Average Error: 13.4 → 7.7
Time: 1.6m
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{\frac{M}{\frac{d}{D}}}{\frac{\frac{1}{h}}{\frac{\sqrt[3]{\frac{\frac{M}{\sqrt[3]{\frac{d}{D}} \cdot \sqrt[3]{\frac{d}{D}}}}{\sqrt[3]{\frac{d}{D}}}}}{4}} \cdot \frac{\ell}{\sqrt[3]{\frac{M}{\frac{d}{D}}} \cdot \sqrt[3]{\frac{M}{\frac{d}{D}}}}}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \frac{\frac{M}{\frac{d}{D}}}{\frac{\frac{1}{h}}{\frac{\sqrt[3]{\frac{\frac{M}{\sqrt[3]{\frac{d}{D}} \cdot \sqrt[3]{\frac{d}{D}}}}{\sqrt[3]{\frac{d}{D}}}}}{4}} \cdot \frac{\ell}{\sqrt[3]{\frac{M}{\frac{d}{D}}} \cdot \sqrt[3]{\frac{M}{\frac{d}{D}}}}}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r8304233 = w0;
        double r8304234 = 1.0;
        double r8304235 = M;
        double r8304236 = D;
        double r8304237 = r8304235 * r8304236;
        double r8304238 = 2.0;
        double r8304239 = d;
        double r8304240 = r8304238 * r8304239;
        double r8304241 = r8304237 / r8304240;
        double r8304242 = pow(r8304241, r8304238);
        double r8304243 = h;
        double r8304244 = l;
        double r8304245 = r8304243 / r8304244;
        double r8304246 = r8304242 * r8304245;
        double r8304247 = r8304234 - r8304246;
        double r8304248 = sqrt(r8304247);
        double r8304249 = r8304233 * r8304248;
        return r8304249;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r8304250 = w0;
        double r8304251 = 1.0;
        double r8304252 = M;
        double r8304253 = d;
        double r8304254 = D;
        double r8304255 = r8304253 / r8304254;
        double r8304256 = r8304252 / r8304255;
        double r8304257 = h;
        double r8304258 = r8304251 / r8304257;
        double r8304259 = cbrt(r8304255);
        double r8304260 = r8304259 * r8304259;
        double r8304261 = r8304252 / r8304260;
        double r8304262 = r8304261 / r8304259;
        double r8304263 = cbrt(r8304262);
        double r8304264 = 4.0;
        double r8304265 = r8304263 / r8304264;
        double r8304266 = r8304258 / r8304265;
        double r8304267 = l;
        double r8304268 = cbrt(r8304256);
        double r8304269 = r8304268 * r8304268;
        double r8304270 = r8304267 / r8304269;
        double r8304271 = r8304266 * r8304270;
        double r8304272 = r8304256 / r8304271;
        double r8304273 = r8304251 - r8304272;
        double r8304274 = sqrt(r8304273);
        double r8304275 = r8304250 * r8304274;
        return r8304275;
}

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 13.4

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

  2. Simplified11.4

    \[\leadsto \color{blue}{\sqrt{1 - \frac{\frac{M}{\frac{d}{D}}}{\frac{\frac{\ell}{h}}{\frac{\frac{M}{\frac{d}{D}}}{4}}}} \cdot w0}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity11.4

    \[\leadsto \sqrt{1 - \frac{\frac{M}{\frac{d}{D}}}{\frac{\frac{\ell}{h}}{\frac{\frac{M}{\frac{d}{D}}}{\color{blue}{1 \cdot 4}}}}} \cdot w0\]

  5. Applied add-cube-cbrt11.4

    \[\leadsto \sqrt{1 - \frac{\frac{M}{\frac{d}{D}}}{\frac{\frac{\ell}{h}}{\frac{\color{blue}{\left(\sqrt[3]{\frac{M}{\frac{d}{D}}} \cdot \sqrt[3]{\frac{M}{\frac{d}{D}}}\right) \cdot \sqrt[3]{\frac{M}{\frac{d}{D}}}}}{1 \cdot 4}}}} \cdot w0\]

