Henrywood and Agarwal, Equation (12)

Average Error: 26.5 → 14.0

Time: 1.0m

Precision: 64

Math

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

↓

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

C

double f(double d, double h, double l, double M, double D) {
        double r7638064 = d;
        double r7638065 = h;
        double r7638066 = r7638064 / r7638065;
        double r7638067 = 1.0;
        double r7638068 = 2.0;
        double r7638069 = r7638067 / r7638068;
        double r7638070 = pow(r7638066, r7638069);
        double r7638071 = l;
        double r7638072 = r7638064 / r7638071;
        double r7638073 = pow(r7638072, r7638069);
        double r7638074 = r7638070 * r7638073;
        double r7638075 = M;
        double r7638076 = D;
        double r7638077 = r7638075 * r7638076;
        double r7638078 = r7638068 * r7638064;
        double r7638079 = r7638077 / r7638078;
        double r7638080 = pow(r7638079, r7638068);
        double r7638081 = r7638069 * r7638080;
        double r7638082 = r7638065 / r7638071;
        double r7638083 = r7638081 * r7638082;
        double r7638084 = r7638067 - r7638083;
        double r7638085 = r7638074 * r7638084;
        return r7638085;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r7638086 = d;
        double r7638087 = cbrt(r7638086);
        double r7638088 = r7638087 * r7638087;
        double r7638089 = 1.0;
        double r7638090 = 2.0;
        double r7638091 = r7638089 / r7638090;
        double r7638092 = pow(r7638088, r7638091);
        double r7638093 = cbrt(r7638088);
        double r7638094 = l;
        double r7638095 = cbrt(r7638094);
        double r7638096 = r7638095 * r7638095;
        double r7638097 = r7638093 / r7638096;
        double r7638098 = pow(r7638097, r7638091);
        double r7638099 = cbrt(r7638087);
        double r7638100 = r7638099 / r7638095;
        double r7638101 = pow(r7638100, r7638091);
        double r7638102 = r7638098 * r7638101;
        double r7638103 = r7638092 * r7638102;
        double r7638104 = h;
        double r7638105 = cbrt(r7638104);
        double r7638106 = r7638086 / r7638105;
        double r7638107 = pow(r7638106, r7638091);
        double r7638108 = 1.0;
        double r7638109 = r7638105 * r7638105;
        double r7638110 = r7638108 / r7638109;
        double r7638111 = pow(r7638110, r7638091);
        double r7638112 = r7638107 * r7638111;
        double r7638113 = r7638103 * r7638112;
        double r7638114 = r7638105 / r7638095;
        double r7638115 = D;
        double r7638116 = M;
        double r7638117 = r7638115 * r7638116;
        double r7638118 = r7638086 * r7638090;
        double r7638119 = r7638117 / r7638118;
        double r7638120 = pow(r7638119, r7638090);
        double r7638121 = r7638114 * r7638120;
        double r7638122 = r7638121 * r7638114;
        double r7638123 = r7638122 * r7638091;
        double r7638124 = r7638123 * r7638114;
        double r7638125 = r7638089 - r7638124;
        double r7638126 = r7638113 * r7638125;
        return r7638126;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

1. Initial program 26.5

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

3. Applied add-cube-cbrt 26.8

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

4. Applied *-un-lft-identity 26.8

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

5. Applied times-frac 26.8

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

6. Applied unpow-prod-down 21.8

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

8. Applied add-cube-cbrt 21.9

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

9. Applied add-cube-cbrt 21.9

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

10. Applied times-frac 21.9

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

11. Applied associate-*r* 20.1

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

12. Simplified 19.7

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

14. Applied *-un-lft-identity 19.7

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

15. Applied add-cube-cbrt 19.9

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

16. Applied times-frac 19.9

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

17. Applied unpow-prod-down 15.0

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

18. Simplified 15.0

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

20. Applied add-cube-cbrt 15.1

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

21. Applied add-cube-cbrt 15.1

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

22. Applied cbrt-prod 15.1

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

23. Applied times-frac 15.1

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

24. Applied unpow-prod-down 14.0

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

25. Final simplification 14.0

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

Reproduce

herbie shell --seed 2019172 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  (* (* (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)))))