Average Error: 25.3 → 9.7
Time: 1.0m
Precision: 64
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \left(1 - \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{1}{2}\right) \cdot \left(\frac{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{M}{2} \cdot D\right)}{d} \cdot \frac{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{M}{2} \cdot D\right)}{d}\right)\right)\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\]
\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(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \left(1 - \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \frac{1}{2}\right) \cdot \left(\frac{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{M}{2} \cdot D\right)}{d} \cdot \frac{\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\frac{M}{2} \cdot D\right)}{d}\right)\right)\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)
double f(double d, double h, double l, double M, double D) {
        double r8985372 = d;
        double r8985373 = h;
        double r8985374 = r8985372 / r8985373;
        double r8985375 = 1.0;
        double r8985376 = 2.0;
        double r8985377 = r8985375 / r8985376;
        double r8985378 = pow(r8985374, r8985377);
        double r8985379 = l;
        double r8985380 = r8985372 / r8985379;
        double r8985381 = pow(r8985380, r8985377);
        double r8985382 = r8985378 * r8985381;
        double r8985383 = M;
        double r8985384 = D;
        double r8985385 = r8985383 * r8985384;
        double r8985386 = r8985376 * r8985372;
        double r8985387 = r8985385 / r8985386;
        double r8985388 = pow(r8985387, r8985376);
        double r8985389 = r8985377 * r8985388;
        double r8985390 = r8985373 / r8985379;
        double r8985391 = r8985389 * r8985390;
        double r8985392 = r8985375 - r8985391;
        double r8985393 = r8985382 * r8985392;
        return r8985393;
}
↓
double f(double d, double h, double l, double M, double D) {
        double r8985394 = d;
        double r8985395 = cbrt(r8985394);
        double r8985396 = l;
        double r8985397 = cbrt(r8985396);
        double r8985398 = r8985395 / r8985397;
        double r8985399 = fabs(r8985398);
        double r8985400 = sqrt(r8985398);
        double r8985401 = r8985399 * r8985400;
        double r8985402 = 1.0;
        double r8985403 = h;
        double r8985404 = cbrt(r8985403);
        double r8985405 = r8985404 / r8985397;
        double r8985406 = 0.5;
        double r8985407 = r8985405 * r8985406;
        double r8985408 = M;
        double r8985409 = 2.0;
        double r8985410 = r8985408 / r8985409;
        double r8985411 = D;
        double r8985412 = r8985410 * r8985411;
        double r8985413 = r8985405 * r8985412;
        double r8985414 = r8985413 / r8985394;
        double r8985415 = r8985414 * r8985414;
        double r8985416 = r8985407 * r8985415;
        double r8985417 = r8985402 - r8985416;
        double r8985418 = r8985401 * r8985417;
        double r8985419 = r8985395 / r8985404;
        double r8985420 = fabs(r8985419);
        double r8985421 = sqrt(r8985419);
        double r8985422 = r8985420 * r8985421;
        double r8985423 = r8985418 * r8985422;
        return r8985423;
}

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 25.3

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

  3. Applied add-cube-cbrt25.5

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

  4. Applied add-cube-cbrt25.6

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

  5. Applied times-frac25.6

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

  6. Applied unpow-prod-down20.9

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

  7. Simplified20.2

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

  8. Simplified20.2

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

  10. Applied add-cube-cbrt20.3

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

  11. Applied add-cube-cbrt20.3

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

  12. Applied times-frac20.3

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

  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. Simplified16.5

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

  16. Applied add-cube-cbrt16.7

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

  17. Applied add-cube-cbrt16.8

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

  18. Applied times-frac16.8

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

  19. Applied unpow-prod-down10.8

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

  20. Simplified10.8

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

  21. Simplified10.8

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

  23. Applied associate-*l*10.5

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

  24. Simplified9.7

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

  25. Final simplification9.7

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

Reproduce

herbie shell --seed 2019164 
(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)))))