Average Error: 14.2 → 9.0
Time: 34.0s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\left(\sqrt[3]{\frac{D}{d} \cdot \frac{M}{2}} \cdot \sqrt[3]{\frac{D}{d} \cdot \frac{M}{2}}\right) \cdot \sqrt[3]{\frac{D}{d} \cdot \frac{M}{2}}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right) \cdot \frac{1}{\ell}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \left({\left(\frac{D}{d} \cdot \frac{M}{2}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\left(\sqrt[3]{\frac{D}{d} \cdot \frac{M}{2}} \cdot \sqrt[3]{\frac{D}{d} \cdot \frac{M}{2}}\right) \cdot \sqrt[3]{\frac{D}{d} \cdot \frac{M}{2}}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)\right) \cdot \frac{1}{\ell}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r6106184 = w0;
        double r6106185 = 1.0;
        double r6106186 = M;
        double r6106187 = D;
        double r6106188 = r6106186 * r6106187;
        double r6106189 = 2.0;
        double r6106190 = d;
        double r6106191 = r6106189 * r6106190;
        double r6106192 = r6106188 / r6106191;
        double r6106193 = pow(r6106192, r6106189);
        double r6106194 = h;
        double r6106195 = l;
        double r6106196 = r6106194 / r6106195;
        double r6106197 = r6106193 * r6106196;
        double r6106198 = r6106185 - r6106197;
        double r6106199 = sqrt(r6106198);
        double r6106200 = r6106184 * r6106199;
        return r6106200;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r6106201 = w0;
        double r6106202 = 1.0;
        double r6106203 = D;
        double r6106204 = d;
        double r6106205 = r6106203 / r6106204;
        double r6106206 = M;
        double r6106207 = 2.0;
        double r6106208 = r6106206 / r6106207;
        double r6106209 = r6106205 * r6106208;
        double r6106210 = 2.0;
        double r6106211 = r6106207 / r6106210;
        double r6106212 = pow(r6106209, r6106211);
        double r6106213 = cbrt(r6106209);
        double r6106214 = r6106213 * r6106213;
        double r6106215 = r6106214 * r6106213;
        double r6106216 = pow(r6106215, r6106211);
        double r6106217 = h;
        double r6106218 = r6106216 * r6106217;
        double r6106219 = r6106212 * r6106218;
        double r6106220 = 1.0;
        double r6106221 = l;
        double r6106222 = r6106220 / r6106221;
        double r6106223 = r6106219 * r6106222;
        double r6106224 = r6106202 - r6106223;
        double r6106225 = sqrt(r6106224);
        double r6106226 = r6106201 * r6106225;
        return r6106226;
}

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 14.2

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

  3. Applied div-inv14.2

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

  4. Applied associate-*r*10.5

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

  6. Applied times-frac10.5

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

  8. Applied sqr-pow10.5

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

  9. Applied associate-*l*8.9

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

  11. Applied add-cube-cbrt9.0

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

  12. Final simplification9.0

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

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(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))))))