Average Error: 26.7 → 10.0
Time: 4.7m
Precision: 64
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\left(1 - \left(h \cdot \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{2}}}{\sqrt[3]{\ell}}\right) \cdot \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{2}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{2}}\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\frac{1}{2}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right|\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\frac{1}{2}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right)\right)\]
\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(1 - \left(h \cdot \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{2}}}{\sqrt[3]{\ell}}\right) \cdot \frac{\frac{\frac{M \cdot D}{2 \cdot d}}{\sqrt[3]{\ell} \cdot \sqrt[3]{2}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{2}}\right) \cdot \left(\left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right)}^{\frac{1}{2}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right|\right) \cdot \left({\left(\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right)}^{\frac{1}{2}} \cdot \left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right|\right)\right)
double f(double d, double h, double l, double M, double D) {
        double r53754231 = d;
        double r53754232 = h;
        double r53754233 = r53754231 / r53754232;
        double r53754234 = 1.0;
        double r53754235 = 2.0;
        double r53754236 = r53754234 / r53754235;
        double r53754237 = pow(r53754233, r53754236);
        double r53754238 = l;
        double r53754239 = r53754231 / r53754238;
        double r53754240 = pow(r53754239, r53754236);
        double r53754241 = r53754237 * r53754240;
        double r53754242 = M;
        double r53754243 = D;
        double r53754244 = r53754242 * r53754243;
        double r53754245 = r53754235 * r53754231;
        double r53754246 = r53754244 / r53754245;
        double r53754247 = pow(r53754246, r53754235);
        double r53754248 = r53754236 * r53754247;
        double r53754249 = r53754232 / r53754238;
        double r53754250 = r53754248 * r53754249;
        double r53754251 = r53754234 - r53754250;
        double r53754252 = r53754241 * r53754251;
        return r53754252;
}
↓
double f(double d, double h, double l, double M, double D) {
        double r53754253 = 1.0;
        double r53754254 = h;
        double r53754255 = M;
        double r53754256 = D;
        double r53754257 = r53754255 * r53754256;
        double r53754258 = 2.0;
        double r53754259 = d;
        double r53754260 = r53754258 * r53754259;
        double r53754261 = r53754257 / r53754260;
        double r53754262 = cbrt(r53754258);
        double r53754263 = r53754261 / r53754262;
        double r53754264 = l;
        double r53754265 = cbrt(r53754264);
        double r53754266 = r53754263 / r53754265;
        double r53754267 = r53754254 * r53754266;
        double r53754268 = r53754265 * r53754262;
        double r53754269 = r53754261 / r53754268;
        double r53754270 = r53754269 / r53754268;
        double r53754271 = r53754267 * r53754270;
        double r53754272 = r53754253 - r53754271;
        double r53754273 = cbrt(r53754259);
        double r53754274 = r53754273 / r53754265;
        double r53754275 = 0.5;
        double r53754276 = pow(r53754274, r53754275);
        double r53754277 = fabs(r53754274);
        double r53754278 = r53754276 * r53754277;
        double r53754279 = cbrt(r53754254);
        double r53754280 = r53754273 / r53754279;
        double r53754281 = pow(r53754280, r53754275);
        double r53754282 = fabs(r53754280);
        double r53754283 = r53754281 * r53754282;
        double r53754284 = r53754278 * r53754283;
        double r53754285 = r53754272 * r53754284;
        return r53754285;
}

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 26.7

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

  3. Applied add-cube-cbrt26.9

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

  4. Applied add-cube-cbrt27.0

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

  5. Applied times-frac27.0

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

  6. Applied unpow-prod-down21.9

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

  7. Simplified21.2

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

  9. Applied add-cube-cbrt21.3

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

  10. Applied add-cube-cbrt21.5

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

  11. Applied times-frac21.4

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

  12. Applied unpow-prod-down17.2

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

  13. Simplified17.0

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

  15. Applied pow117.0

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

  16. Applied pow117.0

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

  17. Applied pow-prod-down17.0

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

  18. Simplified13.6

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

  20. Applied add-cube-cbrt13.6

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

  21. Applied add-cube-cbrt13.6

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

  22. Applied times-frac13.6

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

  23. Applied times-frac11.5

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

  24. Applied associate-*l*10.0

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

  25. Simplified10.0

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

  26. Final simplification10.0

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

Reproduce

herbie shell --seed 2019120 
(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)))))