Average Error: 13.4 → 8.4
Time: 24.6s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\frac{M}{2} \cdot \frac{D}{d}} \cdot \frac{\sqrt[3]{\ell}}{h}}} \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\sqrt{1 - \frac{\frac{M}{2} \cdot \frac{D}{d}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\frac{M}{2} \cdot \frac{D}{d}} \cdot \frac{\sqrt[3]{\ell}}{h}}} \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r2744919 = w0;
        double r2744920 = 1.0;
        double r2744921 = M;
        double r2744922 = D;
        double r2744923 = r2744921 * r2744922;
        double r2744924 = 2.0;
        double r2744925 = d;
        double r2744926 = r2744924 * r2744925;
        double r2744927 = r2744923 / r2744926;
        double r2744928 = pow(r2744927, r2744924);
        double r2744929 = h;
        double r2744930 = l;
        double r2744931 = r2744929 / r2744930;
        double r2744932 = r2744928 * r2744931;
        double r2744933 = r2744920 - r2744932;
        double r2744934 = sqrt(r2744933);
        double r2744935 = r2744919 * r2744934;
        return r2744935;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r2744936 = 1.0;
        double r2744937 = M;
        double r2744938 = 2.0;
        double r2744939 = r2744937 / r2744938;
        double r2744940 = D;
        double r2744941 = d;
        double r2744942 = r2744940 / r2744941;
        double r2744943 = r2744939 * r2744942;
        double r2744944 = l;
        double r2744945 = cbrt(r2744944);
        double r2744946 = r2744945 * r2744945;
        double r2744947 = r2744946 / r2744943;
        double r2744948 = h;
        double r2744949 = r2744945 / r2744948;
        double r2744950 = r2744947 * r2744949;
        double r2744951 = r2744943 / r2744950;
        double r2744952 = r2744936 - r2744951;
        double r2744953 = sqrt(r2744952);
        double r2744954 = w0;
        double r2744955 = r2744953 * r2744954;
        return r2744955;
}

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 13.4

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

  2. Simplified11.5

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

  4. Applied associate-/l/7.8

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

  6. Applied times-frac8.7

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

  8. Applied add-cube-cbrt8.7

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

  9. Applied times-frac9.3

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

  10. Simplified8.4

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

  11. Final simplification8.4

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

Reproduce

herbie shell --seed 2019138 +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))))))