Average Error: 13.6 → 8.3
Time: 2.4m
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{\ell}\right) \cdot h\right) \cdot \frac{M \cdot D}{2 \cdot d}} \cdot w0\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
\sqrt{1 - \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{1}{\ell}\right) \cdot h\right) \cdot \frac{M \cdot D}{2 \cdot d}} \cdot w0
double f(double w0, double M, double D, double h, double l, double d) {
        double r37971310 = w0;
        double r37971311 = 1.0;
        double r37971312 = M;
        double r37971313 = D;
        double r37971314 = r37971312 * r37971313;
        double r37971315 = 2.0;
        double r37971316 = d;
        double r37971317 = r37971315 * r37971316;
        double r37971318 = r37971314 / r37971317;
        double r37971319 = pow(r37971318, r37971315);
        double r37971320 = h;
        double r37971321 = l;
        double r37971322 = r37971320 / r37971321;
        double r37971323 = r37971319 * r37971322;
        double r37971324 = r37971311 - r37971323;
        double r37971325 = sqrt(r37971324);
        double r37971326 = r37971310 * r37971325;
        return r37971326;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r37971327 = 1.0;
        double r37971328 = M;
        double r37971329 = D;
        double r37971330 = r37971328 * r37971329;
        double r37971331 = 2.0;
        double r37971332 = d;
        double r37971333 = r37971331 * r37971332;
        double r37971334 = r37971330 / r37971333;
        double r37971335 = l;
        double r37971336 = r37971327 / r37971335;
        double r37971337 = r37971334 * r37971336;
        double r37971338 = h;
        double r37971339 = r37971337 * r37971338;
        double r37971340 = r37971339 * r37971334;
        double r37971341 = r37971327 - r37971340;
        double r37971342 = sqrt(r37971341);
        double r37971343 = w0;
        double r37971344 = r37971342 * r37971343;
        return r37971344;
}

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 13.6

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

  2. Simplified12.2

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

  4. Applied div-inv12.2

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

  5. Applied associate-*l*8.3

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

  6. Final simplification8.3

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

Reproduce

herbie shell --seed 2019120 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))