Average Error: 1.6 → 1.3
Time: 27.8s
Precision: 64
\[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]
↓
\[\sqrt{\frac{1}{2} + \frac{\frac{\frac{1}{2}}{\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}} \cdot \sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}}}{\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}}}\]
\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}
↓
\sqrt{\frac{1}{2} + \frac{\frac{\frac{1}{2}}{\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}} \cdot \sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}}}{\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}}}
double f(double l, double Om, double kx, double ky) {
        double r766195 = 1.0;
        double r766196 = 2.0;
        double r766197 = r766195 / r766196;
        double r766198 = l;
        double r766199 = r766196 * r766198;
        double r766200 = Om;
        double r766201 = r766199 / r766200;
        double r766202 = pow(r766201, r766196);
        double r766203 = kx;
        double r766204 = sin(r766203);
        double r766205 = pow(r766204, r766196);
        double r766206 = ky;
        double r766207 = sin(r766206);
        double r766208 = pow(r766207, r766196);
        double r766209 = r766205 + r766208;
        double r766210 = r766202 * r766209;
        double r766211 = r766195 + r766210;
        double r766212 = sqrt(r766211);
        double r766213 = r766195 / r766212;
        double r766214 = r766195 + r766213;
        double r766215 = r766197 * r766214;
        double r766216 = sqrt(r766215);
        return r766216;
}
↓
double f(double l, double Om, double kx, double ky) {
        double r766217 = 0.5;
        double r766218 = 2.0;
        double r766219 = l;
        double r766220 = r766218 * r766219;
        double r766221 = Om;
        double r766222 = r766220 / r766221;
        double r766223 = ky;
        double r766224 = sin(r766223);
        double r766225 = r766224 * r766224;
        double r766226 = kx;
        double r766227 = sin(r766226);
        double r766228 = r766227 * r766227;
        double r766229 = r766225 + r766228;
        double r766230 = r766222 * r766229;
        double r766231 = r766230 * r766222;
        double r766232 = 1.0;
        double r766233 = r766231 + r766232;
        double r766234 = sqrt(r766233);
        double r766235 = cbrt(r766234);
        double r766236 = r766235 * r766235;
        double r766237 = r766217 / r766236;
        double r766238 = r766237 / r766235;
        double r766239 = r766217 + r766238;
        double r766240 = sqrt(r766239);
        return r766240;
}

Error

Bits error versus l

Bits error versus Om

Bits error versus kx

Bits error versus ky

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 1.6

    \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\left(\sin kx\right)}^{2} + {\left(\sin ky\right)}^{2}\right)}}\right)}\]

  2. Simplified1.6

    \[\leadsto \color{blue}{\sqrt{\frac{\frac{1}{2}}{\sqrt{\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \left(\frac{\ell \cdot 2}{Om} \cdot \frac{\ell \cdot 2}{Om}\right) + 1}} + \frac{1}{2}}}\]
  3. Using strategy rm

  4. Applied associate-*r*1.3

    \[\leadsto \sqrt{\frac{\frac{1}{2}}{\sqrt{\color{blue}{\left(\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \frac{\ell \cdot 2}{Om}\right) \cdot \frac{\ell \cdot 2}{Om}} + 1}} + \frac{1}{2}}\]
  5. Using strategy rm

  6. Applied add-cube-cbrt1.3

    \[\leadsto \sqrt{\frac{\frac{1}{2}}{\color{blue}{\left(\sqrt[3]{\sqrt{\left(\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \frac{\ell \cdot 2}{Om}\right) \cdot \frac{\ell \cdot 2}{Om} + 1}} \cdot \sqrt[3]{\sqrt{\left(\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \frac{\ell \cdot 2}{Om}\right) \cdot \frac{\ell \cdot 2}{Om} + 1}}\right) \cdot \sqrt[3]{\sqrt{\left(\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \frac{\ell \cdot 2}{Om}\right) \cdot \frac{\ell \cdot 2}{Om} + 1}}}} + \frac{1}{2}}\]

  7. Applied associate-/r*1.3

    \[\leadsto \sqrt{\color{blue}{\frac{\frac{\frac{1}{2}}{\sqrt[3]{\sqrt{\left(\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \frac{\ell \cdot 2}{Om}\right) \cdot \frac{\ell \cdot 2}{Om} + 1}} \cdot \sqrt[3]{\sqrt{\left(\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \frac{\ell \cdot 2}{Om}\right) \cdot \frac{\ell \cdot 2}{Om} + 1}}}}{\sqrt[3]{\sqrt{\left(\left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right) \cdot \frac{\ell \cdot 2}{Om}\right) \cdot \frac{\ell \cdot 2}{Om} + 1}}}} + \frac{1}{2}}\]

  8. Final simplification1.3

    \[\leadsto \sqrt{\frac{1}{2} + \frac{\frac{\frac{1}{2}}{\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}} \cdot \sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}}}{\sqrt[3]{\sqrt{\left(\frac{2 \cdot \ell}{Om} \cdot \left(\sin ky \cdot \sin ky + \sin kx \cdot \sin kx\right)\right) \cdot \frac{2 \cdot \ell}{Om} + 1}}}}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  (sqrt (* (/ 1 2) (+ 1 (/ 1 (sqrt (+ 1 (* (pow (/ (* 2 l) Om) 2) (+ (pow (sin kx) 2) (pow (sin ky) 2))))))))))