Average Error: 13.6 → 8.3
Time: 42.1s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{\frac{D \cdot M}{d \cdot 2} \cdot h}{\frac{\ell}{\frac{D \cdot M}{d \cdot 2}}}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \frac{\frac{D \cdot M}{d \cdot 2} \cdot h}{\frac{\ell}{\frac{D \cdot M}{d \cdot 2}}}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r5229994 = w0;
        double r5229995 = 1.0;
        double r5229996 = M;
        double r5229997 = D;
        double r5229998 = r5229996 * r5229997;
        double r5229999 = 2.0;
        double r5230000 = d;
        double r5230001 = r5229999 * r5230000;
        double r5230002 = r5229998 / r5230001;
        double r5230003 = pow(r5230002, r5229999);
        double r5230004 = h;
        double r5230005 = l;
        double r5230006 = r5230004 / r5230005;
        double r5230007 = r5230003 * r5230006;
        double r5230008 = r5229995 - r5230007;
        double r5230009 = sqrt(r5230008);
        double r5230010 = r5229994 * r5230009;
        return r5230010;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r5230011 = w0;
        double r5230012 = 1.0;
        double r5230013 = D;
        double r5230014 = M;
        double r5230015 = r5230013 * r5230014;
        double r5230016 = d;
        double r5230017 = 2.0;
        double r5230018 = r5230016 * r5230017;
        double r5230019 = r5230015 / r5230018;
        double r5230020 = h;
        double r5230021 = r5230019 * r5230020;
        double r5230022 = l;
        double r5230023 = r5230022 / r5230019;
        double r5230024 = r5230021 / r5230023;
        double r5230025 = r5230012 - r5230024;
        double r5230026 = sqrt(r5230025);
        double r5230027 = r5230011 * r5230026;
        return r5230027;
}

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 13.6

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

  2. Simplified13.1

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

  4. Applied add-cbrt-cube23.4

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

  5. Applied add-cbrt-cube31.2

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

  6. Applied add-cbrt-cube31.2

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

  7. Applied cbrt-unprod31.2

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

  8. Applied add-cbrt-cube37.9

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

  9. Applied add-cbrt-cube44.2

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

  10. Applied cbrt-unprod44.6

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

  11. Applied cbrt-undiv44.6

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

  12. Applied add-cbrt-cube44.6

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

  13. Applied add-cbrt-cube44.6

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

  14. Applied cbrt-unprod44.6

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

  15. Applied add-cbrt-cube44.6

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

  16. Applied cbrt-undiv44.6

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

  17. Applied cbrt-unprod44.7

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

  18. Applied cbrt-undiv44.8

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

  19. Simplified12.0

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

  21. Applied cube-unmult12.0

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

  22. Applied rem-cbrt-cube8.9

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

  24. Applied associate-*l/8.3

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

  25. Final simplification8.3

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

Reproduce

herbie shell --seed 2019134 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))