Average Error: 13.3 → 8.0
Time: 1.4m
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{\frac{h}{\frac{\frac{d}{\frac{M \cdot D}{2}}}{\frac{1}{\ell}}}}{d \cdot \frac{1}{\frac{M \cdot D}{2}}}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \frac{\frac{h}{\frac{\frac{d}{\frac{M \cdot D}{2}}}{\frac{1}{\ell}}}}{d \cdot \frac{1}{\frac{M \cdot D}{2}}}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r7159162 = w0;
        double r7159163 = 1.0;
        double r7159164 = M;
        double r7159165 = D;
        double r7159166 = r7159164 * r7159165;
        double r7159167 = 2.0;
        double r7159168 = d;
        double r7159169 = r7159167 * r7159168;
        double r7159170 = r7159166 / r7159169;
        double r7159171 = pow(r7159170, r7159167);
        double r7159172 = h;
        double r7159173 = l;
        double r7159174 = r7159172 / r7159173;
        double r7159175 = r7159171 * r7159174;
        double r7159176 = r7159163 - r7159175;
        double r7159177 = sqrt(r7159176);
        double r7159178 = r7159162 * r7159177;
        return r7159178;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r7159179 = w0;
        double r7159180 = 1.0;
        double r7159181 = h;
        double r7159182 = d;
        double r7159183 = M;
        double r7159184 = D;
        double r7159185 = r7159183 * r7159184;
        double r7159186 = 2.0;
        double r7159187 = r7159185 / r7159186;
        double r7159188 = r7159182 / r7159187;
        double r7159189 = l;
        double r7159190 = r7159180 / r7159189;
        double r7159191 = r7159188 / r7159190;
        double r7159192 = r7159181 / r7159191;
        double r7159193 = r7159180 / r7159187;
        double r7159194 = r7159182 * r7159193;
        double r7159195 = r7159192 / r7159194;
        double r7159196 = r7159180 - r7159195;
        double r7159197 = sqrt(r7159196);
        double r7159198 = r7159179 * r7159197;
        return r7159198;
}

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 13.3

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

  2. Simplified11.8

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

  4. Applied div-inv11.8

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

  5. Applied associate-/l*8.0

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

  7. Applied div-inv8.0

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

  8. Final simplification8.0

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

Reproduce

herbie shell --seed 2019162 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))