Henrywood and Agarwal, Equation (12)

Average Error: 25.8 → 11.6

Time: 2.6m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;d \le -8.665730107906399 \cdot 10^{-36}:\\ \;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\ell}} \cdot \left|\sqrt[3]{d}\right|\right)\right) \cdot \mathsf{fma}\left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\left(D \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \frac{M}{d}\right) \cdot \left(\left(D \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \frac{M}{d}\right)\right)\right), \frac{-1}{8}, 1\right)\\ \mathbf{elif}\;d \le 5.562719586734957 \cdot 10^{-259}:\\ \;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\ell}} \cdot \left|\sqrt[3]{d}\right|\right)\right) \cdot \mathsf{fma}\left(\left(\left(\frac{M \cdot D}{d} \cdot \frac{h}{\ell}\right) \cdot \frac{M \cdot D}{d}\right), \frac{-1}{8}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\ell}} \cdot \left|\sqrt[3]{d}\right|\right)\right) \cdot \mathsf{fma}\left(\left(\left(\frac{M \cdot D}{d} \cdot \frac{\sqrt{h}}{\sqrt{\ell}}\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{\sqrt{h}}{\sqrt{\ell}}\right)\right), \frac{-1}{8}, 1\right)\\ \end{array}\]

C

double f(double d, double h, double l, double M, double D) {
        double r86859351 = d;
        double r86859352 = h;
        double r86859353 = r86859351 / r86859352;
        double r86859354 = 1.0;
        double r86859355 = 2.0;
        double r86859356 = r86859354 / r86859355;
        double r86859357 = pow(r86859353, r86859356);
        double r86859358 = l;
        double r86859359 = r86859351 / r86859358;
        double r86859360 = pow(r86859359, r86859356);
        double r86859361 = r86859357 * r86859360;
        double r86859362 = M;
        double r86859363 = D;
        double r86859364 = r86859362 * r86859363;
        double r86859365 = r86859355 * r86859351;
        double r86859366 = r86859364 / r86859365;
        double r86859367 = pow(r86859366, r86859355);
        double r86859368 = r86859356 * r86859367;
        double r86859369 = r86859352 / r86859358;
        double r86859370 = r86859368 * r86859369;
        double r86859371 = r86859354 - r86859370;
        double r86859372 = r86859361 * r86859371;
        return r86859372;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r86859373 = d;
        double r86859374 = -8.665730107906399e-36;
        bool r86859375 = r86859373 <= r86859374;
        double r86859376 = cbrt(r86859373);
        double r86859377 = h;
        double r86859378 = cbrt(r86859377);
        double r86859379 = r86859376 / r86859378;
        double r86859380 = fabs(r86859379);
        double r86859381 = sqrt(r86859379);
        double r86859382 = r86859380 * r86859381;
        double r86859383 = l;
        double r86859384 = r86859376 / r86859383;
        double r86859385 = sqrt(r86859384);
        double r86859386 = fabs(r86859376);
        double r86859387 = r86859385 * r86859386;
        double r86859388 = r86859382 * r86859387;
        double r86859389 = cbrt(r86859383);
        double r86859390 = r86859378 / r86859389;
        double r86859391 = D;
        double r86859392 = r86859391 * r86859390;
        double r86859393 = M;
        double r86859394 = r86859393 / r86859373;
        double r86859395 = r86859392 * r86859394;
        double r86859396 = r86859395 * r86859395;
        double r86859397 = r86859390 * r86859396;
        double r86859398 = -0.125;
        double r86859399 = 1.0;
        double r86859400 = fma(r86859397, r86859398, r86859399);
        double r86859401 = r86859388 * r86859400;
        double r86859402 = 5.562719586734957e-259;
        bool r86859403 = r86859373 <= r86859402;
        double r86859404 = r86859393 * r86859391;
        double r86859405 = r86859404 / r86859373;
        double r86859406 = r86859377 / r86859383;
        double r86859407 = r86859405 * r86859406;
        double r86859408 = r86859407 * r86859405;
        double r86859409 = fma(r86859408, r86859398, r86859399);
        double r86859410 = r86859388 * r86859409;
        double r86859411 = sqrt(r86859377);
        double r86859412 = sqrt(r86859383);
        double r86859413 = r86859411 / r86859412;
        double r86859414 = r86859405 * r86859413;
        double r86859415 = r86859414 * r86859414;
        double r86859416 = fma(r86859415, r86859398, r86859399);
        double r86859417 = r86859388 * r86859416;
        double r86859418 = r86859403 ? r86859410 : r86859417;
        double r86859419 = r86859375 ? r86859401 : r86859418;
        return r86859419;
}

