Henrywood and Agarwal, Equation (12)

Average Error: 25.6 → 15.2

Time: 1.2m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;M \cdot D \le 9.498414440691377 \cdot 10^{-42}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{1}{\sqrt[3]{h}}}{\sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right) \cdot \left(\left(1 - h \cdot \left(\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{\ell}\right)\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}}\right)\right)\\ \mathbf{elif}\;M \cdot D \le 3.662838660216526 \cdot 10^{+105}:\\ \;\;\;\;\left(\left(1 - \frac{\frac{1}{8} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}}\right)\right) \cdot \left(\sqrt{\frac{\frac{1}{\sqrt[3]{h}}}{\sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{1}{\sqrt[3]{h}}}{\sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right) \cdot \left(\left(1 - h \cdot \left(\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{\ell}\right)\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}}\right)\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r2986529 = d;
        double r2986530 = h;
        double r2986531 = r2986529 / r2986530;
        double r2986532 = 1.0;
        double r2986533 = 2.0;
        double r2986534 = r2986532 / r2986533;
        double r2986535 = pow(r2986531, r2986534);
        double r2986536 = l;
        double r2986537 = r2986529 / r2986536;
        double r2986538 = pow(r2986537, r2986534);
        double r2986539 = r2986535 * r2986538;
        double r2986540 = M;
        double r2986541 = D;
        double r2986542 = r2986540 * r2986541;
        double r2986543 = r2986533 * r2986529;
        double r2986544 = r2986542 / r2986543;
        double r2986545 = pow(r2986544, r2986533);
        double r2986546 = r2986534 * r2986545;
        double r2986547 = r2986530 / r2986536;
        double r2986548 = r2986546 * r2986547;
        double r2986549 = r2986532 - r2986548;
        double r2986550 = r2986539 * r2986549;
        return r2986550;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r2986551 = M;
        double r2986552 = D;
        double r2986553 = r2986551 * r2986552;
        double r2986554 = 9.498414440691377e-42;
        bool r2986555 = r2986553 <= r2986554;
        double r2986556 = 1.0;
        double r2986557 = h;
        double r2986558 = cbrt(r2986557);
        double r2986559 = r2986556 / r2986558;
        double r2986560 = r2986559 / r2986558;
        double r2986561 = sqrt(r2986560);
        double r2986562 = d;
        double r2986563 = r2986562 / r2986558;
        double r2986564 = sqrt(r2986563);
        double r2986565 = r2986561 * r2986564;
        double r2986566 = 2.0;
        double r2986567 = r2986551 / r2986566;
        double r2986568 = r2986552 / r2986562;
        double r2986569 = r2986567 * r2986568;
        double r2986570 = r2986569 * r2986569;
        double r2986571 = 0.5;
        double r2986572 = r2986570 * r2986571;
        double r2986573 = l;
        double r2986574 = r2986556 / r2986573;
        double r2986575 = r2986572 * r2986574;
        double r2986576 = r2986557 * r2986575;
        double r2986577 = r2986556 - r2986576;
        double r2986578 = cbrt(r2986562);
        double r2986579 = fabs(r2986578);
        double r2986580 = r2986578 / r2986573;
        double r2986581 = sqrt(r2986580);
        double r2986582 = r2986579 * r2986581;
        double r2986583 = r2986577 * r2986582;
        double r2986584 = r2986565 * r2986583;
        double r2986585 = 3.662838660216526e+105;
        bool r2986586 = r2986553 <= r2986585;
        double r2986587 = 0.125;
        double r2986588 = r2986553 * r2986553;
        double r2986589 = r2986588 * r2986557;
        double r2986590 = r2986587 * r2986589;
        double r2986591 = r2986562 * r2986562;
        double r2986592 = r2986591 * r2986573;
        double r2986593 = r2986590 / r2986592;
        double r2986594 = r2986556 - r2986593;
        double r2986595 = r2986594 * r2986582;
        double r2986596 = r2986595 * r2986565;
        double r2986597 = r2986586 ? r2986596 : r2986584;
        double r2986598 = r2986555 ? r2986584 : r2986597;
        return r2986598;
}

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 (* M D) < 9.498414440691377e-42 or 3.662838660216526e+105 < (* M D)

   1. Initial program 25.1

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

   3. Applied *-un-lft-identity 25.1

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

   4. Applied add-cube-cbrt 25.3

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

   5. Applied times-frac 25.3

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

   6. Applied unpow-prod-down 21.6

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

   7. Simplified 21.6

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

   8. Simplified 21.6

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

   10. Applied add-cube-cbrt 21.7

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

   11. Applied *-un-lft-identity 21.7

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

   12. Applied times-frac 21.7

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

   13. Applied unpow-prod-down 17.1

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

   14. Simplified 17.0

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

   15. Simplified 17.0

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

   17. Applied associate-*l* 16.7

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

   18. Simplified 16.9

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

   20. Applied div-inv 16.9

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

   21. Applied associate-*l* 14.2

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

   if 9.498414440691377e-42 < (* M D) < 3.662838660216526e+105

   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. Using strategy rm

   3. Applied *-un-lft-identity 29.8

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

   4. Applied add-cube-cbrt 30.0

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

   5. Applied times-frac 30.0

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

   6. Applied unpow-prod-down 27.4

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

   7. Simplified 27.4

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

   8. Simplified 27.4

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

   10. Applied add-cube-cbrt 27.5

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

   11. Applied *-un-lft-identity 27.5

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

   12. Applied times-frac 27.5

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

   13. Applied unpow-prod-down 22.8

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

   14. Simplified 22.8

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

   15. Simplified 22.8

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

   17. Applied associate-*l* 22.5

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

   18. Simplified 24.3

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

   19. Taylor expanded around inf 44.9

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

   20. Simplified 22.8

       \[\leadsto \left(\sqrt{\frac{\frac{1}{\sqrt[3]{h}}}{\sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right) \cdot \left(\left(\left|\sqrt[3]{d}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}}\right) \cdot \left(1 - \color{blue}{\frac{\frac{1}{8} \cdot \left(\left(\left(D \cdot M\right) \cdot \left(D \cdot M\right)\right) \cdot h\right)}{\ell \cdot \left(d \cdot d\right)}}\right)\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 15.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;M \cdot D \le 9.498414440691377 \cdot 10^{-42}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{1}{\sqrt[3]{h}}}{\sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right) \cdot \left(\left(1 - h \cdot \left(\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{\ell}\right)\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}}\right)\right)\\ \mathbf{elif}\;M \cdot D \le 3.662838660216526 \cdot 10^{+105}:\\ \;\;\;\;\left(\left(1 - \frac{\frac{1}{8} \cdot \left(\left(\left(M \cdot D\right) \cdot \left(M \cdot D\right)\right) \cdot h\right)}{\left(d \cdot d\right) \cdot \ell}\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}}\right)\right) \cdot \left(\sqrt{\frac{\frac{1}{\sqrt[3]{h}}}{\sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{1}{\sqrt[3]{h}}}{\sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right) \cdot \left(\left(1 - h \cdot \left(\left(\left(\left(\frac{M}{2} \cdot \frac{D}{d}\right) \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{\ell}\right)\right) \cdot \left(\left|\sqrt[3]{d}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(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)))))