Henrywood and Agarwal, Equation (12)

Average Error: 26.6 → 15.1

Time: 23.7s

Precision: 64

Math

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

↓

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

C

double f(double d, double h, double l, double M, double D) {
        double r271738 = d;
        double r271739 = h;
        double r271740 = r271738 / r271739;
        double r271741 = 1.0;
        double r271742 = 2.0;
        double r271743 = r271741 / r271742;
        double r271744 = pow(r271740, r271743);
        double r271745 = l;
        double r271746 = r271738 / r271745;
        double r271747 = pow(r271746, r271743);
        double r271748 = r271744 * r271747;
        double r271749 = M;
        double r271750 = D;
        double r271751 = r271749 * r271750;
        double r271752 = r271742 * r271738;
        double r271753 = r271751 / r271752;
        double r271754 = pow(r271753, r271742);
        double r271755 = r271743 * r271754;
        double r271756 = r271739 / r271745;
        double r271757 = r271755 * r271756;
        double r271758 = r271741 - r271757;
        double r271759 = r271748 * r271758;
        return r271759;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r271760 = d;
        double r271761 = cbrt(r271760);
        double r271762 = r271761 * r271761;
        double r271763 = 1.0;
        double r271764 = r271762 / r271763;
        double r271765 = 1.0;
        double r271766 = 2.0;
        double r271767 = r271765 / r271766;
        double r271768 = pow(r271764, r271767);
        double r271769 = h;
        double r271770 = cbrt(r271769);
        double r271771 = r271770 * r271770;
        double r271772 = r271763 / r271771;
        double r271773 = pow(r271772, r271767);
        double r271774 = r271761 / r271770;
        double r271775 = pow(r271774, r271767);
        double r271776 = r271773 * r271775;
        double r271777 = r271768 * r271776;
        double r271778 = l;
        double r271779 = cbrt(r271778);
        double r271780 = r271779 * r271779;
        double r271781 = r271763 / r271780;
        double r271782 = pow(r271781, r271767);
        double r271783 = r271760 / r271779;
        double r271784 = pow(r271783, r271767);
        double r271785 = r271782 * r271784;
        double r271786 = r271777 * r271785;
        double r271787 = M;
        double r271788 = D;
        double r271789 = r271787 * r271788;
        double r271790 = r271766 * r271760;
        double r271791 = r271789 / r271790;
        double r271792 = pow(r271791, r271766);
        double r271793 = r271792 * r271771;
        double r271794 = r271767 * r271793;
        double r271795 = r271770 / r271778;
        double r271796 = r271794 * r271795;
        double r271797 = r271765 - r271796;
        double r271798 = r271786 * r271797;
        return r271798;
}

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 26.6

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

3. Applied *-un-lft-identity 26.6

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

4. Applied add-cube-cbrt 26.8

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

5. Applied times-frac 26.8

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

6. Applied unpow-prod-down 22.1

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

8. Applied add-cube-cbrt 22.2

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

9. Applied *-un-lft-identity 22.2

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

10. Applied times-frac 22.2

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

11. Applied unpow-prod-down 18.6

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

13. Applied *-un-lft-identity 18.6

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

14. Applied add-cube-cbrt 18.6

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

15. Applied times-frac 18.6

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

16. Applied associate-*r* 16.7

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

17. Simplified 16.7

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

19. Applied add-cube-cbrt 16.7

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

20. Applied *-un-lft-identity 16.7

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

21. Applied times-frac 16.7

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

22. Applied unpow-prod-down 15.1

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

23. Final simplification 15.1

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

Reproduce

herbie shell --seed 2020020 
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  :precision binary64
  (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))))