Henrywood and Agarwal, Equation (12)

Average Error: 25.9 → 11.5

Time: 58.3s

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)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le -2.452445300667083 \cdot 10^{+80}:\\ \;\;\;\;\left(1 - \left(\left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)\\ \mathbf{elif}\;\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)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le 0.0:\\ \;\;\;\;\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(\sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}} \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\sqrt[3]{\ell}}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\sqrt[3]{\ell}}}}\right)\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right) \cdot \left(\sqrt{\frac{d}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\sqrt[3]{\ell}}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\sqrt[3]{\ell}}}\right) \cdot \left(\frac{\frac{M}{\frac{2 \cdot d}{D}}}{\sqrt[3]{\ell} \cdot 2} \cdot \frac{\frac{M}{\frac{2 \cdot d}{D}}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\sqrt[3]{\ell}}}\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r7893192 = d;
        double r7893193 = h;
        double r7893194 = r7893192 / r7893193;
        double r7893195 = 1.0;
        double r7893196 = 2.0;
        double r7893197 = r7893195 / r7893196;
        double r7893198 = pow(r7893194, r7893197);
        double r7893199 = l;
        double r7893200 = r7893192 / r7893199;
        double r7893201 = pow(r7893200, r7893197);
        double r7893202 = r7893198 * r7893201;
        double r7893203 = M;
        double r7893204 = D;
        double r7893205 = r7893203 * r7893204;
        double r7893206 = r7893196 * r7893192;
        double r7893207 = r7893205 / r7893206;
        double r7893208 = pow(r7893207, r7893196);
        double r7893209 = r7893197 * r7893208;
        double r7893210 = r7893193 / r7893199;
        double r7893211 = r7893209 * r7893210;
        double r7893212 = r7893195 - r7893211;
        double r7893213 = r7893202 * r7893212;
        return r7893213;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r7893214 = 1.0;
        double r7893215 = h;
        double r7893216 = l;
        double r7893217 = r7893215 / r7893216;
        double r7893218 = M;
        double r7893219 = D;
        double r7893220 = r7893218 * r7893219;
        double r7893221 = 2.0;
        double r7893222 = d;
        double r7893223 = r7893221 * r7893222;
        double r7893224 = r7893220 / r7893223;
        double r7893225 = pow(r7893224, r7893221);
        double r7893226 = 0.5;
        double r7893227 = r7893225 * r7893226;
        double r7893228 = r7893217 * r7893227;
        double r7893229 = r7893214 - r7893228;
        double r7893230 = r7893222 / r7893216;
        double r7893231 = pow(r7893230, r7893226);
        double r7893232 = r7893222 / r7893215;
        double r7893233 = pow(r7893232, r7893226);
        double r7893234 = r7893231 * r7893233;
        double r7893235 = r7893229 * r7893234;
        double r7893236 = -2.452445300667083e+80;
        bool r7893237 = r7893235 <= r7893236;
        double r7893238 = cbrt(r7893215);
        double r7893239 = cbrt(r7893216);
        double r7893240 = r7893238 / r7893239;
        double r7893241 = r7893218 / r7893222;
        double r7893242 = r7893219 / r7893221;
        double r7893243 = r7893241 * r7893242;
        double r7893244 = r7893240 * r7893243;
        double r7893245 = r7893244 * r7893244;
        double r7893246 = r7893245 * r7893226;
        double r7893247 = r7893246 * r7893240;
        double r7893248 = r7893214 - r7893247;
        double r7893249 = r7893248 * r7893234;
        double r7893250 = 0.0;
        bool r7893251 = r7893235 <= r7893250;
        double r7893252 = r7893214 / r7893239;
        double r7893253 = r7893252 / r7893239;
        double r7893254 = sqrt(r7893253);
        double r7893255 = cbrt(r7893222);
        double r7893256 = cbrt(r7893239);
        double r7893257 = r7893255 / r7893256;
        double r7893258 = fabs(r7893257);
        double r7893259 = sqrt(r7893257);
        double r7893260 = r7893258 * r7893259;
        double r7893261 = r7893254 * r7893260;
        double r7893262 = r7893255 / r7893238;
        double r7893263 = sqrt(r7893262);
        double r7893264 = fabs(r7893262);
        double r7893265 = r7893263 * r7893264;
        double r7893266 = r7893261 * r7893265;
        double r7893267 = r7893229 * r7893266;
        double r7893268 = r7893222 / r7893239;
        double r7893269 = sqrt(r7893268);
        double r7893270 = r7893269 * r7893254;
        double r7893271 = r7893265 * r7893270;
        double r7893272 = r7893238 / r7893256;
        double r7893273 = r7893272 * r7893272;
        double r7893274 = r7893223 / r7893219;
        double r7893275 = r7893218 / r7893274;
        double r7893276 = r7893239 * r7893221;
        double r7893277 = r7893275 / r7893276;
        double r7893278 = r7893275 / r7893239;
        double r7893279 = r7893277 * r7893278;
        double r7893280 = r7893273 * r7893279;
        double r7893281 = r7893280 * r7893272;
        double r7893282 = r7893214 - r7893281;
        double r7893283 = r7893271 * r7893282;
        double r7893284 = r7893251 ? r7893267 : r7893283;
        double r7893285 = r7893237 ? r7893249 : r7893284;
        return r7893285;
}

