Henrywood and Agarwal, Equation (9a)

Average Error: 13.4 → 7.9

Time: 23.1s

Precision: 64

Math

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

↓

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

C

double f(double w0, double M, double D, double h, double l, double d) {
        double r2056065 = w0;
        double r2056066 = 1.0;
        double r2056067 = M;
        double r2056068 = D;
        double r2056069 = r2056067 * r2056068;
        double r2056070 = 2.0;
        double r2056071 = d;
        double r2056072 = r2056070 * r2056071;
        double r2056073 = r2056069 / r2056072;
        double r2056074 = pow(r2056073, r2056070);
        double r2056075 = h;
        double r2056076 = l;
        double r2056077 = r2056075 / r2056076;
        double r2056078 = r2056074 * r2056077;
        double r2056079 = r2056066 - r2056078;
        double r2056080 = sqrt(r2056079);
        double r2056081 = r2056065 * r2056080;
        return r2056081;
}

↓

double f(double w0, double M, double D, double h, double l, double d) {
        double r2056082 = 1.0;
        double r2056083 = M;
        double r2056084 = d;
        double r2056085 = r2056083 / r2056084;
        double r2056086 = D;
        double r2056087 = r2056085 * r2056086;
        double r2056088 = l;
        double r2056089 = cbrt(r2056088);
        double r2056090 = 2.0;
        double r2056091 = r2056089 * r2056090;
        double r2056092 = r2056087 / r2056091;
        double r2056093 = h;
        double r2056094 = r2056092 * r2056093;
        double r2056095 = r2056094 / r2056089;
        double r2056096 = r2056095 / r2056089;
        double r2056097 = r2056090 / r2056087;
        double r2056098 = r2056096 / r2056097;
        double r2056099 = r2056082 - r2056098;
        double r2056100 = sqrt(r2056099);
        double r2056101 = w0;
        double r2056102 = r2056100 * r2056101;
        return r2056102;
}

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 11.4

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

4. Applied *-un-lft-identity 11.4

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

5. Applied add-cube-cbrt 11.4

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

6. Applied times-frac 11.4

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

7. Applied *-un-lft-identity 11.4

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

8. Applied add-cube-cbrt 11.4

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

9. Applied times-frac 11.4

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

10. Applied times-frac 8.7

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

12. Applied *-un-lft-identity 8.7

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

13. Applied sqrt-prod 8.7

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

14. Simplified 8.7

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

15. Simplified 8.6

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

17. Applied associate-*l/ 8.6

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

18. Applied associate-/r/ 7.9

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

19. Final simplification 7.9

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

Reproduce

herbie shell --seed 2019156 +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))))))