Henrywood and Agarwal, Equation (9a)

Average Error: 13.4 → 8.1

Time: 59.0s

Precision: 64

Math

\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

↓

\[\sqrt{1 - \left(\left(\frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt[3]{\ell}} \cdot \frac{\frac{\frac{M \cdot D}{d}}{\sqrt{\sqrt{2}}}}{\frac{1}{h}}\right) \cdot \frac{\frac{1}{\sqrt{2}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\frac{M \cdot D}{d}}{2}} \cdot w0\]

C

double f(double w0, double M, double D, double h, double l, double d) {
        double r2761775 = w0;
        double r2761776 = 1.0;
        double r2761777 = M;
        double r2761778 = D;
        double r2761779 = r2761777 * r2761778;
        double r2761780 = 2.0;
        double r2761781 = d;
        double r2761782 = r2761780 * r2761781;
        double r2761783 = r2761779 / r2761782;
        double r2761784 = pow(r2761783, r2761780);
        double r2761785 = h;
        double r2761786 = l;
        double r2761787 = r2761785 / r2761786;
        double r2761788 = r2761784 * r2761787;
        double r2761789 = r2761776 - r2761788;
        double r2761790 = sqrt(r2761789);
        double r2761791 = r2761775 * r2761790;
        return r2761791;
}

↓

double f(double w0, double M, double D, double h, double l, double d) {
        double r2761792 = 1.0;
        double r2761793 = 2.0;
        double r2761794 = sqrt(r2761793);
        double r2761795 = sqrt(r2761794);
        double r2761796 = r2761792 / r2761795;
        double r2761797 = l;
        double r2761798 = cbrt(r2761797);
        double r2761799 = r2761796 / r2761798;
        double r2761800 = M;
        double r2761801 = D;
        double r2761802 = r2761800 * r2761801;
        double r2761803 = d;
        double r2761804 = r2761802 / r2761803;
        double r2761805 = r2761804 / r2761795;
        double r2761806 = h;
        double r2761807 = r2761792 / r2761806;
        double r2761808 = r2761805 / r2761807;
        double r2761809 = r2761799 * r2761808;
        double r2761810 = r2761792 / r2761794;
        double r2761811 = r2761798 * r2761798;
        double r2761812 = r2761810 / r2761811;
        double r2761813 = r2761809 * r2761812;
        double r2761814 = r2761804 / r2761793;
        double r2761815 = r2761813 * r2761814;
        double r2761816 = r2761792 - r2761815;
        double r2761817 = sqrt(r2761816);
        double r2761818 = w0;
        double r2761819 = r2761817 * r2761818;
        return r2761819;
}

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

Bits error versus d

Derivation

1. Initial program 13.4

   \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

2. Simplified 11.5

   \[\leadsto \color{blue}{\sqrt{1 - \frac{\frac{\frac{M \cdot D}{d}}{2}}{\frac{\ell}{h}} \cdot \frac{\frac{M \cdot D}{d}}{2}} \cdot w0}\]
3. Using strategy rm

4. Applied *-un-lft-identity 11.5

   \[\leadsto \sqrt{1 - \frac{\frac{\frac{M \cdot D}{d}}{2}}{\frac{\ell}{\color{blue}{1 \cdot h}}} \cdot \frac{\frac{M \cdot D}{d}}{2}} \cdot w0\]

5. Applied add-cube-cbrt 11.6

   \[\leadsto \sqrt{1 - \frac{\frac{\frac{M \cdot D}{d}}{2}}{\frac{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}{1 \cdot h}} \cdot \frac{\frac{M \cdot D}{d}}{2}} \cdot w0\]

6. Applied times-frac 11.6

   \[\leadsto \sqrt{1 - \frac{\frac{\frac{M \cdot D}{d}}{2}}{\color{blue}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{1} \cdot \frac{\sqrt[3]{\ell}}{h}}} \cdot \frac{\frac{M \cdot D}{d}}{2}} \cdot w0\]

7. Applied add-sqr-sqrt 11.6

   \[\leadsto \sqrt{1 - \frac{\frac{\frac{M \cdot D}{d}}{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{1} \cdot \frac{\sqrt[3]{\ell}}{h}} \cdot \frac{\frac{M \cdot D}{d}}{2}} \cdot w0\]

8. Applied *-un-lft-identity 11.6

   \[\leadsto \sqrt{1 - \frac{\frac{\color{blue}{1 \cdot \frac{M \cdot D}{d}}}{\sqrt{2} \cdot \sqrt{2}}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{1} \cdot \frac{\sqrt[3]{\ell}}{h}} \cdot \frac{\frac{M \cdot D}{d}}{2}} \cdot w0\]