  6. Applied times-frac11.4

    \[\leadsto \sqrt{1 - \frac{\frac{M}{\frac{d}{D}}}{\frac{\frac{\ell}{h}}{\color{blue}{\frac{\sqrt[3]{\frac{M}{\frac{d}{D}}} \cdot \sqrt[3]{\frac{M}{\frac{d}{D}}}}{1} \cdot \frac{\sqrt[3]{\frac{M}{\frac{d}{D}}}}{4}}}}} \cdot w0\]

  7. Applied div-inv11.4

    \[\leadsto \sqrt{1 - \frac{\frac{M}{\frac{d}{D}}}{\frac{\color{blue}{\ell \cdot \frac{1}{h}}}{\frac{\sqrt[3]{\frac{M}{\frac{d}{D}}} \cdot \sqrt[3]{\frac{M}{\frac{d}{D}}}}{1} \cdot \frac{\sqrt[3]{\frac{M}{\frac{d}{D}}}}{4}}}} \cdot w0\]

  8. Applied times-frac7.7

    \[\leadsto \sqrt{1 - \frac{\frac{M}{\frac{d}{D}}}{\color{blue}{\frac{\ell}{\frac{\sqrt[3]{\frac{M}{\frac{d}{D}}} \cdot \sqrt[3]{\frac{M}{\frac{d}{D}}}}{1}} \cdot \frac{\frac{1}{h}}{\frac{\sqrt[3]{\frac{M}{\frac{d}{D}}}}{4}}}}} \cdot w0\]

  9. Simplified7.7

    \[\leadsto \sqrt{1 - \frac{\frac{M}{\frac{d}{D}}}{\color{blue}{\frac{\ell}{\sqrt[3]{\frac{M}{\frac{d}{D}}} \cdot \sqrt[3]{\frac{M}{\frac{d}{D}}}}} \cdot \frac{\frac{1}{h}}{\frac{\sqrt[3]{\frac{M}{\frac{d}{D}}}}{4}}}} \cdot w0\]
  10. Using strategy rm

  11. Applied add-cube-cbrt7.7

    \[\leadsto \sqrt{1 - \frac{\frac{M}{\frac{d}{D}}}{\frac{\ell}{\sqrt[3]{\frac{M}{\frac{d}{D}}} \cdot \sqrt[3]{\frac{M}{\frac{d}{D}}}} \cdot \frac{\frac{1}{h}}{\frac{\sqrt[3]{\frac{M}{\color{blue}{\left(\sqrt[3]{\frac{d}{D}} \cdot \sqrt[3]{\frac{d}{D}}\right) \cdot \sqrt[3]{\frac{d}{D}}}}}}{4}}}} \cdot w0\]

  12. Applied associate-/r*7.7

    \[\leadsto \sqrt{1 - \frac{\frac{M}{\frac{d}{D}}}{\frac{\ell}{\sqrt[3]{\frac{M}{\frac{d}{D}}} \cdot \sqrt[3]{\frac{M}{\frac{d}{D}}}} \cdot \frac{\frac{1}{h}}{\frac{\sqrt[3]{\color{blue}{\frac{\frac{M}{\sqrt[3]{\frac{d}{D}} \cdot \sqrt[3]{\frac{d}{D}}}}{\sqrt[3]{\frac{d}{D}}}}}}{4}}}} \cdot w0\]

  13. Final simplification7.7

    \[\leadsto w0 \cdot \sqrt{1 - \frac{\frac{M}{\frac{d}{D}}}{\frac{\frac{1}{h}}{\frac{\sqrt[3]{\frac{\frac{M}{\sqrt[3]{\frac{d}{D}} \cdot \sqrt[3]{\frac{d}{D}}}}{\sqrt[3]{\frac{d}{D}}}}}{4}} \cdot \frac{\ell}{\sqrt[3]{\frac{M}{\frac{d}{D}}} \cdot \sqrt[3]{\frac{M}{\frac{d}{D}}}}}}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))