 / 2.0)))) * ((double) pow(((double) (((double) cbrt(d)) / ((double) cbrt(h)))), ((double) (1.0 / 2.0)))))) * ((double) (((double) (((double) pow(((double) (1.0 / ((double) (((double) cbrt(l)) * ((double) cbrt(l)))))), ((double) (1.0 / 2.0)))) * ((double) pow(((double) (d / ((double) cbrt(l)))), ((double) (1.0 / 2.0)))))) * ((double) (1.0 + ((double) (h * ((double) (((double) (1.0 / 2.0)) * ((double) (((double) pow(((double) (M * ((double) (D / ((double) (d * 2.0)))))), ((double) (2.0 / 2.0)))) * ((double) (((double) pow(((double) (M * ((double) (D / ((double) (d * 2.0)))))), ((double) (2.0 / 2.0)))) * ((double) (-1.0 / l))))))))))))))));
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 26.2

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

  2. Simplified25.7

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

  4. Applied add-cube-cbrt26.0

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

  5. Applied add-cube-cbrt26.1

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

  6. Applied times-frac26.1

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

  7. Applied unpow-prod-down20.2

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

  8. Simplified20.2

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

  10. Applied add-cube-cbrt20.3

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

  11. Applied *-un-lft-identity20.3

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

  12. Applied times-frac20.3

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

  13. Applied unpow-prod-down15.3

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

  15. Applied *-un-lft-identity15.3

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

  16. Applied sqr-pow15.3

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

  17. Applied times-frac13.1

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

  18. Simplified13.1

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

  19. Simplified13.1

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

  21. Applied div-inv13.1

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

  22. Final simplification13.1

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

Reproduce

herbie shell --seed 2020185 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1.0 2.0)) (pow (/ d l) (/ 1.0 2.0))) (- 1.0 (* (* (/ 1.0 2.0) (pow (/ (* M D) (* 2.0 d)) 2.0)) (/ h l)))))