9. Applied times-frac 11.6

   \[\leadsto \sqrt{1 - \frac{\color{blue}{\frac{1}{\sqrt{2}} \cdot \frac{\frac{M \cdot D}{d}}{\sqrt{2}}}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{1} \cdot \frac{\sqrt[3]{\ell}}{h}} \cdot \frac{\frac{M \cdot D}{d}}{2}} \cdot w0\]

10. Applied times-frac 8.8

    \[\leadsto \sqrt{1 - \color{blue}{\left(\frac{\frac{1}{\sqrt{2}}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{1}} \cdot \frac{\frac{\frac{M \cdot D}{d}}{\sqrt{2}}}{\frac{\sqrt[3]{\ell}}{h}}\right)} \cdot \frac{\frac{M \cdot D}{d}}{2}} \cdot w0\]

11. Simplified 8.8

    \[\leadsto \sqrt{1 - \left(\color{blue}{\frac{\frac{1}{\sqrt{2}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}} \cdot \frac{\frac{\frac{M \cdot D}{d}}{\sqrt{2}}}{\frac{\sqrt[3]{\ell}}{h}}\right) \cdot \frac{\frac{M \cdot D}{d}}{2}} \cdot w0\]
12. Using strategy rm

13. Applied div-inv 8.8

    \[\leadsto \sqrt{1 - \left(\frac{\frac{1}{\sqrt{2}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{\frac{M \cdot D}{d}}{\sqrt{2}}}{\color{blue}{\sqrt[3]{\ell} \cdot \frac{1}{h}}}\right) \cdot \frac{\frac{M \cdot D}{d}}{2}} \cdot w0\]

14. Applied add-sqr-sqrt 8.8

    \[\leadsto \sqrt{1 - \left(\frac{\frac{1}{\sqrt{2}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{\frac{M \cdot D}{d}}{\sqrt{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}}}{\sqrt[3]{\ell} \cdot \frac{1}{h}}\right) \cdot \frac{\frac{M \cdot D}{d}}{2}} \cdot w0\]

15. Applied sqrt-prod 8.8

    \[\leadsto \sqrt{1 - \left(\frac{\frac{1}{\sqrt{2}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{\frac{M \cdot D}{d}}{\color{blue}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}}}{\sqrt[3]{\ell} \cdot \frac{1}{h}}\right) \cdot \frac{\frac{M \cdot D}{d}}{2}} \cdot w0\]

16. Applied *-un-lft-identity 8.8

    \[\leadsto \sqrt{1 - \left(\frac{\frac{1}{\sqrt{2}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\frac{\color{blue}{1 \cdot \frac{M \cdot D}{d}}}{\sqrt{\sqrt{2}} \cdot \sqrt{\sqrt{2}}}}{\sqrt[3]{\ell} \cdot \frac{1}{h}}\right) \cdot \frac{\frac{M \cdot D}{d}}{2}} \cdot w0\]

17. Applied times-frac 8.8

    \[\leadsto \sqrt{1 - \left(\frac{\frac{1}{\sqrt{2}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{\color{blue}{\frac{1}{\sqrt{\sqrt{2}}} \cdot \frac{\frac{M \cdot D}{d}}{\sqrt{\sqrt{2}}}}}{\sqrt[3]{\ell} \cdot \frac{1}{h}}\right) \cdot \frac{\frac{M \cdot D}{d}}{2}} \cdot w0\]

18. Applied times-frac 8.1

    \[\leadsto \sqrt{1 - \left(\frac{\frac{1}{\sqrt{2}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \color{blue}{\left(\frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt[3]{\ell}} \cdot \frac{\frac{\frac{M \cdot D}{d}}{\sqrt{\sqrt{2}}}}{\frac{1}{h}}\right)}\right) \cdot \frac{\frac{M \cdot D}{d}}{2}} \cdot w0\]

19. Final simplification 8.1

    \[\leadsto \sqrt{1 - \left(\left(\frac{\frac{1}{\sqrt{\sqrt{2}}}}{\sqrt[3]{\ell}} \cdot \frac{\frac{\frac{M \cdot D}{d}}{\sqrt{\sqrt{2}}}}{\frac{1}{h}}\right) \cdot \frac{\frac{1}{\sqrt{2}}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right) \cdot \frac{\frac{M \cdot D}{d}}{2}} \cdot w0\]

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))