Average Error: 25.8 → 10.2
Time: 5.6m
Precision: 64
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{M}{2} \cdot \left(\frac{D}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right) \cdot \left(\frac{M}{2} \cdot \left(\frac{D}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right)}{2} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right)\]
\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}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \left(1 - \frac{\left(\frac{M}{2} \cdot \left(\frac{D}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right) \cdot \left(\frac{M}{2} \cdot \left(\frac{D}{d} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right)}{2} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right)
double f(double d, double h, double l, double M, double D) {
        double r92841485 = d;
        double r92841486 = h;
        double r92841487 = r92841485 / r92841486;
        double r92841488 = 1.0;
        double r92841489 = 2.0;
        double r92841490 = r92841488 / r92841489;
        double r92841491 = pow(r92841487, r92841490);
        double r92841492 = l;
        double r92841493 = r92841485 / r92841492;
        double r92841494 = pow(r92841493, r92841490);
        double r92841495 = r92841491 * r92841494;
        double r92841496 = M;
        double r92841497 = D;
        double r92841498 = r92841496 * r92841497;
        double r92841499 = r92841489 * r92841485;
        double r92841500 = r92841498 / r92841499;
        double r92841501 = pow(r92841500, r92841489);
        double r92841502 = r92841490 * r92841501;
        double r92841503 = r92841486 / r92841492;
        double r92841504 = r92841502 * r92841503;
        double r92841505 = r92841488 - r92841504;
        double r92841506 = r92841495 * r92841505;
        return r92841506;
}
↓
double f(double d, double h, double l, double M, double D) {
        double r92841507 = d;
        double r92841508 = cbrt(r92841507);
        double r92841509 = h;
        double r92841510 = cbrt(r92841509);
        double r92841511 = r92841508 / r92841510;
        double r92841512 = fabs(r92841511);
        double r92841513 = sqrt(r92841511);
        double r92841514 = r92841512 * r92841513;
        double r92841515 = l;
        double r92841516 = cbrt(r92841515);
        double r92841517 = r92841508 / r92841516;
        double r92841518 = fabs(r92841517);
        double r92841519 = sqrt(r92841517);
        double r92841520 = r92841518 * r92841519;
        double r92841521 = 1.0;
        double r92841522 = M;
        double r92841523 = 2.0;
        double r92841524 = r92841522 / r92841523;
        double r92841525 = D;
        double r92841526 = r92841525 / r92841507;
        double r92841527 = r92841510 / r92841516;
        double r92841528 = r92841526 * r92841527;
        double r92841529 = r92841524 * r92841528;
        double r92841530 = r92841529 * r92841529;
        double r92841531 = r92841530 / r92841523;
        double r92841532 = r92841531 * r92841527;
        double r92841533 = r92841521 - r92841532;
        double r92841534 = r92841520 * r92841533;
        double r92841535 = r92841514 * r92841534;
        return r92841535;
}

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 25.8

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

  3. Applied add-cube-cbrt26.0

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

  4. Applied add-cube-cbrt26.1

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

  5. Applied times-frac26.1

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

  6. Applied unpow-prod-down21.0

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

  7. Simplified20.4

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

  8. Simplified20.4

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

  10. Applied add-cube-cbrt20.4

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

  11. Applied add-cube-cbrt20.4

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

  12. Applied times-frac20.4

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

  13. Applied associate-*r*18.6

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

  14. Simplified16.4

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

  16. Applied add-cube-cbrt16.6

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

  17. Applied add-cube-cbrt16.7

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

  18. Applied times-frac16.7

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

  19. Applied unpow-prod-down11.2

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

  20. Simplified10.6

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

  21. Simplified10.6

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

  23. Applied associate-*l*10.2

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

  24. Final simplification10.2

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

Reproduce

herbie shell --seed 2019125 
(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)))))