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 d < -8.665730107906399e-36

   1. Initial program 20.3

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

   2. Simplified 20.0

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

   4. Applied add-cube-cbrt 20.3

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

   5. Applied add-cube-cbrt 20.5

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

   6. Applied times-frac 20.5

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

   7. Applied sqrt-prod 13.2

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

   8. Simplified 12.2

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

   10. Applied *-un-lft-identity 12.2

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

   11. Applied add-cube-cbrt 12.4

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

   12. Applied times-frac 12.4

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

   13. Applied sqrt-prod 10.1

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

   14. Simplified 10.1

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

   15. Taylor expanded around -inf 26.0

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

   16. Simplified 10.5

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

   18. Applied add-cube-cbrt 10.5

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

   19. Applied add-cube-cbrt 10.5

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

   20. Applied times-frac 10.5

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

   21. Applied associate-*r* 6.9

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

   22. Simplified 4.4

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

   if -8.665730107906399e-36 < d < 5.562719586734957e-259

   1. Initial program 33.2

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

   2. Simplified 34.1

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

   4. Applied add-cube-cbrt 34.3

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

   5. Applied add-cube-cbrt 34.3

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

   6. Applied times-frac 34.3

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

   7. Applied sqrt-prod 32.4

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

   8. Simplified 32.4

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

   10. Applied *-un-lft-identity 32.4

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

   11. Applied add-cube-cbrt 32.6

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

   12. Applied times-frac 32.6

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

   13. Applied sqrt-prod 28.7

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

   14. Simplified 28.7

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

   15. Taylor expanded around -inf 49.8

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

   16. Simplified 27.4

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

   18. Applied associate-*l* 24.1

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

   if 5.562719586734957e-259 < 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. Simplified 25.2

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

   4. Applied add-cube-cbrt 25.5

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

   5. Applied add-cube-cbrt 25.6

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

   6. Applied times-frac 25.6

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

   7. Applied sqrt-prod 19.8

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

   8. Simplified 19.0

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

   10. Applied *-un-lft-identity 19.0

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

   11. Applied add-cube-cbrt 19.2

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

   12. Applied times-frac 19.2

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

   13. Applied sqrt-prod 15.8

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

   14. Simplified 15.8

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

   15. Taylor expanded around -inf 32.7

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

   16. Simplified 15.9

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

   18. Applied add-sqr-sqrt 15.9

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

   19. Applied add-sqr-sqrt 15.9

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

   20. Applied times-frac 15.9

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

   21. Applied unswap-sqr 9.2

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

4. Final simplification 11.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;d \le -8.665730107906399 \cdot 10^{-36}:\\ \;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\ell}} \cdot \left|\sqrt[3]{d}\right|\right)\right) \cdot \mathsf{fma}\left(\left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\left(D \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \frac{M}{d}\right) \cdot \left(\left(D \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right) \cdot \frac{M}{d}\right)\right)\right), \frac{-1}{8}, 1\right)\\ \mathbf{elif}\;d \le 5.562719586734957 \cdot 10^{-259}:\\ \;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\ell}} \cdot \left|\sqrt[3]{d}\right|\right)\right) \cdot \mathsf{fma}\left(\left(\left(\frac{M \cdot D}{d} \cdot \frac{h}{\ell}\right) \cdot \frac{M \cdot D}{d}\right), \frac{-1}{8}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{\sqrt[3]{d}}{\ell}} \cdot \left|\sqrt[3]{d}\right|\right)\right) \cdot \mathsf{fma}\left(\left(\left(\frac{M \cdot D}{d} \cdot \frac{\sqrt{h}}{\sqrt{\ell}}\right) \cdot \left(\frac{M \cdot D}{d} \cdot \frac{\sqrt{h}}{\sqrt{\ell}}\right)\right), \frac{-1}{8}, 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 +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)))))