Average Error: 14.1 → 8.4
Time: 32.9s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\left(\sqrt[3]{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \sqrt[3]{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}}\right)\right) \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right)} \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\sqrt{1 - \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \left(\left(\left(\sqrt[3]{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}} \cdot \sqrt[3]{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}}\right) \cdot \left(\frac{\sqrt[3]{h}}{\sqrt[3]{\ell}} \cdot \sqrt[3]{{\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)}}\right)\right) \cdot \left({\left(\frac{M \cdot D}{d \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot \frac{\sqrt[3]{h}}{\sqrt[3]{\ell}}\right)\right)} \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r5021109 = w0;
        double r5021110 = 1.0;
        double r5021111 = M;
        double r5021112 = D;
        double r5021113 = r5021111 * r5021112;
        double r5021114 = 2.0;
        double r5021115 = d;
        double r5021116 = r5021114 * r5021115;
        double r5021117 = r5021113 / r5021116;
        double r5021118 = pow(r5021117, r5021114);
        double r5021119 = h;
        double r5021120 = l;
        double r5021121 = r5021119 / r5021120;
        double r5021122 = r5021118 * r5021121;
        double r5021123 = r5021110 - r5021122;
        double r5021124 = sqrt(r5021123);
        double r5021125 = r5021109 * r5021124;
        return r5021125;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r5021126 = 1.0;
        double r5021127 = h;
        double r5021128 = cbrt(r5021127);
        double r5021129 = l;
        double r5021130 = cbrt(r5021129);
        double r5021131 = r5021128 / r5021130;
        double r5021132 = M;
        double r5021133 = D;
        double r5021134 = r5021132 * r5021133;
        double r5021135 = d;
        double r5021136 = 2.0;
        double r5021137 = r5021135 * r5021136;
        double r5021138 = r5021134 / r5021137;
        double r5021139 = 2.0;
        double r5021140 = r5021136 / r5021139;
        double r5021141 = pow(r5021138, r5021140);
        double r5021142 = cbrt(r5021141);
        double r5021143 = r5021142 * r5021142;
        double r5021144 = r5021131 * r5021142;
        double r5021145 = r5021143 * r5021144;
        double r5021146 = r5021141 * r5021131;
        double r5021147 = r5021145 * r5021146;
        double r5021148 = r5021131 * r5021147;
        double r5021149 = r5021126 - r5021148;
        double r5021150 = sqrt(r5021149);
        double r5021151 = w0;
        double r5021152 = r5021150 * r5021151;
        return r5021152;
}

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 add-cube-cbrt14.1

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

  4. Applied add-cube-cbrt14.1

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

  5. Applied times-frac14.1

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

  6. Applied associate-*r*10.9

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

  7. Simplified10.1

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

  9. Applied sqr-pow10.1

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

  10. Applied associate-*l*9.1

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

  12. Applied *-un-lft-identity9.1

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

  13. Applied sqrt-prod9.1

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

  14. Simplified9.1

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

  15. Simplified8.4

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

  17. Applied add-cube-cbrt8.4

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

  18. Applied associate-*l*8.4

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

  19. Final simplification8.4

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

Reproduce

herbie shell --seed 2019171 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))