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 3 regimes

2. if (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))) < -2.452445300667083e+80

   1. Initial program 35.9

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

   3. Applied add-cube-cbrt 36.2

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

   4. Applied add-cube-cbrt 36.2

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

   5. Applied times-frac 36.2

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

   6. Applied associate-*r* 32.3

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

   7. Simplified 24.5

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

   if -2.452445300667083e+80 < (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))) < 0.0

   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 26.5

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

   4. Applied add-cube-cbrt 26.6

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

   5. Applied times-frac 26.6

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

   6. Applied unpow-prod-down 21.5

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

   7. Simplified 21.5

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

   8. Simplified 21.5

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

   10. Applied add-cube-cbrt 21.6

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

   11. Applied *-un-lft-identity 21.6

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

   12. Applied times-frac 21.6

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

   13. Applied unpow-prod-down 7.7

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

   14. Simplified 7.7

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

   15. Simplified 7.7

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

   17. Applied add-cube-cbrt 7.8

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

   18. Applied add-cube-cbrt 7.9

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

   19. Applied times-frac 7.9

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

   20. Applied sqrt-prod 5.2

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

   21. Simplified 5.2

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

   if 0.0 < (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l))))

   1. Initial program 24.0

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

   3. Applied add-cube-cbrt 24.4

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

   4. Applied add-cube-cbrt 24.5

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

   5. Applied times-frac 24.5

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

   6. Applied unpow-prod-down 18.8

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

   7. Simplified 17.8

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

   8. Simplified 17.8

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

   10. Applied add-cube-cbrt 17.9

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

   11. Applied *-un-lft-identity 17.9

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

   12. Applied times-frac 17.9

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

   13. Applied unpow-prod-down 15.9

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

   14. Simplified 15.9

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

   15. Simplified 15.9

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

   17. Applied add-cube-cbrt 15.9

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

   18. Applied *-un-lft-identity 15.9

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

   19. Applied times-frac 15.9

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

   20. Applied associate-*r* 13.6

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

   21. Simplified 13.4

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

   23. Applied add-cube-cbrt 13.4

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

   24. Applied add-cube-cbrt 13.4

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

   25. Applied times-frac 13.4

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

   26. Applied associate-*r* 12.2

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

   27. Simplified 10.4

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

4. Final simplification 11.5

   \[\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)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le -2.452445300667083 \cdot 10^{+80}:\\ \;\;\;\;\left(1 - \left(\left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{M}{d} \cdot \frac{D}{2}\right)\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \left({\left(\frac{d}{\ell}\right)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right)\\ \mathbf{elif}\;\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)}^{\frac{1}{2}} \cdot {\left(\frac{d}{h}\right)}^{\frac{1}{2}}\right) \le 0.0:\\ \;\;\;\;\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(\sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}} \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\sqrt[3]{\ell}}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\sqrt[3]{\ell}}}}\right)\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right) \cdot \left(\sqrt{\frac{d}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{\frac{1}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}}\right)\right) \cdot \left(1 - \left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\sqrt[3]{\ell}}} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\sqrt[3]{\ell}}}\right) \cdot \left(\frac{\frac{M}{\frac{2 \cdot d}{D}}}{\sqrt[3]{\ell} \cdot 2} \cdot \frac{\frac{M}{\frac{2 \cdot d}{D}}}{\sqrt[3]{\ell}}\right)\right) \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\sqrt[3]{\ell}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))))