Average Error: 14.0 → 8.5
Time: 32.6s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left(h \cdot {\left(\frac{1}{\frac{2}{M \cdot \frac{D}{d}}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(\frac{\frac{D}{d}}{\frac{2}{M}}\right)}^{\left(\frac{2}{2}\right)}}{\ell}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \left(h \cdot {\left(\frac{1}{\frac{2}{M \cdot \frac{D}{d}}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{{\left(\frac{\frac{D}{d}}{\frac{2}{M}}\right)}^{\left(\frac{2}{2}\right)}}{\ell}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r127094 = w0;
        double r127095 = 1.0;
        double r127096 = M;
        double r127097 = D;
        double r127098 = r127096 * r127097;
        double r127099 = 2.0;
        double r127100 = d;
        double r127101 = r127099 * r127100;
        double r127102 = r127098 / r127101;
        double r127103 = pow(r127102, r127099);
        double r127104 = h;
        double r127105 = l;
        double r127106 = r127104 / r127105;
        double r127107 = r127103 * r127106;
        double r127108 = r127095 - r127107;
        double r127109 = sqrt(r127108);
        double r127110 = r127094 * r127109;
        return r127110;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r127111 = w0;
        double r127112 = 1.0;
        double r127113 = h;
        double r127114 = 1.0;
        double r127115 = 2.0;
        double r127116 = M;
        double r127117 = D;
        double r127118 = d;
        double r127119 = r127117 / r127118;
        double r127120 = r127116 * r127119;
        double r127121 = r127115 / r127120;
        double r127122 = r127114 / r127121;
        double r127123 = 2.0;
        double r127124 = r127115 / r127123;
        double r127125 = pow(r127122, r127124);
        double r127126 = r127113 * r127125;
        double r127127 = r127115 / r127116;
        double r127128 = r127119 / r127127;
        double r127129 = pow(r127128, r127124);
        double r127130 = l;
        double r127131 = r127129 / r127130;
        double r127132 = r127126 * r127131;
        double r127133 = r127112 - r127132;
        double r127134 = sqrt(r127133);
        double r127135 = r127111 * r127134;
        return r127135;
}

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 14.0

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

  2. Simplified10.8

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

  4. Applied sqr-pow10.8

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

  5. Applied associate-*r*9.2

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

  6. Simplified10.1

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

  8. Applied *-un-lft-identity10.1

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

  9. Applied times-frac9.6

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

  10. Simplified9.6

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

  11. Simplified8.5

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

  13. Applied clear-num8.5

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

  14. Simplified8.5

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

  15. Final simplification8.5

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

Reproduce

herbie shell --seed 2019196 
(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))))))