Average Error: 14.4 → 9.1
Time: 16.8s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}{\ell}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
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)}{\ell}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r173215 = w0;
        double r173216 = 1.0;
        double r173217 = M;
        double r173218 = D;
        double r173219 = r173217 * r173218;
        double r173220 = 2.0;
        double r173221 = d;
        double r173222 = r173220 * r173221;
        double r173223 = r173219 / r173222;
        double r173224 = pow(r173223, r173220);
        double r173225 = h;
        double r173226 = l;
        double r173227 = r173225 / r173226;
        double r173228 = r173224 * r173227;
        double r173229 = r173216 - r173228;
        double r173230 = sqrt(r173229);
        double r173231 = r173215 * r173230;
        return r173231;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r173232 = w0;
        double r173233 = 1.0;
        double r173234 = M;
        double r173235 = D;
        double r173236 = r173234 * r173235;
        double r173237 = 2.0;
        double r173238 = d;
        double r173239 = r173237 * r173238;
        double r173240 = r173236 / r173239;
        double r173241 = 2.0;
        double r173242 = r173237 / r173241;
        double r173243 = pow(r173240, r173242);
        double r173244 = h;
        double r173245 = r173243 * r173244;
        double r173246 = r173243 * r173245;
        double r173247 = l;
        double r173248 = r173246 / r173247;
        double r173249 = r173233 - r173248;
        double r173250 = sqrt(r173249);
        double r173251 = r173232 * r173250;
        return r173251;
}

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 14.4

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

  3. Applied associate-*r/10.8

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

  5. Applied sqr-pow10.8

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

  6. Applied associate-*l*9.1

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

  7. Final simplification9.1

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(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))))))