Henrywood and Agarwal, Equation (12)

Average Error: 26.1 → 11.7

Time: 1.7m

Precision: 64

Math

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

↓

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

C

double f(double d, double h, double l, double M, double D) {
        double r7830011 = d;
        double r7830012 = h;
        double r7830013 = r7830011 / r7830012;
        double r7830014 = 1.0;
        double r7830015 = 2.0;
        double r7830016 = r7830014 / r7830015;
        double r7830017 = pow(r7830013, r7830016);
        double r7830018 = l;
        double r7830019 = r7830011 / r7830018;
        double r7830020 = pow(r7830019, r7830016);
        double r7830021 = r7830017 * r7830020;
        double r7830022 = M;
        double r7830023 = D;
        double r7830024 = r7830022 * r7830023;
        double r7830025 = r7830015 * r7830011;
        double r7830026 = r7830024 / r7830025;
        double r7830027 = pow(r7830026, r7830015);
        double r7830028 = r7830016 * r7830027;
        double r7830029 = r7830012 / r7830018;
        double r7830030 = r7830028 * r7830029;
        double r7830031 = r7830014 - r7830030;
        double r7830032 = r7830021 * r7830031;
        return r7830032;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r7830033 = 1.0;
        double r7830034 = h;
        double r7830035 = M;
        double r7830036 = d;
        double r7830037 = D;
        double r7830038 = r7830036 / r7830037;
        double r7830039 = r7830035 / r7830038;
        double r7830040 = 2.0;
        double r7830041 = r7830039 / r7830040;
        double r7830042 = r7830034 * r7830041;
        double r7830043 = l;
        double r7830044 = r7830040 * r7830043;
        double r7830045 = r7830044 / r7830041;
        double r7830046 = r7830042 / r7830045;
        double r7830047 = r7830033 - r7830046;
        double r7830048 = cbrt(r7830036);
        double r7830049 = cbrt(r7830034);
        double r7830050 = r7830048 / r7830049;
        double r7830051 = sqrt(r7830050);
        double r7830052 = r7830050 * r7830050;
        double r7830053 = sqrt(r7830052);
        double r7830054 = r7830051 * r7830053;
        double r7830055 = cbrt(r7830043);
        double r7830056 = r7830048 / r7830055;
        double r7830057 = 0.5;
        double r7830058 = pow(r7830056, r7830057);
        double r7830059 = r7830056 * r7830056;
        double r7830060 = pow(r7830059, r7830057);
        double r7830061 = r7830058 * r7830060;
        double r7830062 = r7830054 * r7830061;
        double r7830063 = r7830047 * r7830062;
        return r7830063;
}

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 26.1

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

3. Applied add-cube-cbrt 26.3

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

4. Applied add-cube-cbrt 26.4

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

5. Applied times-frac 26.4

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

6. Applied unpow-prod-down 21.2

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

7. Simplified 21.2

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

8. Simplified 21.2

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

10. Applied associate-*l/ 21.2

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

11. Applied frac-times 20.2

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

12. Simplified 18.9

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

14. Applied add-cube-cbrt 19.0

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

15. Applied add-cube-cbrt 19.1

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

16. Applied times-frac 19.1

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

17. Applied unpow-prod-down 13.3

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

18. Simplified 13.3

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

19. Simplified 13.3

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

21. Applied associate-/l* 11.7

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

22. Final simplification 11.7

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

Reproduce

herbie shell --seed 2019149 
(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)))))