Henrywood and Agarwal, Equation (12)

Average Error: 27.0 → 13.2

Time: 1.0m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \le -1.332074362493885807282809284139451105714 \cdot 10^{-71}:\\ \;\;\;\;\left(\left(1 - \frac{1}{2} \cdot \left({\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\frac{\ell}{h}}\right)\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(h \cdot \left(\frac{\frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}} \cdot \frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}}\right)\right)\right) \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r224226 = d;
        double r224227 = h;
        double r224228 = r224226 / r224227;
        double r224229 = 1.0;
        double r224230 = 2.0;
        double r224231 = r224229 / r224230;
        double r224232 = pow(r224228, r224231);
        double r224233 = l;
        double r224234 = r224226 / r224233;
        double r224235 = pow(r224234, r224231);
        double r224236 = r224232 * r224235;
        double r224237 = M;
        double r224238 = D;
        double r224239 = r224237 * r224238;
        double r224240 = r224230 * r224226;
        double r224241 = r224239 / r224240;
        double r224242 = pow(r224241, r224230);
        double r224243 = r224231 * r224242;
        double r224244 = r224227 / r224233;
        double r224245 = r224243 * r224244;
        double r224246 = r224229 - r224245;
        double r224247 = r224236 * r224246;
        return r224247;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r224248 = 1.0;
        double r224249 = h;
        double r224250 = l;
        double r224251 = r224249 / r224250;
        double r224252 = M;
        double r224253 = D;
        double r224254 = r224252 * r224253;
        double r224255 = 2.0;
        double r224256 = d;
        double r224257 = r224255 * r224256;
        double r224258 = r224254 / r224257;
        double r224259 = pow(r224258, r224255);
        double r224260 = r224248 / r224255;
        double r224261 = r224259 * r224260;
        double r224262 = r224251 * r224261;
        double r224263 = r224248 - r224262;
        double r224264 = r224256 / r224250;
        double r224265 = pow(r224264, r224260);
        double r224266 = r224256 / r224249;
        double r224267 = pow(r224266, r224260);
        double r224268 = r224265 * r224267;
        double r224269 = r224263 * r224268;
        double r224270 = -1.3320743624938858e-71;
        bool r224271 = r224269 <= r224270;
        double r224272 = r224256 / r224253;
        double r224273 = r224255 * r224272;
        double r224274 = r224252 / r224273;
        double r224275 = 2.0;
        double r224276 = r224255 / r224275;
        double r224277 = pow(r224274, r224276);
        double r224278 = r224252 / r224257;
        double r224279 = r224253 * r224278;
        double r224280 = pow(r224279, r224276);
        double r224281 = r224250 / r224249;
        double r224282 = r224280 / r224281;
        double r224283 = r224277 * r224282;
        double r224284 = r224260 * r224283;
        double r224285 = r224248 - r224284;
        double r224286 = r224285 * r224265;
        double r224287 = r224286 * r224267;
        double r224288 = cbrt(r224256);
        double r224289 = cbrt(r224249);
        double r224290 = r224288 / r224289;
        double r224291 = pow(r224290, r224260);
        double r224292 = r224288 * r224288;
        double r224293 = r224289 * r224289;
        double r224294 = r224292 / r224293;
        double r224295 = pow(r224294, r224260);
        double r224296 = r224291 * r224295;
        double r224297 = cbrt(r224250);
        double r224298 = r224280 / r224297;
        double r224299 = r224298 / r224297;
        double r224300 = r224299 * r224298;
        double r224301 = r224249 * r224300;
        double r224302 = r224260 * r224301;
        double r224303 = r224248 - r224302;
        double r224304 = r224297 * r224297;
        double r224305 = r224292 / r224304;
        double r224306 = pow(r224305, r224260);
        double r224307 = r224288 / r224297;
        double r224308 = pow(r224307, r224260);
        double r224309 = r224306 * r224308;
        double r224310 = r224303 * r224309;
        double r224311 = r224296 * r224310;
        double r224312 = r224271 ? r224287 : r224311;
        return r224312;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

1. Split input into 2 regimes

2. if (* (* (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)))) < -1.3320743624938858e-71

   1. Initial program 29.8

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

   2. Simplified 31.9

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

   4. Applied *-un-lft-identity 31.9

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

   5. Applied sqr-pow 31.9

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

   6. Applied times-frac 27.0

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

   7. Applied associate-*l* 21.7

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

   8. Simplified 26.5

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

   if -1.3320743624938858e-71 < (* (* (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))))

   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. Simplified 25.4

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

   4. Applied add-cube-cbrt 25.7

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

   5. Applied add-cube-cbrt 25.8

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

   6. Applied times-frac 25.8

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

   7. Applied unpow-prod-down 19.0

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

   9. Applied add-cube-cbrt 19.0

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

   10. Applied add-cube-cbrt 19.2

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

   11. Applied times-frac 19.2

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

   12. Applied unpow-prod-down 12.3

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

   14. Applied add-cube-cbrt 12.4

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

   15. Applied sqr-pow 12.4

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

   16. Applied times-frac 10.3

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

   17. Simplified 11.4

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

   18. Simplified 10.5

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

4. Final simplification 13.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - \frac{h}{\ell} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{1}{2}\right)\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\right) \le -1.332074362493885807282809284139451105714 \cdot 10^{-71}:\\ \;\;\;\;\left(\left(1 - \frac{1}{2} \cdot \left({\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\frac{\ell}{h}}\right)\right) \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot {\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{h} \cdot \sqrt[3]{h}}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(\left(1 - \frac{1}{2} \cdot \left(h \cdot \left(\frac{\frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}} \cdot \frac{{\left(D \cdot \frac{M}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}}\right)\right)\right) \cdot \left({\left(\frac{\sqrt[3]{d} \cdot \sqrt[3]{d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\left(\frac{1}{2}\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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)))))