Average Error: 25.9 → 12.9
Time: 1.9m
Precision: 64
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[(\left(\frac{\left(\frac{M \cdot D}{d} \cdot h\right) \cdot \frac{M \cdot D}{d}}{\ell}\right) \cdot \frac{-1}{8} + 1)_* \cdot \left(\left(\left|\sqrt[3]{d}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\right)\]
double f(double d, double h, double l, double M, double D) {
        double r47923788 = d;
        double r47923789 = h;
        double r47923790 = r47923788 / r47923789;
        double r47923791 = 1.0;
        double r47923792 = 2.0;
        double r47923793 = r47923791 / r47923792;
        double r47923794 = pow(r47923790, r47923793);
        double r47923795 = l;
        double r47923796 = r47923788 / r47923795;
        double r47923797 = pow(r47923796, r47923793);
        double r47923798 = r47923794 * r47923797;
        double r47923799 = M;
        double r47923800 = D;
        double r47923801 = r47923799 * r47923800;
        double r47923802 = r47923792 * r47923788;
        double r47923803 = r47923801 / r47923802;
        double r47923804 = pow(r47923803, r47923792);
        double r47923805 = r47923793 * r47923804;
        double r47923806 = r47923789 / r47923795;
        double r47923807 = r47923805 * r47923806;
        double r47923808 = r47923791 - r47923807;
        double r47923809 = r47923798 * r47923808;
        return r47923809;
}
↓
double f(double d, double h, double l, double M, double D) {
        double r47923810 = M;
        double r47923811 = D;
        double r47923812 = r47923810 * r47923811;
        double r47923813 = d;
        double r47923814 = r47923812 / r47923813;
        double r47923815 = h;
        double r47923816 = r47923814 * r47923815;
        double r47923817 = r47923816 * r47923814;
        double r47923818 = l;
        double r47923819 = r47923817 / r47923818;
        double r47923820 = -0.125;
        double r47923821 = 1.0;
        double r47923822 = fma(r47923819, r47923820, r47923821);
        double r47923823 = cbrt(r47923813);
        double r47923824 = fabs(r47923823);
        double r47923825 = r47923823 / r47923818;
        double r47923826 = sqrt(r47923825);
        double r47923827 = r47923824 * r47923826;
        double r47923828 = cbrt(r47923815);
        double r47923829 = r47923823 / r47923828;
        double r47923830 = fabs(r47923829);
        double r47923831 = sqrt(r47923829);
        double r47923832 = r47923830 * r47923831;
        double r47923833 = r47923827 * r47923832;
        double r47923834 = r47923822 * r47923833;
        return r47923834;
}


\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(\frac{\left(\frac{M \cdot D}{d} \cdot h\right) \cdot \frac{M \cdot D}{d}}{\ell}\right) \cdot \frac{-1}{8} + 1)_* \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)

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

  1. Initial program 25.9

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

  2. Simplified26.1

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

  4. Applied add-cube-cbrt26.4

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

  5. Applied add-cube-cbrt26.5

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

  6. Applied times-frac26.5

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

  7. Applied sqrt-prod21.3

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

  8. Simplified20.6

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

  10. Applied *-un-lft-identity20.6

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

  11. Applied add-cube-cbrt20.8

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

  12. Applied times-frac20.8

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

  13. Applied sqrt-prod17.5

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

  14. Simplified17.5

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

  15. Taylor expanded around -inf 35.8

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

  16. Simplified15.1

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

  18. Applied associate-*l*12.9

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

  19. Final simplification12.9

    \[\leadsto (\left(\frac{\left(\frac{M \cdot D}{d} \cdot h\right) \cdot \frac{M \cdot D}{d}}{\ell}\right) \cdot \frac{-1}{8} + 1)_* \cdot \left(\left(\left|\sqrt[3]{d}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\ell}}\right) \cdot \left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{h}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{h}}}\right)\right)\]

Reproduce

herbie shell --seed 2019102 +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)))))