Average Error: 14.1 → 8.4
Time: 10.7s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{{\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{\left(\frac{2}{2}\right)}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{{\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{\left(\frac{2}{2}\right)}}} \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\sqrt[3]{\ell}}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \frac{{\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}} \cdot \sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{\left(\frac{2}{2}\right)}}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{{\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{\left(\frac{2}{2}\right)}}} \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\sqrt[3]{\ell}}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r259090 = w0;
        double r259091 = 1.0;
        double r259092 = M;
        double r259093 = D;
        double r259094 = r259092 * r259093;
        double r259095 = 2.0;
        double r259096 = d;
        double r259097 = r259095 * r259096;
        double r259098 = r259094 / r259097;
        double r259099 = pow(r259098, r259095);
        double r259100 = h;
        double r259101 = l;
        double r259102 = r259100 / r259101;
        double r259103 = r259099 * r259102;
        double r259104 = r259091 - r259103;
        double r259105 = sqrt(r259104);
        double r259106 = r259090 * r259105;
        return r259106;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r259107 = w0;
        double r259108 = 1.0;
        double r259109 = M;
        double r259110 = D;
        double r259111 = r259109 * r259110;
        double r259112 = 2.0;
        double r259113 = d;
        double r259114 = r259112 * r259113;
        double r259115 = r259111 / r259114;
        double r259116 = cbrt(r259115);
        double r259117 = r259116 * r259116;
        double r259118 = 2.0;
        double r259119 = r259112 / r259118;
        double r259120 = pow(r259117, r259119);
        double r259121 = l;
        double r259122 = cbrt(r259121);
        double r259123 = r259122 * r259122;
        double r259124 = pow(r259116, r259119);
        double r259125 = r259123 / r259124;
        double r259126 = r259120 / r259125;
        double r259127 = pow(r259115, r259119);
        double r259128 = h;
        double r259129 = r259127 * r259128;
        double r259130 = r259129 / r259122;
        double r259131 = r259126 * r259130;
        double r259132 = r259108 - r259131;
        double r259133 = sqrt(r259132);
        double r259134 = r259107 * r259133;
        return r259134;
}

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 14.1

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

  3. Applied associate-*r/10.7

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

  5. Applied sqr-pow10.7

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

  6. Applied associate-*l*9.3

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

  8. Applied add-cube-cbrt9.3

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

  9. Applied times-frac8.4

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

  11. Applied add-cube-cbrt8.4

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

  12. Applied unpow-prod-down8.4

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

  13. Applied associate-/l*8.4

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

  14. Final simplification8.4

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

Reproduce

herbie shell --seed 2020027 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  :precision binary64
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))